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Lambert W function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the Lambert W function, also called the omega function or product logarithm,<a href="#cite_note-1">[1]</a> is a <a href="Multivalued_function" title="Multivalued function">multivalued function</a>, namely the <a href="Branch_point" title="Branch point">branches</a> of the <a href="Converse_relation" title="Converse relation">converse relation</a> of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(w)=we^{w}}">

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<annotation encoding="application/x-tex">{\displaystyle f(w)=we^{w}}</annotation>

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</math><img src="./7f0f7f9f57e5eb769034c972927e0b9b1d2c7b68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.007ex; height:2.843ex;" alt="{\displaystyle f(w)=we^{w}}" loading="lazy">, where w is any <a href="Complex_number" title="Complex number">complex number</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{w}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{w}}</annotation>

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</math><img src="./efa917845908e76a936367c158e8798c70779a2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.343ex;" alt="{\displaystyle e^{w}}" loading="lazy"> is the <a href="Exponential_function" title="Exponential function">exponential function</a>. The function is named after <a href="Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Lambert</a>, who considered a related problem in 1758. Building on Lambert's work, <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> described the W function per se in 1783.<a href="#cite_note-2">[2]</a>

For each integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> there is one branch, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{k}\left(z\right)}">

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<annotation encoding="application/x-tex">{\displaystyle W_{k}\left(z\right)}</annotation>

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</math><img src="./1cdb7e8f92a9d80dbbe7fcc6be13b3a8a9053922.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.567ex; height:2.843ex;" alt="{\displaystyle W_{k}\left(z\right)}" loading="lazy">, which is a complex-valued function of one complex argument. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}}</annotation>

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</math><img src="./7f541f57fd799ba5137a2e50a1a728dde4306c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{0}}" loading="lazy"> is known as the <a href="Principal_branch" title="Principal branch">principal branch</a>. These functions have the following property: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

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<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

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</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> are any complex numbers, then

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<annotation encoding="application/x-tex">{\displaystyle we^{w}=z}</annotation>

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</math><img src="./ad41339c8ffa64fefe024547ffde6b49897bcdf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.343ex; height:2.343ex;" alt="{\displaystyle we^{w}=z}" loading="lazy"></dd></dl>

holds if and only if

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<mtext> for some integer </mtext>

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<annotation encoding="application/x-tex">{\displaystyle w=W_{k}(z)\ \ {\text{ for some integer }}k.}</annotation>

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</math><img src="./35a2ae3aa67fcc672b67fd57066bd4cb643443a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.1ex; height:2.843ex;" alt="{\displaystyle w=W_{k}(z)\ \ {\text{ for some integer }}k.}" loading="lazy"></dd></dl>

When dealing with real numbers only, the two branches <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}}</annotation>

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</math><img src="./7f541f57fd799ba5137a2e50a1a728dde4306c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{-1}}</annotation>

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</math><img src="./bdf91805ba18586ff58b3035ff4b8fe48d39dd67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.526ex; height:2.509ex;" alt="{\displaystyle W_{-1}}" loading="lazy"> suffice: for real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> the equation

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<annotation encoding="application/x-tex">{\displaystyle ye^{y}=x}</annotation>

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</math><img src="./a92302519e95af77ecdab3ec269db1a74b9a55a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.716ex; height:2.676ex;" alt="{\displaystyle ye^{y}=x}" loading="lazy"></dd></dl>

can be solved for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\geq {\frac {-1}{e}}}">

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<annotation encoding="application/x-tex">{\textstyle x\geq {\frac {-1}{e}}}</annotation>

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</math><img src="./2d7d239560197e49fbc2677235099cced4e0bacd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.365ex; height:3.509ex;" alt="{\textstyle x\geq {\frac {-1}{e}}}" loading="lazy">; yields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=W_{0}\left(x\right)}">

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<annotation encoding="application/x-tex">{\displaystyle y=W_{0}\left(x\right)}</annotation>

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</math><img src="./eadc3feb9227b3b1a996a2e952973f52e43ef81a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.028ex; height:2.843ex;" alt="{\displaystyle y=W_{0}\left(x\right)}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation>

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</math><img src="./a2608e2b392b079f5b763f27bf52883dbee3b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}" loading="lazy"> and the two values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=W_{0}\left(x\right)}">

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<annotation encoding="application/x-tex">{\displaystyle y=W_{0}\left(x\right)}</annotation>

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</math><img src="./eadc3feb9227b3b1a996a2e952973f52e43ef81a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.028ex; height:2.843ex;" alt="{\displaystyle y=W_{0}\left(x\right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=W_{-1}\left(x\right)}">

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<annotation encoding="application/x-tex">{\displaystyle y=W_{-1}\left(x\right)}</annotation>

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</math><img src="./3fda605707d49858f2fa42f8e7c8cfd64d1ac89d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.306ex; height:2.843ex;" alt="{\displaystyle y=W_{-1}\left(x\right)}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {-1}{e}}\leq x<0}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {-1}{e}}\leq x<0}</annotation>

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</math><img src="./bfb6d31ef665be1c7c57862780afeedbd798d1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.626ex; height:3.509ex;" alt="{\textstyle {\frac {-1}{e}}\leq x<0}" loading="lazy">.

The Lambert W function's branches cannot be expressed in terms of <a href="Elementary_function" title="Elementary function">elementary functions</a>.<a href="#cite_note-3">[3]</a> It is useful in <a href="Combinatorics" title="Combinatorics">combinatorics</a>, for instance, in the enumeration of <a href="Tree_graph" class="mw-redirect" title="Tree graph">trees</a>. It can be used to solve various equations involving exponentials (e.g. the maxima of the <a href="Planck's_law" title="Planck's law">Planck</a>, <a href="Bose%E2%80%93Einstein_distribution" class="mw-redirect" title="Bose–Einstein distribution">Bose–Einstein</a>, and <a href="Fermi%E2%80%93Dirac_distribution" class="mw-redirect" title="Fermi–Dirac distribution">Fermi–Dirac</a> distributions) and also occurs in the solution of <a href="Delay_differential_equation" title="Delay differential equation">delay differential equations</a>, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'\left(t\right)=a\ y\left(t-1\right)}">

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<annotation encoding="application/x-tex">{\displaystyle y'\left(t\right)=a\ y\left(t-1\right)}</annotation>

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</math><img src="./fde3d509eadaa8197245c46887b53e41b03bafb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.984ex; height:3.009ex;" alt="{\displaystyle y'\left(t\right)=a\ y\left(t-1\right)}" loading="lazy">. In <a href="Biochemistry" title="Biochemistry">biochemistry</a>, and in particular <a href="Enzyme_kinetics" title="Enzyme kinetics">enzyme kinetics</a>, an opened-form solution for the time-course kinetics analysis of <a href="Michaelis%E2%80%93Menten_kinetics" title="Michaelis–Menten kinetics">Michaelis–Menten kinetics</a> is described in terms of the Lambert W function.

<div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2></div>

The notation convention chosen here (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}}</annotation>

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</math><img src="./7f541f57fd799ba5137a2e50a1a728dde4306c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{-1}}</annotation>

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</math><img src="./bdf91805ba18586ff58b3035ff4b8fe48d39dd67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.526ex; height:2.509ex;" alt="{\displaystyle W_{-1}}" loading="lazy">) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and <a href="Donald_Knuth" title="Donald Knuth">Knuth</a>.<a href="#cite_note-Corless-4">[4]</a>

The name "product logarithm" can be understood as follows: since the <a href="Inverse_function" title="Inverse function">inverse function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(w\right)=e^{w}}">

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<annotation encoding="application/x-tex">{\displaystyle f\left(w\right)=e^{w}}</annotation>

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</math><img src="./99b6daffa82049a9e7a1909a128922b001b60a97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.73ex; height:2.843ex;" alt="{\displaystyle f\left(w\right)=e^{w}}" loading="lazy"> is termed the <a href="Logarithm" title="Logarithm">logarithm</a>, it makes sense to call the inverse "function" of the <a href="Product_(mathematics)" title="Product (mathematics)">product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle we^{w}}">

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<annotation encoding="application/x-tex">{\displaystyle we^{w}}</annotation>

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</math><img src="./69159aa9b8a7142701afcf3b165652db64dd55d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.157ex; height:2.343ex;" alt="{\displaystyle we^{w}}" loading="lazy"> the "product logarithm". (Technical note: like the <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a>, it is multivalued and thus W is described as a <a href="Converse_relation" title="Converse relation">converse relation</a> rather than inverse function.) It is related to the <a href="Omega_constant" title="Omega constant">omega constant</a>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}\left(1\right)}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left(1\right)}</annotation>

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</math><img src="./2f7e7c620f4e1d8390be0c942283aeed245870a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle W_{0}\left(1\right)}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

Lambert first considered the related Lambert's Transcendental Equation in 1758,<a href="#cite_note-5">[5]</a> which led to an article by <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in 1783<a href="#cite_note-6">[6]</a> that discussed the special case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle we^{w}}">

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<annotation encoding="application/x-tex">{\displaystyle we^{w}}</annotation>

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</math><img src="./69159aa9b8a7142701afcf3b165652db64dd55d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.157ex; height:2.343ex;" alt="{\displaystyle we^{w}}" loading="lazy">.

The equation Lambert considered was

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<annotation encoding="application/x-tex">{\displaystyle x=x^{m}+q.}</annotation>

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</math><img src="./dcfee5197bc0b88fecf7593a500ef3cb30a54018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.99ex; height:2.676ex;" alt="{\displaystyle x=x^{m}+q.}" loading="lazy"></dd></dl>

Euler transformed this equation into the form

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<annotation encoding="application/x-tex">{\displaystyle x^{a}-x^{b}=(a-b)cx^{a+b}.}</annotation>

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</math><img src="./167e439f005583eff5d898307b7d100494b97168.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.584ex; height:3.176ex;" alt="{\displaystyle x^{a}-x^{b}=(a-b)cx^{a+b}.}" loading="lazy"></dd></dl>

Both authors derived a series solution for their equations.

Once Euler had solved this equation, he considered the case ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}">

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<annotation encoding="application/x-tex">{\displaystyle a=b}</annotation>

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</math><img src="./1956b03d1314c7071ac1f45ed7b1e29422dcfcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}" loading="lazy">⁠. Taking limits, he derived the equation

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<annotation encoding="application/x-tex">{\displaystyle \ln x=cx^{a}.}</annotation>

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</math><img src="./8cc45ca36ddb3e0af14162b135dbc2fcbb03d0c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.84ex; height:2.343ex;" alt="{\displaystyle \ln x=cx^{a}.}" loading="lazy"></dd></dl>

He then put ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1}">

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<annotation encoding="application/x-tex">{\displaystyle a=1}</annotation>

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</math><img src="./6104442ed30596ef4d7795d3186273f68d796ea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=1}" loading="lazy">⁠ and obtained a convergent series solution for the resulting equation, expressing ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠ in terms of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">⁠.

After taking derivatives with respect to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠ and some manipulation, the standard form of the Lambert function is obtained.

In 1993, it was reported that the Lambert ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">⁠ function provides an exact solution to the quantum-mechanical <a href="Delta_potential#Double_delta_potential" title="Delta potential">double-well Dirac delta function model</a> for equal charges<a href="#cite_note-7">[7]</a>—a fundamental problem in physics. Prompted by this, Rob Corless and developers of the <a href="Maple_software" class="mw-redirect" title="Maple software">Maple</a> <a href="Computer_algebra_system" title="Computer algebra system">computer algebra system</a> realized that "the Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been."<a href="#cite_note-Corless-4">[4]</a><a href="#cite_note-corless_maple-8">[8]</a>

Another example where this function is found is in <a href="Michaelis%E2%80%93Menten_kinetics" title="Michaelis–Menten kinetics">Michaelis–Menten kinetics</a>.<a href="#cite_note-9">[9]</a>

Although it was widely believed that the Lambert ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">⁠ function cannot be expressed in terms of elementary (<a href="Liouvillian_function" title="Liouvillian function">Liouvillian</a>) functions, the first published proof did not appear until 2008.<a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading2"><h2 id="Elementary_properties,_branches_and_range">Elementary properties, branches and range</h2></div>

There are countably many branches of the W function, denoted by Wk(z), for integer k; W0(z) being the main (or principal) branch. W0(z) is defined for all complex numbers z while Wk(z) with k ≠ 0 is defined for all non-zero z. With W0(0) = 0 and limz→0 Wk(z) = −∞ for all k ≠ 0.

The branch point for the principal branch is at z = −<style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠1/e⁠, with a branch cut that extends to −∞ along the negative real axis. This branch cut separates the principal branch from the two branches W−1 and W1. In all branches Wk with k ≠ 0, there is a branch point at z = 0 and a branch cut along the entire negative real axis.

The functions Wk(z), k ∈ Z are all <a href="Injective_function" title="Injective function">injective</a> and their ranges are disjoint. The range of the entire multivalued function W is the complex plane. The image of the real axis is the union of the real axis and the <a href="Quadratrix_of_Hippias" title="Quadratrix of Hippias">quadratrix of Hippias</a>, the parametric curve w = −t cot t + it.

<div class="mw-heading mw-heading3"><h3 id="Inverse">Inverse</h3></div>

The range plot above also delineates the regions in the complex plane where the simple inverse relationship ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(n,ze^{z})=z}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle W(n,ze^{z})=z}</annotation>

</semantics>

</math><img src="./44bea647e5f75e6aa6ea721fc6e82ae261354cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.033ex; height:2.843ex;" alt="{\displaystyle W(n,ze^{z})=z}" loading="lazy">⁠ is true. ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=ze^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=ze^{z}}</annotation>

</semantics>

</math><img src="./7a7917800390a92b884fb7b89fe588991cc26f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.55ex; height:2.676ex;" alt="{\displaystyle f=ze^{z}}" loading="lazy">⁠ implies that there exists an ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠ such that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=W(n,f)=W(n,ze^{z})}">

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=W(n,f)=W(n,ze^{z})}</annotation>

</semantics>

</math><img src="./7638b1747063871193bcf89f749d43940cc6c0fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.083ex; height:2.843ex;" alt="{\displaystyle z=W(n,f)=W(n,ze^{z})}" loading="lazy">⁠, where ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠ depends upon the value of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">⁠. The value of the integer ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠ changes abruptly when ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ze^{z}}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ze^{z}}</annotation>

</semantics>

</math><img src="./be38e336fdae9740f565dacea4054ddfd0cc6311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.173ex; height:2.343ex;" alt="{\displaystyle ze^{z}}" loading="lazy">⁠ is at the branch cut of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(n,ze^{z})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W(n,ze^{z})}</annotation>

</semantics>

</math><img src="./fb66de58888c16bec55aabfe905396a59190dcd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.846ex; height:2.843ex;" alt="{\displaystyle W(n,ze^{z})}" loading="lazy">⁠, which means that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ze^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ze^{z}}</annotation>

</semantics>

</math><img src="./be38e336fdae9740f565dacea4054ddfd0cc6311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.173ex; height:2.343ex;" alt="{\displaystyle ze^{z}}" loading="lazy">⁠ ≤ 0, except for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n=0}</annotation>

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</math><img src="./26819344e55f5e671c76c07c18eb4291fcec85ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}" loading="lazy">⁠ where it is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ze^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ze^{z}}</annotation>

</semantics>

</math><img src="./be38e336fdae9740f565dacea4054ddfd0cc6311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.173ex; height:2.343ex;" alt="{\displaystyle ze^{z}}" loading="lazy">⁠ ≤ −1/⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy">⁠.

Defining ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=x+iy}</annotation>

</semantics>

</math><img src="./08e90bb6b36fef59c6113eed2a08f10d77240741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}" loading="lazy">⁠, where ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">⁠ are real, and expressing ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{z}}</annotation>

</semantics>

</math><img src="./f4772def31b56e642df3e4d1160cadff3d80ba45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.085ex; height:2.343ex;" alt="{\displaystyle e^{z}}" loading="lazy">⁠ in polar coordinates, it is seen that

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}ze^{z}&=(x+iy)e^{x}(\cos y+i\sin y)\\&=e^{x}(x\cos y-y\sin y)+ie^{x}(x\sin y+y\cos y)\\\end{aligned}}}</annotation>

</semantics>

</math><img src="./5d668b1da5121409ba1a86942a706c42716eebb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.1ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}ze^{z}&=(x+iy)e^{x}(\cos y+i\sin y)\\&=e^{x}(x\cos y-y\sin y)+ie^{x}(x\sin y+y\cos y)\\\end{aligned}}}" loading="lazy"></dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\neq 0}</annotation>

</semantics>

</math><img src="./5920e98ff3dd1cb41e01f76243300450c958d5e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.676ex;" alt="{\displaystyle n\neq 0}" loading="lazy">, the branch cut for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(n,ze^{z})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W(n,ze^{z})}</annotation>

</semantics>

</math><img src="./fb66de58888c16bec55aabfe905396a59190dcd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.846ex; height:2.843ex;" alt="{\displaystyle W(n,ze^{z})}" loading="lazy">⁠ is the non-positive real axis, so that

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\sin y+y\cos y=0\Rightarrow x=-y/\tan(y),}</annotation>

</semantics>

</math><img src="./22d2d35673f18dd149f590cbfa430167c80c7ec9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.939ex; height:2.843ex;" alt="{\displaystyle x\sin y+y\cos y=0\Rightarrow x=-y/\tan(y),}" loading="lazy"></dd></dl>

and

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x\cos y-y\sin y)e^{x}\leq 0.}</annotation>

</semantics>

</math><img src="./c2db7c5664466f757624e52e909a7404bd79a41a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.125ex; height:2.843ex;" alt="{\displaystyle (x\cos y-y\sin y)e^{x}\leq 0.}" loading="lazy"></dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=0}</annotation>

</semantics>

</math><img src="./26819344e55f5e671c76c07c18eb4291fcec85ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}" loading="lazy">, the branch cut for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W[n,ze^{z}]}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W[n,ze^{z}]}</annotation>

</semantics>

</math><img src="./fb4be5264a023f670dd1e4d4db4fbbffa4b51573.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.331ex; height:2.843ex;" alt="{\displaystyle W[n,ze^{z}]}" loading="lazy">⁠ is the real axis with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty <z\leq -1/e}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\infty <z\leq -1/e}</annotation>

</semantics>

</math><img src="./a077745950ba9bf004b30e28de4fd2d074b13c10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.633ex; height:2.843ex;" alt="{\displaystyle -\infty <z\leq -1/e}" loading="lazy">, so that the inequality becomes

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x\cos y-y\sin y)e^{x}\leq -1/e.}</annotation>

</semantics>

</math><img src="./6b9f6b929b1cf90d7bdacba6344fd55f1d596bbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.179ex; height:2.843ex;" alt="{\displaystyle (x\cos y-y\sin y)e^{x}\leq -1/e.}" loading="lazy"></dd></dl>

Inside the regions bounded by the above, there are no discontinuous changes in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(n,ze^{z})}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W(n,ze^{z})}</annotation>

</semantics>

</math><img src="./fb66de58888c16bec55aabfe905396a59190dcd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.846ex; height:2.843ex;" alt="{\displaystyle W(n,ze^{z})}" loading="lazy">⁠, and those regions specify where the ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">⁠ function is simply invertible, i.e. ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(n,ze^{z})=z}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W(n,ze^{z})=z}</annotation>

</semantics>

</math><img src="./44bea647e5f75e6aa6ea721fc6e82ae261354cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.033ex; height:2.843ex;" alt="{\displaystyle W(n,ze^{z})=z}" loading="lazy">⁠.

<div class="mw-heading mw-heading2"><h2 id="Calculus">Calculus</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Derivative">Derivative</h3></div>

By <a href="Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">implicit differentiation</a>, one can show that all branches of W satisfy the <a href="Ordinary_differential_equation" title="Ordinary differential equation">differential equation</a>

<mfrac>

<mrow>

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<mrow>

</mrow>

</mfrac>

</mrow>

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<mfrac>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle z(1+W){\frac {dW}{dz}}=W\quad {\text{for }}z\neq -{\frac {1}{e}}.}</annotation>

</semantics>

</math><img src="./d6e11498ef9bcadd7ebfa8f9d0423590c73aeefc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.686ex; height:5.509ex;" alt="{\displaystyle z(1+W){\frac {dW}{dz}}=W\quad {\text{for }}z\neq -{\frac {1}{e}}.}" loading="lazy"></dd></dl>

(W is not <a href="Differentiable_function" title="Differentiable function">differentiable</a> for z = −⁠1/e⁠.) As a consequence, that gets the following formula for the derivative of W:

<mfrac>

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<mrow>

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</mfrac>

</mrow>

<mfrac>

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<mrow>

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</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dW}{dz}}={\frac {W(z)}{z(1+W(z))}}\quad {\text{for }}z\not \in \left\{0,-{\frac {1}{e}}\right\}.}</annotation>

</semantics>

</math><img src="./9fd7247525138ee99341ddd3c6e7439700eaba50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.796ex; height:6.509ex;" alt="{\displaystyle {\frac {dW}{dz}}={\frac {W(z)}{z(1+W(z))}}\quad {\text{for }}z\not \in \left\{0,-{\frac {1}{e}}\right\}.}" loading="lazy"></dd></dl>

Using the identity eW(z) = ⁠z/W(z)⁠, gives the following equivalent formula:

<mfrac>

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</mrow>

<mrow>

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</mfrac>

</mrow>

<mfrac>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dW}{dz}}={\frac {1}{z+e^{W(z)}}}\quad {\text{for }}z\neq -{\frac {1}{e}}.}</annotation>

</semantics>

</math><img src="./c6d6d95e284d6c0f3cbd8b9d5f81c416030465d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.766ex; height:5.843ex;" alt="{\displaystyle {\frac {dW}{dz}}={\frac {1}{z+e^{W(z)}}}\quad {\text{for }}z\neq -{\frac {1}{e}}.}" loading="lazy"></dd></dl>

At the origin we have

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W'_{0}(0)=1.}</annotation>

</semantics>

</math><img src="./97f05108dcc7139c218201543423589bcc18b373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.127ex; height:3.009ex;" alt="{\displaystyle W'_{0}(0)=1.}" loading="lazy"></dd></dl>

The n-th derivative of W is of the form:

<mfrac>

<mrow>

<msup>

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</msup>

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</msup>

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<mfrac>

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<msub>

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</msub>

</mrow>

<mrow>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{n}W}{dz^{n}}}={\frac {P_{n}(W(z))}{(z+e^{W(z)})^{n}(W(z)+1)^{n-1}}}\quad {\text{for }}n>0,\,z\neq -{\frac {1}{e}}.}</annotation>

</semantics>

</math><img src="./e23289cfaf990d74b636d1c4f394eb37725e4d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:57.555ex; height:6.676ex;" alt="{\displaystyle {\frac {d^{n}W}{dz^{n}}}={\frac {P_{n}(W(z))}{(z+e^{W(z)})^{n}(W(z)+1)^{n-1}}}\quad {\text{for }}n>0,\,z\neq -{\frac {1}{e}}.}" loading="lazy"></dd></dl>

Where Pn is a polynomial function with coefficients defined in <a href="https://oeis.org/A042977" class="extiw external" title="oeis:A042977">A042977</a>. If and only if z is a root of Pn then zez is a root of the n-th derivative of W.

Taking the derivative of the n-th derivative of W yields:

<mfrac>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{n+1}W}{dz^{n+1}}}={\frac {(W(z)+1)P_{n}'(W(z))+(1-3n-nW(z))P_{n}(W(z))}{(n+e^{W(z)})^{n+1}(W(z)+1)^{n}}}\quad {\text{for }}n>0,\,z\neq -{\frac {1}{e}}.}</annotation>

</semantics>

</math><img src="./33cd57f640b8b571c4956802b745ac7b1fdcb465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:84.777ex; height:6.676ex;" alt="{\displaystyle {\frac {d^{n+1}W}{dz^{n+1}}}={\frac {(W(z)+1)P_{n}'(W(z))+(1-3n-nW(z))P_{n}(W(z))}{(n+e^{W(z)})^{n+1}(W(z)+1)^{n}}}\quad {\text{for }}n>0,\,z\neq -{\frac {1}{e}}.}" loading="lazy"></dd></dl>

Inductively proving the n-th derivative equation.

<div class="mw-heading mw-heading3"><h3 id="Integral">Integral</h3></div>

The function W(x), and many other expressions involving W(x), can be <a href="Integral" title="Integral">integrated</a> using the <a href="Substitution_rule" class="mw-redirect" title="Substitution rule">substitution</a> w = W(x), i.e. x = wew:

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

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</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int W(x)\,dx&=xW(x)-x+e^{W(x)}+C\\&=x\left(W(x)-1+{\frac {1}{W(x)}}\right)+C.\end{aligned}}}</annotation>

</semantics>

</math><img src="./306bb9dc1cbd1b4d53d1e9f5065a5fc30e0829e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:44.157ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\int W(x)\,dx&=xW(x)-x+e^{W(x)}+C\\&=x\left(W(x)-1+{\frac {1}{W(x)}}\right)+C.\end{aligned}}}" loading="lazy"></dd></dl>

(The last equation is more common in the literature but is undefined at x = 0). One consequence of this (using the fact that W0(e) = 1) is the identity

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{e}W_{0}(x)\,dx=e-1.}</annotation>

</semantics>

</math><img src="./cb54de08d32bd92da9412647bff8acc894c3e79b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.001ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{e}W_{0}(x)\,dx=e-1.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Asymptotic_expansions">Asymptotic expansions</h2></div>

The <a href="Taylor_series" title="Taylor series">Taylor series</a> of W0 around 0 can be found using the <a href="Lagrange_inversion_theorem" title="Lagrange inversion theorem">Lagrange inversion theorem</a> and is given by

<msub>

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</msub>

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</mrow>

</munderover>

<mfrac>

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</msup>

<msup>

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<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}x^{n}=x-x^{2}+{\tfrac {3}{2}}x^{3}-{\tfrac {16}{6}}x^{4}+{\tfrac {125}{24}}x^{5}-\cdots .}</annotation>

</semantics>

</math><img src="./b1a082e34888e547d3f1ebabd04188900376a490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.306ex; height:7.009ex;" alt="{\displaystyle W_{0}(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}x^{n}=x-x^{2}+{\tfrac {3}{2}}x^{3}-{\tfrac {16}{6}}x^{4}+{\tfrac {125}{24}}x^{5}-\cdots .}" loading="lazy"></dd></dl>

The <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> is ⁠1/e⁠, as may be seen by the <a href="Ratio_test" title="Ratio test">ratio test</a>. The function defined by this series can be extended to a <a href="Holomorphic_function" title="Holomorphic function">holomorphic function</a> defined on all complex numbers with a <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a> along the <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> (−∞, −⁠1/e⁠]; this holomorphic function defines the <a href="Principal_branch" title="Principal branch">principal branch</a> of the Lambert W function.

For large values of x, W0 is asymptotic to

<mtr>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W_{0}(x)&=L_{1}-L_{2}+{\frac {L_{2}}{L_{1}}}+{\frac {L_{2}\left(-2+L_{2}\right)}{2L_{1}^{2}}}+{\frac {L_{2}\left(6-9L_{2}+2L_{2}^{2}\right)}{6L_{1}^{3}}}+{\frac {L_{2}\left(-12+36L_{2}-22L_{2}^{2}+3L_{2}^{3}\right)}{12L_{1}^{4}}}+\cdots \\[5pt]&=L_{1}-L_{2}+\sum _{l=0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{l}\left[{\begin{smallmatrix}l+m\\l+1\end{smallmatrix}}\right]}{m!}}L_{1}^{-l-m}L_{2}^{m},\end{aligned}}}</annotation>

</semantics>

</math><img src="./f5e7cb8d231b7fabe03c2fc5d8845d1c9e18f467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:105.668ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}W_{0}(x)&=L_{1}-L_{2}+{\frac {L_{2}}{L_{1}}}+{\frac {L_{2}\left(-2+L_{2}\right)}{2L_{1}^{2}}}+{\frac {L_{2}\left(6-9L_{2}+2L_{2}^{2}\right)}{6L_{1}^{3}}}+{\frac {L_{2}\left(-12+36L_{2}-22L_{2}^{2}+3L_{2}^{3}\right)}{12L_{1}^{4}}}+\cdots \\[5pt]&=L_{1}-L_{2}+\sum _{l=0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{l}\left[{\begin{smallmatrix}l+m\\l+1\end{smallmatrix}}\right]}{m!}}L_{1}^{-l-m}L_{2}^{m},\end{aligned}}}" loading="lazy"></dd></dl>

where L1 = ln x, L2 = ln ln x, and <big><big>[</big></big>l + m l + 1<big><big>]</big></big> is a non-negative <a href="Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling number of the first kind</a>.<a href="#cite_note-Corless-4">[4]</a> Keeping only the first two terms of the expansion,

<msub>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)=\ln x-\ln \ln x+{\mathcal {o}}(1).}</annotation>

</semantics>

</math><img src="./b3100ccb9158198246554a066e1ea2927cf81888.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.551ex; height:2.843ex;" alt="{\displaystyle W_{0}(x)=\ln x-\ln \ln x+{\mathcal {o}}(1).}" loading="lazy"></dd></dl>

The other real branch, W−1, defined in the interval [−⁠1/e⁠, 0), has an approximation of the same form as x approaches zero, with in this case L1 = ln(−x) and L2 = ln(−ln(−x)).<a href="#cite_note-Corless-4">[4]</a>

<div class="mw-heading mw-heading3"><h3 id="Integer_and_complex_powers">Integer and complex powers</h3></div>

Integer powers of W0 also admit simple <a href="Taylor_series" title="Taylor series">Taylor</a> (or <a href="Laurent_series" title="Laurent series">Laurent</a>) series expansions at zero:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)^{2}=\sum _{n=2}^{\infty }{\frac {-2\left(-n\right)^{n-3}}{(n-2)!}}x^{n}=x^{2}-2x^{3}+4x^{4}-{\tfrac {25}{3}}x^{5}+18x^{6}-\cdots .}</annotation>

</semantics>

</math><img src="./065dad231dc1911e1b8c9554ac9bcb957e4c06b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:69.075ex; height:7.176ex;" alt="{\displaystyle W_{0}(x)^{2}=\sum _{n=2}^{\infty }{\frac {-2\left(-n\right)^{n-3}}{(n-2)!}}x^{n}=x^{2}-2x^{3}+4x^{4}-{\tfrac {25}{3}}x^{5}+18x^{6}-\cdots .}" loading="lazy"></dd></dl>

More generally, for r ∈ Z, the <a href="Lagrange_inversion_theorem" title="Lagrange inversion theorem">Lagrange inversion formula</a> gives

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)^{r}=\sum _{n=r}^{\infty }{\frac {-r\left(-n\right)^{n-r-1}}{(n-r)!}}x^{n},}</annotation>

</semantics>

</math><img src="./365b5d7e95aba4df61b47c528c5d72a0aa633b45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.44ex; height:7.176ex;" alt="{\displaystyle W_{0}(x)^{r}=\sum _{n=r}^{\infty }{\frac {-r\left(-n\right)^{n-r-1}}{(n-r)!}}x^{n},}" loading="lazy"></dd></dl>

which is, in general, a Laurent series of order r. Equivalently, the latter can be written in the form of a Taylor expansion of powers of W0(x) / x:

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {W_{0}(x)}{x}}\right)^{r}=e^{-rW_{0}(x)}=\sum _{n=0}^{\infty }{\frac {r\left(n+r\right)^{n-1}}{n!}}\left(-x\right)^{n},}</annotation>

</semantics>

</math><img src="./ee03e351f282958338ce03db15342e0d1881ef0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.604ex; height:7.176ex;" alt="{\displaystyle \left({\frac {W_{0}(x)}{x}}\right)^{r}=e^{-rW_{0}(x)}=\sum _{n=0}^{\infty }{\frac {r\left(n+r\right)^{n-1}}{n!}}\left(-x\right)^{n},}" loading="lazy"></dd></dl>

which holds for any r ∈ C and |x| < ⁠1/e⁠.

<div class="mw-heading mw-heading2"><h2 id="Bounds_and_inequalities">Bounds and inequalities</h2></div>

A number of non-asymptotic bounds are known for the Lambert function.

<div class="mw-heading mw-heading3"><h3 id="Principal_branch">Principal branch</h3></div>

Hoorfar and Hassani<a href="#cite_note-:0-11">[11]</a> showed that the following bound holds for x ≥ e:

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<annotation encoding="application/x-tex">{\displaystyle \ln x-\ln \ln x+{\frac {\ln \ln x}{2\ln x}}\leq W_{0}(x)\leq \ln x-\ln \ln x+{\frac {e}{e-1}}{\frac {\ln \ln x}{\ln x}}.}</annotation>

</semantics>

</math><img src="./51a61ec340c1a7712379e961292278725e25f5d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:63.43ex; height:5.509ex;" alt="{\displaystyle \ln x-\ln \ln x+{\frac {\ln \ln x}{2\ln x}}\leq W_{0}(x)\leq \ln x-\ln \ln x+{\frac {e}{e-1}}{\frac {\ln \ln x}{\ln x}}.}" loading="lazy"></dd></dl>

Roberto Iacono and John P. Boyd<a href="#cite_note-doi.org-12">[12]</a> enhanced the bounds for x ≥ e as follows:

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<annotation encoding="application/x-tex">{\displaystyle \ln \left({\frac {x}{\ln x}}\right)-{\frac {\ln \left({\frac {x}{\ln x}}\right)}{1+\ln \left({\frac {x}{\ln x}}\right)}}\ln \left(1-{\frac {\ln \ln x}{\ln x}}\right)\leq W_{0}(x)\leq \ln \left({\frac {x}{\ln x}}\right)-\ln \left(\left(1-{\frac {\ln \ln x}{\ln x}}\right)\left(1-{\frac {\ln \left(1-{\frac {\ln \ln x}{\ln x}}\right)}{1+\ln \left({\frac {x}{\ln x}}\right)}}\right)\right).}</annotation>

</semantics>

</math><img src="./c9be1cb9665154688e0097c7c55e6aa6cac247a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:110.536ex; height:10.176ex;" alt="{\displaystyle \ln \left({\frac {x}{\ln x}}\right)-{\frac {\ln \left({\frac {x}{\ln x}}\right)}{1+\ln \left({\frac {x}{\ln x}}\right)}}\ln \left(1-{\frac {\ln \ln x}{\ln x}}\right)\leq W_{0}(x)\leq \ln \left({\frac {x}{\ln x}}\right)-\ln \left(\left(1-{\frac {\ln \ln x}{\ln x}}\right)\left(1-{\frac {\ln \left(1-{\frac {\ln \ln x}{\ln x}}\right)}{1+\ln \left({\frac {x}{\ln x}}\right)}}\right)\right).}" loading="lazy"></dd></dl>

Hoorfar and Hassani<a href="#cite_note-:0-11">[11]</a> also showed the general bound

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)\leq \ln \left({\frac {x+y}{1+\ln(y)}}\right),}</annotation>

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</math><img src="./75f5d19b119d702b0ba652a5b3caa99efe691a65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.236ex; height:6.343ex;" alt="{\displaystyle W_{0}(x)\leq \ln \left({\frac {x+y}{1+\ln(y)}}\right),}" loading="lazy"></dd></dl>

for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y>1/e}">

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</math><img src="./5da08c735da9c3b908a8e8207f7d13a3c624787b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle y>1/e}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq -1/e}">

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<annotation encoding="application/x-tex">{\displaystyle x\geq -1/e}</annotation>

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</math><img src="./f1ae2cfc5ffaa05584125975340052250cd9ee3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.645ex; height:2.843ex;" alt="{\displaystyle x\geq -1/e}" loading="lazy">, with equality only for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y\ln(y)}">

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<annotation encoding="application/x-tex">{\displaystyle x=y\ln(y)}</annotation>

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</math><img src="./e52a2b98d889529f09470eb9ea603cddc5e7933e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.875ex; height:2.843ex;" alt="{\displaystyle x=y\ln(y)}" loading="lazy">.

The bound allows many other bounds to be derived, such as taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x+1}">

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</math><img src="./14d4839f41dc3d34ae434dfad1ab18479cc6a41b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle y=x+1}" loading="lazy"> which gives the bound

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)\leq \ln \left({\frac {2x+1}{1+\ln(x+1)}}\right).}</annotation>

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</math><img src="./d42617ffa7b36c0d3e587214b4120d29ec29ab08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.413ex; height:6.343ex;" alt="{\displaystyle W_{0}(x)\leq \ln \left({\frac {2x+1}{1+\ln(x+1)}}\right).}" loading="lazy"></dd></dl>

Bounds for the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}(-xe^{-x})}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(-xe^{-x})}</annotation>

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</math><img src="./e7695d046d6c48daf6f7582a99c45fea8bb8391a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.73ex; height:3.009ex;" alt="{\displaystyle W_{0}(-xe^{-x})}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle x\geq 1}</annotation>

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</math><img src="./5ca3ced43f1713577888a8a7ade2d0aaf8354a4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 1}" loading="lazy"> are obtained by Stewart.<a href="#cite_note-13">[13]</a>

<div class="mw-heading mw-heading3"><h3 id="Secondary_branch">Secondary branch</h3></div>

The branch W−1 can be bounded as follows:<a href="#cite_note-Chatzigeorgiou-14">[14]</a>

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<annotation encoding="application/x-tex">{\displaystyle -1-{\sqrt {2u}}-u<W_{-1}\left(-e^{-u-1}\right)<-1-{\sqrt {2u}}-{\tfrac {2}{3}}u\quad {\text{for }}u>0.}</annotation>

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</math><img src="./4cf2cde487b2f1f1ee7175a456648358e0bb4c69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:63.086ex; height:3.676ex;" alt="{\displaystyle -1-{\sqrt {2u}}-u<W_{-1}\left(-e^{-u-1}\right)<-1-{\sqrt {2u}}-{\tfrac {2}{3}}u\quad {\text{for }}u>0.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Identities">Identities</h2></div>

A few identities follow from the definition:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W_{0}(xe^{x})&=x&{\text{for }}x&\geq -1,\\W_{-1}(xe^{x})&=x&{\text{for }}x&\leq -1.\end{aligned}}}</annotation>

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</math><img src="./981c1760f1de8eb973aee158570dc31331726312.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.158ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}W_{0}(xe^{x})&=x&{\text{for }}x&\geq -1,\\W_{-1}(xe^{x})&=x&{\text{for }}x&\leq -1.\end{aligned}}}" loading="lazy"></dd></dl>

Since f(x) = xex is not <a href="Injective" class="mw-redirect" title="Injective">injective</a>, it does not always hold that W(f(x)) = x, much like with the <a href="Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse trigonometric functions</a>. For fixed x < 0 and x ≠ −1, the equation xex = yey has two real solutions in y, one of which is of course y = x. Then, for i = 0 and x < −1, as well as for i = −1 and x ∈ (−1, 0), y = Wi(xex) is the other solution.

Some other identities:<a href="#cite_note-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&W(x)e^{W(x)}=x,\quad {\text{therefore:}}\\[5pt]&e^{W(x)}={\frac {x}{W(x)}},\qquad e^{-W(x)}={\frac {W(x)}{x}},\qquad e^{nW(x)}=\left({\frac {x}{W(x)}}\right)^{n}.\end{aligned}}}</annotation>

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</math><img src="./db7a28b4f31f15aa4730b353d5ef11b206fa43b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:63.959ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}&W(x)e^{W(x)}=x,\quad {\text{therefore:}}\\[5pt]&e^{W(x)}={\frac {x}{W(x)}},\qquad e^{-W(x)}={\frac {W(x)}{x}},\qquad e^{nW(x)}=\left({\frac {x}{W(x)}}\right)^{n}.\end{aligned}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \ln W_{0}(x)=\ln x-W_{0}(x)\quad {\text{for }}x>0.}</annotation>

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</math><img src="./1df4e8b60b91680f39b59692181bb02cf9b07334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.622ex; height:2.843ex;" alt="{\displaystyle \ln W_{0}(x)=\ln x-W_{0}(x)\quad {\text{for }}x>0.}" loading="lazy"><a href="#cite_note-16">[16]</a></dd>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left(x\ln x\right)=\ln x\quad {\text{and}}\quad e^{W_{0}\left(x\ln x\right)}=x\quad {\text{for }}{\frac {1}{e}}\leq x.}</annotation>

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</math><img src="./03d2e2924513724ec00d0d0e2f8ce98b4f4cd6bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.159ex; height:5.176ex;" alt="{\displaystyle W_{0}\left(x\ln x\right)=\ln x\quad {\text{and}}\quad e^{W_{0}\left(x\ln x\right)}=x\quad {\text{for }}{\frac {1}{e}}\leq x.}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle W_{-1}\left(x\ln x\right)=\ln x\quad {\text{and}}\quad e^{W_{-1}\left(x\ln x\right)}=x\quad {\text{for }}0<x\leq {\frac {1}{e}}.}</annotation>

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</math><img src="./43bcb86e6fb0bdcf6df5042fdaf41402fe5824b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.737ex; height:5.176ex;" alt="{\displaystyle W_{-1}\left(x\ln x\right)=\ln x\quad {\text{and}}\quad e^{W_{-1}\left(x\ln x\right)}=x\quad {\text{for }}0<x\leq {\frac {1}{e}}.}" loading="lazy"></dd>

<dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&W(x)=\ln {\frac {x}{W(x)}}&&{\text{for }}x\geq -{\frac {1}{e}},\\[5pt]&W\left({\frac {nx^{n}}{W\left(x\right)^{n-1}}}\right)=nW(x)&&{\text{for }}n,x>0\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&W(x)=\ln {\frac {x}{W(x)}}&&{\text{for }}x\geq -{\frac {1}{e}},\\[5pt]&W\left({\frac {nx^{n}}{W\left(x\right)^{n-1}}}\right)=nW(x)&&{\text{for }}n,x>0\end{aligned}}}</annotation>

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</math><img src="./e834c16fb0e25cf530b84a39bdcc0bba7c75d762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.581ex; margin-bottom: -0.257ex; width:43.945ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}&W(x)=\ln {\frac {x}{W(x)}}&&{\text{for }}x\geq -{\frac {1}{e}},\\[5pt]&W\left({\frac {nx^{n}}{W\left(x\right)^{n-1}}}\right)=nW(x)&&{\text{for }}n,x>0\end{aligned}}}" loading="lazy">

<dl><dd>(which can be extended to other n and x if the correct branch is chosen).</dd></dl></dd>

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<annotation encoding="application/x-tex">{\displaystyle W(x)+W(y)=W\left(xy\left({\frac {1}{W(x)}}+{\frac {1}{W(y)}}\right)\right)\quad {\text{for }}x,y>0.}</annotation>

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</math><img src="./a81c1f873a7678c1fa948811300b339935ce0749.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.44ex; height:6.343ex;" alt="{\displaystyle W(x)+W(y)=W\left(xy\left({\frac {1}{W(x)}}+{\frac {1}{W(y)}}\right)\right)\quad {\text{for }}x,y>0.}" loading="lazy"></dd></dl>

Substituting −ln x in the definition:<a href="#cite_note-17">[17]</a>

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}W_{0}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}0&<x\leq e,\\[5pt]W_{-1}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}x&>e.\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W_{0}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}0&<x\leq e,\\[5pt]W_{-1}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}x&>e.\end{aligned}}}</annotation>

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</math><img src="./108810da107aa426c8708fe1c3eaa34a2da9575e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:42.935ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}W_{0}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}0&<x\leq e,\\[5pt]W_{-1}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}x&>e.\end{aligned}}}" loading="lazy"></dd></dl>

With Euler's iterated exponential h(x):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h(x)&=e^{-W(-\ln x)}\\&={\frac {W(-\ln x)}{-\ln x}}\quad {\text{for }}x\neq 1.\end{aligned}}}</annotation>

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</math><img src="./7510b63b91578e7a554c9092c932aef8bf9a256b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:31.187ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}h(x)&=e^{-W(-\ln x)}\\&={\frac {W(-\ln x)}{-\ln x}}\quad {\text{for }}x\neq 1.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Special_values">Special values</h2></div>

The following are special values of the principal branch:

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left(-{\frac {\pi }{2}}\right)={\frac {i\pi }{2}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left(-{\frac {1}{e}}\right)=-1}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left(2\ln 2\right)=\ln 2}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left(x\ln x\right)=\ln x\quad \left(x\geqslant {\tfrac {1}{e}}\approx 0.36788\right)}</annotation>

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<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}\left(x^{x+1}\ln x\right)=x\ln x\quad \left(x>0\right)}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left(x^{x+1}\ln x\right)=x\ln x\quad \left(x>0\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(0)=0}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(1)=\Omega =\left(\int _{-\infty }^{\infty }{\frac {dt}{\left(e^{t}-t\right)^{2}+\pi ^{2}}}\right)^{\!-1}\!\!\!\!-\,1\approx 0.56714329\quad }</annotation>

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</math><img src="./03b85d12934c4570fb9cbacfb15de5930a0244f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.445ex; height:8.009ex;" alt="{\displaystyle W_{0}(1)=\Omega =\left(\int _{-\infty }^{\infty }{\frac {dt}{\left(e^{t}-t\right)^{2}+\pi ^{2}}}\right)^{\!-1}\!\!\!\!-\,1\approx 0.56714329\quad }" loading="lazy"> (the <a href="Omega_constant" title="Omega constant">omega constant</a>)</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(1)=e^{-W_{0}(1)}=\ln {\frac {1}{W_{0}(1)}}=-\ln W_{0}(1)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(e)=1}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left(e^{1+e}\right)=e}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left({\frac {\sqrt {e}}{2}}\right)={\frac {1}{2}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\left({\frac {\sqrt[{n}]{e}}{n}}\right)={\frac {1}{n}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(-1)\approx -0.31813+1.33723i}</annotation>

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Special values of the branch W−1:

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<annotation encoding="application/x-tex">{\displaystyle W_{-1}\left(-{\frac {\ln 2}{2}}\right)=-\ln 4}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2></div>

The principal branch of the Lambert function can be represented by a proper integral, due to Poisson:<a href="#cite_note-Finch-18">[18]</a>

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {\pi }{2}}W_{0}(-x)=\int _{0}^{\pi }{\frac {\sin \left({\tfrac {3}{2}}t\right)-xe^{\cos t}\sin \left({\tfrac {5}{2}}t-\sin t\right)}{1-2xe^{\cos t}\cos(t-\sin t)+x^{2}e^{2\cos t}}}\sin \left({\tfrac {1}{2}}t\right)\,dt\quad {\text{for }}|x|<{\frac {1}{e}}.}</annotation>

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</math><img src="./07c88a1fa9d2c5d022886689e5ca9b711d5af866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:79.492ex; height:8.343ex;" alt="{\displaystyle -{\frac {\pi }{2}}W_{0}(-x)=\int _{0}^{\pi }{\frac {\sin \left({\tfrac {3}{2}}t\right)-xe^{\cos t}\sin \left({\tfrac {5}{2}}t-\sin t\right)}{1-2xe^{\cos t}\cos(t-\sin t)+x^{2}e^{2\cos t}}}\sin \left({\tfrac {1}{2}}t\right)\,dt\quad {\text{for }}|x|<{\frac {1}{e}}.}" loading="lazy"></dd></dl>

Another representation of the principal branch was found by Kalugin–Jeffrey–Corless:<a href="#cite_note-19">[19]</a>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\ln \left(1+x{\frac {\sin t}{t}}e^{t\cot t}\right)dt.}</annotation>

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</math><img src="./d07ba1a260f0280a88e9170b7223af26482571da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.849ex; height:6.176ex;" alt="{\displaystyle W_{0}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\ln \left(1+x{\frac {\sin t}{t}}e^{t\cot t}\right)dt.}" loading="lazy"></dd></dl>

The following <a href="Continued_fraction" title="Continued fraction">continued fraction</a> representation also holds for the principal branch:<a href="#cite_note-Dubinov-20">[20]</a>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)={\cfrac {x}{1+{\cfrac {x}{1+{\cfrac {x}{2+{\cfrac {5x}{3+{\cfrac {17x}{10+{\cfrac {133x}{17+{\cfrac {1927x}{190+{\cfrac {13582711x}{94423+\ddots }}}}}}}}}}}}}}}}.}</annotation>

</semantics>

</math><img src="./b17f35c9a33ba89e7bd6ba44e884c8627cfa681d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -29.671ex; width:61.122ex; height:33.843ex;" alt="{\displaystyle W_{0}(x)={\cfrac {x}{1+{\cfrac {x}{1+{\cfrac {x}{2+{\cfrac {5x}{3+{\cfrac {17x}{10+{\cfrac {133x}{17+{\cfrac {1927x}{190+{\cfrac {13582711x}{94423+\ddots }}}}}}}}}}}}}}}}.}" loading="lazy"></dd></dl>

Also, if |W0(x)| < 1:<a href="#cite_note-21">[21]</a>

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)={\cfrac {x}{\exp {\cfrac {x}{\exp {\cfrac {x}{\ddots }}}}}}.}</annotation>

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</math><img src="./2f63a5dffbb454d9468b4f909899f35737331ce6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:23.498ex; height:15.843ex;" alt="{\displaystyle W_{0}(x)={\cfrac {x}{\exp {\cfrac {x}{\exp {\cfrac {x}{\ddots }}}}}}.}" loading="lazy"></dd></dl>

In turn, if |W0(x)| > 1, then

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x)=\ln {\cfrac {x}{\ln {\cfrac {x}{\ln {\cfrac {x}{\ddots }}}}}}.}</annotation>

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</math><img src="./63c97b3a9b168b2e1b7923731adae59d391f9807.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:22.599ex; height:15.843ex;" alt="{\displaystyle W_{0}(x)=\ln {\cfrac {x}{\ln {\cfrac {x}{\ln {\cfrac {x}{\ddots }}}}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Other_formulas">Other formulas</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definite_integrals">Definite integrals</h3></div>

There are several useful definite integral formulas involving the principal branch of the W function, including the following:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\int _{0}^{\pi }W_{0}\left(2\cot ^{2}x\right)\sec ^{2}x\,dx=4{\sqrt {\pi }},\\[5pt]&\int _{0}^{\infty }{\frac {W_{0}(x)}{x{\sqrt {x}}}}\,dx=2{\sqrt {2\pi }},\\[5pt]&\int _{0}^{\infty }W_{0}\left({\frac {1}{x^{2}}}\right)\,dx={\sqrt {2\pi }},{\text{ and more generally}}\\[5pt]&\int _{0}^{\infty }W_{0}\left({\frac {1}{x^{N}}}\right)\,dx=N^{1-{\frac {1}{N}}}\Gamma \left(1-{\frac {1}{N}}\right)\qquad {\text{for }}N>0\end{aligned}}}">

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<mtext> and more generally</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\int _{0}^{\pi }W_{0}\left(2\cot ^{2}x\right)\sec ^{2}x\,dx=4{\sqrt {\pi }},\\[5pt]&\int _{0}^{\infty }{\frac {W_{0}(x)}{x{\sqrt {x}}}}\,dx=2{\sqrt {2\pi }},\\[5pt]&\int _{0}^{\infty }W_{0}\left({\frac {1}{x^{2}}}\right)\,dx={\sqrt {2\pi }},{\text{ and more generally}}\\[5pt]&\int _{0}^{\infty }W_{0}\left({\frac {1}{x^{N}}}\right)\,dx=N^{1-{\frac {1}{N}}}\Gamma \left(1-{\frac {1}{N}}\right)\qquad {\text{for }}N>0\end{aligned}}}</annotation>

</semantics>

</math><img src="./15cb1080e5d711b28682984c7747b35b993f946a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:56.561ex; height:28.509ex;" alt="{\displaystyle {\begin{aligned}&\int _{0}^{\pi }W_{0}\left(2\cot ^{2}x\right)\sec ^{2}x\,dx=4{\sqrt {\pi }},\\[5pt]&\int _{0}^{\infty }{\frac {W_{0}(x)}{x{\sqrt {x}}}}\,dx=2{\sqrt {2\pi }},\\[5pt]&\int _{0}^{\infty }W_{0}\left({\frac {1}{x^{2}}}\right)\,dx={\sqrt {2\pi }},{\text{ and more generally}}\\[5pt]&\int _{0}^{\infty }W_{0}\left({\frac {1}{x^{N}}}\right)\,dx=N^{1-{\frac {1}{N}}}\Gamma \left(1-{\frac {1}{N}}\right)\qquad {\text{for }}N>0\end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> denotes the <a href="Gamma_function" title="Gamma function">gamma function</a>.

The first identity can be found by writing the <a href="Gaussian_integral" title="Gaussian integral">Gaussian integral</a> in <a href="Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>.

The second identity can be derived by making the substitution u = W0(x), which gives

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=ue^{u},\\[5pt]{\frac {dx}{du}}&=(u+1)e^{u}.\end{aligned}}}</annotation>

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</math><img src="./6d9f83a1f16b84fd3408ce93ce89f1165e8c455e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:17.276ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}x&=ue^{u},\\[5pt]{\frac {dx}{du}}&=(u+1)e^{u}.\end{aligned}}}" loading="lazy"></dd></dl>

Thus

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {W_{0}(x)}{x{\sqrt {x}}}}\,dx&=\int _{0}^{\infty }{\frac {u}{ue^{u}{\sqrt {ue^{u}}}}}(u+1)e^{u}\,du\\[5pt]&=\int _{0}^{\infty }{\frac {u+1}{\sqrt {ue^{u}}}}du\\[5pt]&=\int _{0}^{\infty }{\frac {u+1}{\sqrt {u}}}{\frac {1}{\sqrt {e^{u}}}}du\\[5pt]&=\int _{0}^{\infty }u^{\tfrac {1}{2}}e^{-{\frac {u}{2}}}du+\int _{0}^{\infty }u^{-{\tfrac {1}{2}}}e^{-{\frac {u}{2}}}du\\[5pt]&=2\int _{0}^{\infty }(2w)^{\tfrac {1}{2}}e^{-w}\,dw+2\int _{0}^{\infty }(2w)^{-{\tfrac {1}{2}}}e^{-w}\,dw&&\quad (u=2w)\\[5pt]&=2{\sqrt {2}}\int _{0}^{\infty }w^{\tfrac {1}{2}}e^{-w}\,dw+{\sqrt {2}}\int _{0}^{\infty }w^{-{\tfrac {1}{2}}}e^{-w}\,dw\\[5pt]&=2{\sqrt {2}}\cdot \Gamma \left({\tfrac {3}{2}}\right)+{\sqrt {2}}\cdot \Gamma \left({\tfrac {1}{2}}\right)\\[5pt]&=2{\sqrt {2}}\left({\tfrac {1}{2}}{\sqrt {\pi }}\right)+{\sqrt {2}}\left({\sqrt {\pi }}\right)\\[5pt]&=2{\sqrt {2\pi }}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./f836d0b33ed568739bf7e32a1ded6e95b08c4429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -29.005ex; width:77.79ex; height:59.176ex;" alt="{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {W_{0}(x)}{x{\sqrt {x}}}}\,dx&=\int _{0}^{\infty }{\frac {u}{ue^{u}{\sqrt {ue^{u}}}}}(u+1)e^{u}\,du\\[5pt]&=\int _{0}^{\infty }{\frac {u+1}{\sqrt {ue^{u}}}}du\\[5pt]&=\int _{0}^{\infty }{\frac {u+1}{\sqrt {u}}}{\frac {1}{\sqrt {e^{u}}}}du\\[5pt]&=\int _{0}^{\infty }u^{\tfrac {1}{2}}e^{-{\frac {u}{2}}}du+\int _{0}^{\infty }u^{-{\tfrac {1}{2}}}e^{-{\frac {u}{2}}}du\\[5pt]&=2\int _{0}^{\infty }(2w)^{\tfrac {1}{2}}e^{-w}\,dw+2\int _{0}^{\infty }(2w)^{-{\tfrac {1}{2}}}e^{-w}\,dw&&\quad (u=2w)\\[5pt]&=2{\sqrt {2}}\int _{0}^{\infty }w^{\tfrac {1}{2}}e^{-w}\,dw+{\sqrt {2}}\int _{0}^{\infty }w^{-{\tfrac {1}{2}}}e^{-w}\,dw\\[5pt]&=2{\sqrt {2}}\cdot \Gamma \left({\tfrac {3}{2}}\right)+{\sqrt {2}}\cdot \Gamma \left({\tfrac {1}{2}}\right)\\[5pt]&=2{\sqrt {2}}\left({\tfrac {1}{2}}{\sqrt {\pi }}\right)+{\sqrt {2}}\left({\sqrt {\pi }}\right)\\[5pt]&=2{\sqrt {2\pi }}.\end{aligned}}}" loading="lazy"></dd></dl>

The third identity may be derived from the second by making the substitution u = x−2 and the first can also be derived from the third by the substitution z = ⁠1/√2⁠ tan x. Deriving its generalization, the fourth identity, is only slightly more involved and can be done by substituting, in turn, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=x^{\frac {1}{N}}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle u=x^{\frac {1}{N}}}</annotation>

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</math><img src="./2948a90cfd9122bc3d2957eb60a6211d0c5e23fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.011ex; height:3.509ex;" alt="{\displaystyle u=x^{\frac {1}{N}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=W_{0}(u)}">

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<annotation encoding="application/x-tex">{\displaystyle t=W_{0}(u)}</annotation>

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</math><img src="./ff4afaad1e48047eddb9470202baef9c9c1f1ffb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.325ex; height:2.843ex;" alt="{\displaystyle t=W_{0}(u)}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z={\frac {t}{N}}}">

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<annotation encoding="application/x-tex">{\displaystyle z={\frac {t}{N}}}</annotation>

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</math><img src="./128105b6a03f0bfd2eaeed5e07f21b8597de8d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.086ex; height:5.176ex;" alt="{\displaystyle z={\frac {t}{N}}}" loading="lazy">, observing that one obtains two integrals matching the definition of the gamma function, and finally using the properties of the gamma function to collect terms and simplify.

Except for z along the branch cut (−∞, −⁠1/e⁠] (where the integral does not converge), the principal branch of the Lambert W function can be computed by the following integral:<a href="#cite_note-22">[22]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W_{0}(z)&={\frac {z}{2\pi }}\int _{-\pi }^{\pi }{\frac {\left(1-\nu \cot \nu \right)^{2}+\nu ^{2}}{z+\nu \csc \left(\nu \right)e^{-\nu \cot \nu }}}\,d\nu \\[5pt]&={\frac {z}{\pi }}\int _{0}^{\pi }{\frac {\left(1-\nu \cot \nu \right)^{2}+\nu ^{2}}{z+\nu \csc \left(\nu \right)e^{-\nu \cot \nu }}}\,d\nu ,\end{aligned}}}</annotation>

</semantics>

</math><img src="./7260cf04ec55826cd7b9c3e6efab9908b6dedd79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:40.389ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}W_{0}(z)&={\frac {z}{2\pi }}\int _{-\pi }^{\pi }{\frac {\left(1-\nu \cot \nu \right)^{2}+\nu ^{2}}{z+\nu \csc \left(\nu \right)e^{-\nu \cot \nu }}}\,d\nu \\[5pt]&={\frac {z}{\pi }}\int _{0}^{\pi }{\frac {\left(1-\nu \cot \nu \right)^{2}+\nu ^{2}}{z+\nu \csc \left(\nu \right)e^{-\nu \cot \nu }}}\,d\nu ,\end{aligned}}}" loading="lazy"></dd></dl>

where the two integral expressions are equivalent due to the symmetry of the integrand.

<div class="mw-heading mw-heading3"><h3 id="Indefinite_integrals">Indefinite integrals</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {W(x)^{2}}{2}}+W(x)+C}</annotation>

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</math>

/* start https://en.wikipedia.org/ */

.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}

/* end https://en.wikipedia.org/ */

</style><div class="math_proof" style="">1st proof

Introduce substitution variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=W(x)\rightarrow ue^{u}=x\;\;\;\;{\frac {d}{du}}ue^{u}=(u+1)e^{u}}">

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<annotation encoding="application/x-tex">{\displaystyle u=W(x)\rightarrow ue^{u}=x\;\;\;\;{\frac {d}{du}}ue^{u}=(u+1)e^{u}}</annotation>

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</math><img src="./aa92a9b10e20c8752c319d4fb57380b036e75a12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.675ex; height:5.509ex;" alt="{\displaystyle u=W(x)\rightarrow ue^{u}=x\;\;\;\;{\frac {d}{du}}ue^{u}=(u+1)e^{u}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int {\frac {u}{ue^{u}}}(u+1)e^{u}\,du}</annotation>

</semantics>

</math><img src="./ccd13eeb1c47e3f71e06e7fbcd481d0f61dff63b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.646ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int {\frac {u}{ue^{u}}}(u+1)e^{u}\,du}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int {\frac {\cancel {\color {OliveGreen}{u}}}{{\cancel {\color {OliveGreen}{u}}}{\cancel {\color {BrickRed}{e^{u}}}}}}\left(u+1\right){\cancel {\color {BrickRed}{e^{u}}}}\,du}</annotation>

</semantics>

</math><img src="./02bd93d455d260f6c61b9c55c5d37909138e3189.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.252ex; height:6.843ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int {\frac {\cancel {\color {OliveGreen}{u}}}{{\cancel {\color {OliveGreen}{u}}}{\cancel {\color {BrickRed}{e^{u}}}}}}\left(u+1\right){\cancel {\color {BrickRed}{e^{u}}}}\,du}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int (u+1)\,du}</annotation>

</semantics>

</math><img src="./ffe2f16ef671c890f019b53667f58e7128f9dfc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.581ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int (u+1)\,du}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {u^{2}}{2}}+u+C}</annotation>

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</math><img src="./d6d6377840b9930afa25a5a0dd26c48499cc12a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.309ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {u^{2}}{2}}+u+C}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle u=W(x)}</annotation>

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</math><img src="./53e78c5bb180833f480d0927415773b9bde3d6bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.002ex; height:2.843ex;" alt="{\displaystyle u=W(x)}" loading="lazy"></dd></dl></dd>

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {W(x)^{2}}{2}}+W(x)+C}</annotation>

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</math><img src="./ea93fa470db6742e23dc723c5fd86bdb87240415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.798ex; height:6.343ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {W(x)^{2}}{2}}+W(x)+C}" loading="lazy"></dd></dl>

</div>

<div class="math_proof" style="">2nd proof

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<annotation encoding="application/x-tex">{\displaystyle W(x)e^{W(x)}=x\rightarrow {\frac {W(x)}{x}}=e^{-W(x)}}</annotation>

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</math><img src="./b4ab107f6c240468905ecbbd0a71f615b99bc0f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.918ex; height:5.676ex;" alt="{\displaystyle W(x)e^{W(x)}=x\rightarrow {\frac {W(x)}{x}}=e^{-W(x)}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int e^{-W(x)}\,dx}</annotation>

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</math><img src="./cbf710c1bf1605c420003d17122dcb65a0496f46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.362ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int e^{-W(x)}\,dx}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle u=W(x)\rightarrow ue^{u}=x\;\;\;\;{\frac {d}{\,du}}ue^{u}=\left(u+1\right)e^{u}}</annotation>

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</math><img src="./3f7eceb4f312fa18314ca915b60a5d447de888cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.449ex; height:5.509ex;" alt="{\displaystyle u=W(x)\rightarrow ue^{u}=x\;\;\;\;{\frac {d}{\,du}}ue^{u}=\left(u+1\right)e^{u}}" loading="lazy"></dd></dl></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int e^{-u}(u+1)e^{u}\,du}</annotation>

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</math><img src="./67089fe4e73b3192edae4e420ba559acfb1afd5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.758ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int e^{-u}(u+1)e^{u}\,du}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int {\cancel {\color {OliveGreen}{e^{-u}}}}\left(u+1\right){\cancel {\color {OliveGreen}{e^{u}}}}\,du}</annotation>

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</math><img src="./db78de27edde05f75eba5013b821f7a990a1b3f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.087ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int {\cancel {\color {OliveGreen}{e^{-u}}}}\left(u+1\right){\cancel {\color {OliveGreen}{e^{u}}}}\,du}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int (u+1)\,du}</annotation>

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</math><img src="./ffe2f16ef671c890f019b53667f58e7128f9dfc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.581ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;\int (u+1)\,du}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {u^{2}}{2}}+u+C}</annotation>

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</math><img src="./d6d6377840b9930afa25a5a0dd26c48499cc12a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.309ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {u^{2}}{2}}+u+C}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle u=W(x)}</annotation>

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</math><img src="./53e78c5bb180833f480d0927415773b9bde3d6bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.002ex; height:2.843ex;" alt="{\displaystyle u=W(x)}" loading="lazy"></dd></dl></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {W(x)^{2}}{2}}+W(x)+C}</annotation>

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</math><img src="./ea93fa470db6742e23dc723c5fd86bdb87240415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.798ex; height:6.343ex;" alt="{\displaystyle \int {\frac {W(x)}{x}}\,dx\;=\;{\frac {W(x)^{2}}{2}}+W(x)+C}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Ae^{Bx}\right)^{2}}{2B}}+{\frac {W\left(Ae^{Bx}\right)}{B}}+C}</annotation>

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</math>

<div class="math_proof" style="">Proof

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<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;\int W\left(Ae^{Bx}\right)\,dx}</annotation>

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</math><img src="./cb813552bfcd2057b4bea625469f376925c5112d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.587ex; height:5.676ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;\int W\left(Ae^{Bx}\right)\,dx}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle u=Bx\rightarrow {\frac {u}{B}}=x\;\;\;\;{\frac {d}{du}}{\frac {u}{B}}={\frac {1}{B}}}</annotation>

</semantics>

</math><img src="./49a42bb657063ed1be4ac6d186fbd084b5eb0683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.425ex; height:5.509ex;" alt="{\displaystyle u=Bx\rightarrow {\frac {u}{B}}=x\;\;\;\;{\frac {d}{du}}{\frac {u}{B}}={\frac {1}{B}}}" loading="lazy"></dd></dl></dd></dl>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;\int W\left(Ae^{u}\right){\frac {1}{B}}du}</annotation>

</semantics>

</math><img src="./47604610291d4adecf81363512b224aa6446c62f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.232ex; height:5.676ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;\int W\left(Ae^{u}\right){\frac {1}{B}}du}" loading="lazy">

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

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<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v=e^{u}\rightarrow \ln \left(v\right)=u\;\;\;\;{\frac {d}{dv}}\ln \left(v\right)={\frac {1}{v}}}</annotation>

</semantics>

</math><img src="./7bbc5def732e32e8084a23f4957ab7d06f288d3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.521ex; height:5.509ex;" alt="{\displaystyle v=e^{u}\rightarrow \ln \left(v\right)=u\;\;\;\;{\frac {d}{dv}}\ln \left(v\right)={\frac {1}{v}}}" loading="lazy"></dd></dl></dd></dl>

<mrow>

<mrow>

<msup>

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</msup>

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</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {W\left(Av\right)}{v}}dv}</annotation>

</semantics>

</math><img src="./b807c25a7ddf633223cf8cfe3853384869b86a42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.738ex; height:6.176ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {W\left(Av\right)}{v}}dv}" loading="lazy">

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=Av\rightarrow {\frac {w}{A}}=v\;\;\;\;{\frac {d}{dw}}{\frac {w}{A}}={\frac {1}{A}}}</annotation>

</semantics>

</math><img src="./a527e297453f71ad413cee01e30db36d1ac11f34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.606ex; height:5.509ex;" alt="{\displaystyle w=Av\rightarrow {\frac {w}{A}}=v\;\;\;\;{\frac {d}{dw}}{\frac {w}{A}}={\frac {1}{A}}}" loading="lazy"></dd></dl></dd></dl>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mstyle>

</menclose>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</menclose>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {{\cancel {\color {OliveGreen}{A}}}W(w)}{w}}{\cancel {\color {OliveGreen}{\frac {1}{A}}}}dw}</annotation>

</semantics>

</math><img src="./af6cfb97be13570ae6c2457df9b44bfbc85cbe1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.558ex; height:7.009ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {{\cancel {\color {OliveGreen}{A}}}W(w)}{w}}{\cancel {\color {OliveGreen}{\frac {1}{A}}}}dw}" loading="lazy">

<mrow>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=W\left(w\right)\rightarrow te^{t}=w\;\;\;\;{\frac {d}{dt}}te^{t}=\left(t+1\right)e^{t}}</annotation>

</semantics>

</math><img src="./5c1f4a30a87bb4def0bbe5c6d4988699d5cbab7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.628ex; height:5.509ex;" alt="{\displaystyle t=W\left(w\right)\rightarrow te^{t}=w\;\;\;\;{\frac {d}{dt}}te^{t}=\left(t+1\right)e^{t}}" loading="lazy"></dd></dl></dd></dl>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {t}{te^{t}}}\left(t+1\right)e^{t}dt}</annotation>

</semantics>

</math><img src="./2f8c371fe17fe1673fd134c9165077d5992f9b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.032ex; height:5.676ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {t}{te^{t}}}\left(t+1\right)e^{t}dt}" loading="lazy">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mrow>

</mstyle>

</menclose>

<mrow>

</mrow>

</mstyle>

</menclose>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</menclose>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</menclose>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {\cancel {\color {OliveGreen}{t}}}{{\cancel {\color {OliveGreen}{t}}}{\cancel {\color {BrickRed}{e^{t}}}}}}\left(t+1\right){\cancel {\color {BrickRed}{e^{t}}}}dt}</annotation>

</semantics>

</math><img src="./97b5c4444db84ae153885f7ddcd3f832fc6c006e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.864ex; height:7.509ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {\cancel {\color {OliveGreen}{t}}}{{\cancel {\color {OliveGreen}{t}}}{\cancel {\color {BrickRed}{e^{t}}}}}}\left(t+1\right){\cancel {\color {BrickRed}{e^{t}}}}dt}" loading="lazy">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int (t+1)dt}</annotation>

</semantics>

</math><img src="./d28f457d98a95995742d5b29b4923e9168dc8808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.376ex; height:5.676ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int (t+1)dt}" loading="lazy">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {t^{2}}{2B}}+{\frac {t}{B}}+C}</annotation>

</semantics>

</math><img src="./936ab5e5e370031a971c9b1743318ef16ee1ae7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.298ex; height:6.176ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {t^{2}}{2B}}+{\frac {t}{B}}+C}" loading="lazy">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=W\left(w\right)}</annotation>

</semantics>

</math><img src="./a439235f10c82655944090697a984e554f1afee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.234ex; height:2.843ex;" alt="{\displaystyle t=W\left(w\right)}" loading="lazy"></dd></dl></dd></dl>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

<mrow>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(w\right)^{2}}{2B}}+{\frac {W\left(w\right)}{B}}+C}</annotation>

</semantics>

</math><img src="./92196e2ee1cc8932237594e75bff94287db4bb66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.866ex; height:6.509ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(w\right)^{2}}{2B}}+{\frac {W\left(w\right)}{B}}+C}" loading="lazy">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=Av}</annotation>

</semantics>

</math><img src="./59cf2756e56786a772de179bf9bdbf0dbc55d08f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.633ex; height:2.176ex;" alt="{\displaystyle w=Av}" loading="lazy"></dd></dl></dd></dl>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Av\right)^{2}}{2B}}+{\frac {W\left(Av\right)}{B}}+C}</annotation>

</semantics>

</math><img src="./3ee6427e1c8a4fe293f3635d9c0ea8f8cbc17b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.279ex; height:6.509ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Av\right)^{2}}{2B}}+{\frac {W\left(Av\right)}{B}}+C}" loading="lazy">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v=e^{u}}</annotation>

</semantics>

</math><img src="./e52b1f2941829f56f496dd60ee4c7011ee6b0769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.482ex; height:2.343ex;" alt="{\displaystyle v=e^{u}}" loading="lazy"></dd></dl></dd></dl>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Ae^{u}\right)^{2}}{2B}}+{\frac {W\left(Ae^{u}\right)}{B}}+C}</annotation>

</semantics>

</math><img src="./7d0664b14654bb10d086928a5ee168ee3b41c3d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.536ex; height:6.509ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Ae^{u}\right)^{2}}{2B}}+{\frac {W\left(Ae^{u}\right)}{B}}+C}" loading="lazy">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=Bx}</annotation>

</semantics>

</math><img src="./25c2067259df8d67a013ce43721d1188db87c2d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.522ex; height:2.176ex;" alt="{\displaystyle u=Bx}" loading="lazy"></dd></dl></dd></dl>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Ae^{Bx}\right)^{2}}{2B}}+{\frac {W\left(Ae^{Bx}\right)}{B}}+C}</annotation>

</semantics>

</math><img src="./7730c4fb69d7ba2d35e05cc3e796ec5cc2846689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:50.672ex; height:7.009ex;" alt="{\displaystyle \int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Ae^{Bx}\right)^{2}}{2B}}+{\frac {W\left(Ae^{Bx}\right)}{B}}+C}" loading="lazy">

</div>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x^{2}}}\,dx\;=\;\operatorname {Ei} \left(-W(x)\right)-e^{-W(x)}+C}</annotation>

</semantics>

</math>

<div class="math_proof" style="">Proof

Introduce substitution variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=W(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=W(x)}</annotation>

</semantics>

</math><img src="./53e78c5bb180833f480d0927415773b9bde3d6bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.002ex; height:2.843ex;" alt="{\displaystyle u=W(x)}" loading="lazy">, which gives us <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ue^{u}=x}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ue^{u}=x}</annotation>

</semantics>

</math><img src="./e6ff321e86e10b8153e11a9d11321e617e7df97e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.014ex; height:2.343ex;" alt="{\displaystyle ue^{u}=x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{du}}ue^{u}=\left(u+1\right)e^{u}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{du}}ue^{u}=\left(u+1\right)e^{u}}</annotation>

</semantics>

</math><img src="./9f6e1cd84bc3b07f3b14f0300a61dfb94f572cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.851ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{du}}ue^{u}=\left(u+1\right)e^{u}}" loading="lazy">

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int {\frac {W(x)}{x^{2}}}\,dx\;&=\;\int {\frac {u}{\left(ue^{u}\right)^{2}}}\left(u+1\right)e^{u}du\\&=\;\int {\frac {u+1}{ue^{u}}}du\\&=\;\int {\frac {u}{ue^{u}}}du\;+\;\int {\frac {1}{ue^{u}}}du\\&=\;\int e^{-u}du\;+\;\int {\frac {e^{-u}}{u}}du\end{aligned}}}</annotation>

</semantics>

</math><img src="./2bd8aec9e33a7a10b7310b60c44da16a735cc7c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:40.291ex; height:24.676ex;" alt="{\displaystyle {\begin{aligned}\int {\frac {W(x)}{x^{2}}}\,dx\;&=\;\int {\frac {u}{\left(ue^{u}\right)^{2}}}\left(u+1\right)e^{u}du\\&=\;\int {\frac {u+1}{ue^{u}}}du\\&=\;\int {\frac {u}{ue^{u}}}du\;+\;\int {\frac {1}{ue^{u}}}du\\&=\;\int e^{-u}du\;+\;\int {\frac {e^{-u}}{u}}du\end{aligned}}}" loading="lazy">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v=-u\rightarrow -v=u\;\;\;\;{\frac {d}{dv}}-v=-1}</annotation>

</semantics>

</math><img src="./17a59eae11462470fbfd04fa00ca03dc35b4828e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.139ex; height:5.509ex;" alt="{\displaystyle v=-u\rightarrow -v=u\;\;\;\;{\frac {d}{dv}}-v=-1}" loading="lazy"></dd></dl></dd></dl>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x^{2}}}\,dx\;=\;\int e^{v}\left(-1\right)dv\;+\;\int {\frac {e^{-u}}{u}}du}</annotation>

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</math><img src="./f68ac356743e70b34bcaa6362367b3d8ba94bfda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.532ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x^{2}}}\,dx\;=\;\int e^{v}\left(-1\right)dv\;+\;\int {\frac {e^{-u}}{u}}du}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x^{2}}}\,dx\;=\;-e^{v}+\operatorname {Ei} \left(-u\right)+C}</annotation>

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</math><img src="./de75f2cc40113dc06b3a7f2ff79daf90d3979dc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.858ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x^{2}}}\,dx\;=\;-e^{v}+\operatorname {Ei} \left(-u\right)+C}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle v=-u}</annotation>

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</math><img src="./9381f0fe5d189aa22e3113a8b62a3b4a43ea7c14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.364ex; height:2.176ex;" alt="{\displaystyle v=-u}" loading="lazy"></dd></dl></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \int {\frac {W(x)}{x^{2}}}\,dx\;=\;-e^{-u}+\operatorname {Ei} \left(-u\right)+C}</annotation>

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</math><img src="./ed4745880f092c37d93e64efd54186b15c55ccf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.279ex; height:6.176ex;" alt="{\displaystyle \int {\frac {W(x)}{x^{2}}}\,dx\;=\;-e^{-u}+\operatorname {Ei} \left(-u\right)+C}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle u=W(x)}</annotation>

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</math><img src="./53e78c5bb180833f480d0927415773b9bde3d6bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.002ex; height:2.843ex;" alt="{\displaystyle u=W(x)}" loading="lazy"></dd></dl></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int {\frac {W(x)}{x^{2}}}\,dx\;&=\;-e^{-W(x)}+\operatorname {Ei} \left(-W(x)\right)+C\\&=\;\operatorname {Ei} \left(-W(x)\right)-e^{-W(x)}+C\end{aligned}}}</annotation>

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</math><img src="./e1d30e951baf69fb8ae5af6e6ec03b2dc12dd8a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:44.276ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\int {\frac {W(x)}{x^{2}}}\,dx\;&=\;-e^{-W(x)}+\operatorname {Ei} \left(-W(x)\right)+C\\&=\;\operatorname {Ei} \left(-W(x)\right)-e^{-W(x)}+C\end{aligned}}}" loading="lazy">

</div>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Solving_equations">Solving equations</h3></div>

The Lambert W function is used to solve equations in which the unknown quantity occurs both in the base and in the exponent, or both inside and outside of a logarithm. The strategy is to convert such an equation into one of the form zez = w and then to solve for z using the W function.

For example, the equation

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<annotation encoding="application/x-tex">{\displaystyle 3^{x}=2x+2}</annotation>

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</math><img src="./df482d9d65ba3aef450d98aaddb3d0c3d07edf69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.928ex; height:2.509ex;" alt="{\displaystyle 3^{x}=2x+2}" loading="lazy"></dd></dl>

(where x is an unknown <a href="Real_number" title="Real number">real number</a>) can be solved by rewriting it as

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<mtext>multiply by </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&(x+1)\ 3^{-x}={\frac {1}{2}}&({\mbox{multiply by }}3^{-x}/2)\\\Leftrightarrow \ &(-x-1)\ 3^{-x-1}=-{\frac {1}{6}}&({\mbox{multiply by }}{-}1/3)\\\Leftrightarrow \ &(\ln 3)(-x-1)\ e^{(\ln 3)(-x-1)}=-{\frac {\ln 3}{6}}&({\mbox{multiply by }}\ln 3)\end{aligned}}}</annotation>

</semantics>

</math><img src="./bc682971247cc489f0442702ed4c8555dfcaec8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.276ex; margin-bottom: -0.228ex; width:63.803ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}&(x+1)\ 3^{-x}={\frac {1}{2}}&({\mbox{multiply by }}3^{-x}/2)\\\Leftrightarrow \ &(-x-1)\ 3^{-x-1}=-{\frac {1}{6}}&({\mbox{multiply by }}{-}1/3)\\\Leftrightarrow \ &(\ln 3)(-x-1)\ e^{(\ln 3)(-x-1)}=-{\frac {\ln 3}{6}}&({\mbox{multiply by }}\ln 3)\end{aligned}}}" loading="lazy"></dd></dl>

This last equation has the desired form and the solutions for real x are:

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<annotation encoding="application/x-tex">{\displaystyle (\ln 3)(-x-1)=W_{0}\left({\frac {-\ln 3}{6}}\right)\ \ \ {\textrm {or}}\ \ \ (\ln 3)(-x-1)=W_{-1}\left({\frac {-\ln 3}{6}}\right)}</annotation>

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</math><img src="./aedb3eaee558a8188b33fa8bae6773519f7d3a20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:69.07ex; height:6.176ex;" alt="{\displaystyle (\ln 3)(-x-1)=W_{0}\left({\frac {-\ln 3}{6}}\right)\ \ \ {\textrm {or}}\ \ \ (\ln 3)(-x-1)=W_{-1}\left({\frac {-\ln 3}{6}}\right)}" loading="lazy"></dd></dl>

and thus:

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<annotation encoding="application/x-tex">{\displaystyle x=-1-{\frac {W_{0}\left(-{\frac {\ln 3}{6}}\right)}{\ln 3}}=-0.79011\ldots \ \ {\textrm {or}}\ \ x=-1-{\frac {W_{-1}\left(-{\frac {\ln 3}{6}}\right)}{\ln 3}}=1.44456\ldots }</annotation>

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</math><img src="./065db9ed8e3638fc4fcc719069f1e6eaef8091ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:80.952ex; height:7.676ex;" alt="{\displaystyle x=-1-{\frac {W_{0}\left(-{\frac {\ln 3}{6}}\right)}{\ln 3}}=-0.79011\ldots \ \ {\textrm {or}}\ \ x=-1-{\frac {W_{-1}\left(-{\frac {\ln 3}{6}}\right)}{\ln 3}}=1.44456\ldots }" loading="lazy"></dd></dl>

Generally, the solution to

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<annotation encoding="application/x-tex">{\displaystyle x=a+b\,e^{cx}}</annotation>

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</math><img src="./8122b0b4741982d55f7391bc08dcf14bc8037c4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.851ex; height:2.509ex;" alt="{\displaystyle x=a+b\,e^{cx}}" loading="lazy"></dd></dl>

is:

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<annotation encoding="application/x-tex">{\displaystyle x=a-{\frac {1}{c}}W(-bc\,e^{ac})}</annotation>

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</math><img src="./daf94774467d8ddbfc8af3f6fdafa3e50a47c404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.838ex; height:5.176ex;" alt="{\displaystyle x=a-{\frac {1}{c}}W(-bc\,e^{ac})}" loading="lazy"></dd></dl>

where a, b, and c are complex constants, with b and c not equal to zero, and the W function is of any integer order.

<div class="mw-heading mw-heading3"><h3 id="Inviscid_flows">Inviscid flows</h3></div>

Applying the unusual accelerating <a href="Traveling_wave" class="mw-redirect" title="Traveling wave">traveling-wave</a> <a href="Ansatz" title="Ansatz">Ansatz</a> in the form of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\eta )=\rho {\big (}x-{\frac {at^{2}}{2}}{\big )}}">

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<annotation encoding="application/x-tex">{\displaystyle \rho (\eta )=\rho {\big (}x-{\frac {at^{2}}{2}}{\big )}}</annotation>

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</math><img src="./4bc75701bf0ccd5c7db117a34caf292edb698c25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.741ex; height:5.676ex;" alt="{\displaystyle \rho (\eta )=\rho {\big (}x-{\frac {at^{2}}{2}}{\big )}}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

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</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy">, a, x and t are the density, the reduced variable, the acceleration, the spatial and the temporal variables) the fluid <a href="Density" title="Density">density</a> of the corresponding <a href="Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler equation</a> can be given with the help of the W function.<a href="#cite_note-23">[23]</a>

<div class="mw-heading mw-heading3"><h3 id="Viscous_flows">Viscous flows</h3></div>

Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in laboratory experiments can be described by using the Lambert–Euler omega function as follows:

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<annotation encoding="application/x-tex">{\displaystyle H(x)=1+W\left((H(0)-1)e^{(H(0)-1)-{\frac {x}{L}}}\right),}</annotation>

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</math><img src="./4f6f227705ad57c53a9eda6c56db926d256ea5fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.063ex; height:4.843ex;" alt="{\displaystyle H(x)=1+W\left((H(0)-1)e^{(H(0)-1)-{\frac {x}{L}}}\right),}" loading="lazy"></dd></dl>

where H(x) is the debris flow height, x is the channel downstream position, L is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.

In <a href="Pipe_flow" title="Pipe flow">pipe flow</a>, the Lambert W function is part of the explicit formulation of the <a href="Colebrook_equation" class="mw-redirect" title="Colebrook equation">Colebrook equation</a> for finding the <a href="Darcy_friction_factor" class="mw-redirect" title="Darcy friction factor">Darcy friction factor</a>. This factor is used to determine the pressure drop through a straight run of pipe when the flow is <a href="Turbulent" class="mw-redirect" title="Turbulent">turbulent</a>.<a href="#cite_note-AAMore-24">[24]</a>

<div class="mw-heading mw-heading3"><h3 id="Time-dependent_flow_in_simple_branch_hydraulic_systems">Time-dependent flow in simple branch hydraulic systems</h3></div>

The principal branch of the Lambert W function is employed in the field of <a href="Mechanical_engineering" title="Mechanical engineering">mechanical engineering</a>, in the study of time dependent transfer of <a href="Newtonian_fluid" title="Newtonian fluid">Newtonian fluids</a> between two reservoirs with varying free surface levels, using centrifugal pumps.<a href="#cite_note-25">[25]</a> The Lambert W function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Q_{\text{turb}}&={\frac {Q_{i}}{\zeta _{i}}}W_{0}\left[\zeta _{i}\,e^{(\zeta _{i}+\beta t/b)}\right]\\Q_{\text{lam}}&={\frac {Q_{i}}{\xi _{i}}}W_{0}\left[\xi _{i}\,e^{\left(\xi _{i}+\beta t/(b-\Gamma _{1})\right)}\right]\end{aligned}}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{i}}</annotation>

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</math><img src="./b9f7193081d440425e522698e80817b5d558df03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}" loading="lazy"> is the initial flow rate and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is time.

<div class="mw-heading mw-heading3"><h3 id="Neuroimaging">Neuroimaging</h3></div>

The Lambert W function is employed in the field of neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain <a href="Voxel" title="Voxel">voxel</a>, to the corresponding blood oxygenation level dependent (BOLD) signal.<a href="#cite_note-26">[26]</a>

<div class="mw-heading mw-heading3"><h3 id="Chemical_engineering">Chemical engineering</h3></div>

The Lambert W function is employed in the field of chemical engineering for modeling the porous electrode film thickness in a <a href="Glassy_carbon" title="Glassy carbon">glassy carbon</a> based <a href="Supercapacitor" title="Supercapacitor">supercapacitor</a> for electrochemical energy storage. The Lambert W function provides an exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a>

<div class="mw-heading mw-heading3"><h3 id="Crystal_growth">Crystal growth</h3></div>

In the <a href="Crystal_growth" title="Crystal growth">crystal growth</a>, the negative principal of the Lambert W-function can be used to calculate the distribution coefficient, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy">, and solute concentration in the melt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{L}}">

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</math><img src="./05f02d9f85b006129ba87c429e1d3df814531c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\textstyle C_{L}}" loading="lazy">,<a href="#cite_note-29">[29]</a><a href="#cite_note-30">[30]</a> from the <a href="Scheil_equation" title="Scheil equation">Scheil equation</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&k={\frac {W_{0}(Z)}{\ln(1-fs)}}\\&C_{L}={\frac {C_{0}}{(1-fs)}}e^{W_{0}(Z)}\\&Z={\frac {C_{S}}{C_{0}}}(1-fs)\ln(1-fs)\end{aligned}}}</annotation>

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</math><img src="./7ae7eb9d2602b94603a58870898f23b60f29fcfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:28.009ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}&k={\frac {W_{0}(Z)}{\ln(1-fs)}}\\&C_{L}={\frac {C_{0}}{(1-fs)}}e^{W_{0}(Z)}\\&Z={\frac {C_{S}}{C_{0}}}(1-fs)\ln(1-fs)\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Materials_science">Materials science</h3></div>

The Lambert W function is employed in the field of <a href="Epitaxy" title="Epitaxy">epitaxial film growth</a> for the determination of the critical <a href="Dislocation" title="Dislocation">dislocation</a> onset film thickness. This is the calculated thickness of an epitaxial film, where due to thermodynamic principles the film will develop crystallographic dislocations in order to minimise the <a href="Elastic_energy" title="Elastic energy">elastic energy</a> stored in the films. Prior to application of Lambert W for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert W turns it in an explicit equation for analytical handling with ease.<a href="#cite_note-31">[31]</a>

<div class="mw-heading mw-heading3"><h3 id="Semiconductor">Semiconductor</h3></div>

It was shown that a W-function describes the relation between voltage, current and resistance in a diode.<a href="#cite_note-32">[32]</a>

<div class="mw-heading mw-heading3"><h3 id="Porous_media">Porous media</h3></div>

The Lambert W function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneous tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the −1 branch applies if the displacement is unstable with the heavier fluid running underneath the lighter fluid.<a href="#cite_note-33">[33]</a>

<div class="mw-heading mw-heading3"><h3 id="Bernoulli_numbers_and_Todd_genus">Bernoulli numbers and Todd genus</h3></div>

The equation (linked with the generating functions of <a href="Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a> and <a href="Genus_of_a_multiplicative_sequence" title="Genus of a multiplicative sequence">Todd genus</a>):

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<annotation encoding="application/x-tex">{\displaystyle Y={\frac {X}{1-e^{X}}}}</annotation>

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</math><img src="./5a9680f5df7637e52b63efdcd136425ff35997d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.427ex; height:5.676ex;" alt="{\displaystyle Y={\frac {X}{1-e^{X}}}}" loading="lazy"></dd></dl>

can be solved by means of the two real branches W0 and W−1:

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<annotation encoding="application/x-tex">{\displaystyle X(Y)={\begin{cases}W_{-1}\left(Ye^{Y}\right)-W_{0}\left(Ye^{Y}\right)=Y-W_{0}\left(Ye^{Y}\right)&{\text{for }}Y<-1,\\W_{0}\left(Ye^{Y}\right)-W_{-1}\left(Ye^{Y}\right)=Y-W_{-1}\left(Ye^{Y}\right)&{\text{for }}-1<Y<0.\end{cases}}}</annotation>

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</math><img src="./6c493f0d2de166d8399d9453f1dcca29531539ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:74.06ex; height:6.509ex;" alt="{\displaystyle X(Y)={\begin{cases}W_{-1}\left(Ye^{Y}\right)-W_{0}\left(Ye^{Y}\right)=Y-W_{0}\left(Ye^{Y}\right)&{\text{for }}Y<-1,\\W_{0}\left(Ye^{Y}\right)-W_{-1}\left(Ye^{Y}\right)=Y-W_{-1}\left(Ye^{Y}\right)&{\text{for }}-1<Y<0.\end{cases}}}" loading="lazy"></dd></dl>

This application shows that the branch difference of the W function can be employed in order to solve other transcendental equations.<a href="#cite_note-34">[34]</a>

<div class="mw-heading mw-heading3"><h3 id="Statistics">Statistics</h3></div>

The centroid of a set of histograms defined with respect to the symmetrized <a href="Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a> (also called the Jeffreys divergence <a href="#cite_note-35">[35]</a>) has a closed form using the Lambert W function.<a href="#cite_note-36">[36]</a>

<div class="mw-heading mw-heading3"><h3 id="Pooling_of_tests_for_infectious_diseases">Pooling of tests for infectious diseases</h3></div>

Solving for the optimal group size to pool tests so that at least one individual is infected involves the Lambert W function.<a href="#cite_note-37">[37]</a><a href="#cite_note-38">[38]</a><a href="#cite_note-39">[39]</a>

<div class="mw-heading mw-heading3"><h3 id="Exact_solutions_of_the_Schrödinger_equation">Exact solutions of the Schrödinger equation</h3></div>

The Lambert W function appears in a quantum-mechanical potential, which affords the fifth – next to those of the <a href="Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root potential – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {V_{0}}{1+W\left(e^{-{\frac {x}{\sigma }}}\right)}}.}</annotation>

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</math><img src="./01b0c41d5e19b96a3745e972ae938a146c4ee515.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.163ex; height:8.009ex;" alt="{\displaystyle V={\frac {V_{0}}{1+W\left(e^{-{\frac {x}{\sigma }}}\right)}}.}" loading="lazy"></dd></dl>

A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to<a href="#cite_note-40">[40]</a>

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<annotation encoding="application/x-tex">{\displaystyle z=W\left(e^{-{\frac {x}{\sigma }}}\right).}</annotation>

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</math><img src="./edafe8c6367d2c2af8a47bd68c588ccbe785905b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.012ex; height:4.843ex;" alt="{\displaystyle z=W\left(e^{-{\frac {x}{\sigma }}}\right).}" loading="lazy"></dd></dl>

The Lambert W function also appears in the exact solution for the bound state energy of the one dimensional Schrödinger equation with a <a href="Delta_potential#Double_delta_potential" title="Delta potential">Double Delta Potential</a>.

<div class="mw-heading mw-heading3"><h3 id="Exact_solution_of_QCD_coupling_constant">Exact solution of QCD coupling constant</h3></div>

In <a href="Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a>, the <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> of the <a href="Strong_interaction" title="Strong interaction">Strong interaction</a>, the <a href="Coupling_constant" title="Coupling constant">coupling constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{s}}}">

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</math><img src="./eeb7a4350c4570a405da68ebf10fab848b5b7801.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.368ex; height:2.009ex;" alt="{\displaystyle \alpha _{\text{s}}}" loading="lazy"> is computed perturbatively, the order n corresponding to <a href="Feynman_diagrams" class="mw-redirect" title="Feynman diagrams">Feynman diagrams</a> including n quantum loops.<a href="#cite_note-PPNG_review_2016-41">[41]</a> The first order, n = 1, solution is exact (at that order) and analytical. At higher orders, n > 1, there is no exact and analytical solution and one typically uses an <a href="Iterative_method" title="Iterative method">iterative method</a> to furnish an approximate solution. However, for second order, n = 2, the Lambert function provides an exact (if non-analytical) solution.<a href="#cite_note-PPNG_review_2016-41">[41]</a>

<div class="mw-heading mw-heading3"><h3 id="Exact_solutions_of_the_Einstein_vacuum_equations">Exact solutions of the Einstein vacuum equations</h3></div>

In the <a href="Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a> solution of the Einstein vacuum equations, the W function is needed to go from the <a href="Eddington%E2%80%93Finkelstein_coordinates" title="Eddington–Finkelstein coordinates">Eddington–Finkelstein coordinates</a> to the Schwarzschild coordinates. For this reason, it also appears in the construction of the <a href="Kruskal%E2%80%93Szekeres_coordinates" title="Kruskal–Szekeres coordinates">Kruskal–Szekeres coordinates</a>.

<div class="mw-heading mw-heading3"><h3 id="Resonances_of_the_delta-shell_potential">Resonances of the delta-shell potential</h3></div>

The s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert W function.<a href="#cite_note-42">[42]</a>

<div class="mw-heading mw-heading3"><h3 id="Thermodynamic_equilibrium">Thermodynamic equilibrium</h3></div>

If a reaction involves reactants and products having <a href="Heat_capacity" title="Heat capacity">heat capacities</a> that are constant with temperature then the equilibrium constant K obeys

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<annotation encoding="application/x-tex">{\displaystyle \ln K={\frac {a}{T}}+b+c\ln T}</annotation>

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for some constants a, b, and c. When c (equal to ⁠ΔCp/R⁠) is not zero the value or values of T can be found where K equals a given value as follows, where L can be used for ln T.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-a&=(b-\ln K)T+cT\ln T\\&=(b-\ln K)e^{L}+cLe^{L}\\[5pt]-{\frac {a}{c}}&=\left({\frac {b-\ln K}{c}}+L\right)e^{L}\\[5pt]-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}&=\left(L+{\frac {b-\ln K}{c}}\right)e^{L+{\frac {b-\ln K}{c}}}\\[5pt]L&=W\left(-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}\right)+{\frac {\ln K-b}{c}}\\[5pt]T&=\exp \left(W\left(-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}\right)+{\frac {\ln K-b}{c}}\right).\end{aligned}}}</annotation>

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</math><img src="./3eb5cee9fdd7908a30ded5a772ec49f985ebaa1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.005ex; width:49.621ex; height:35.176ex;" alt="{\displaystyle {\begin{aligned}-a&=(b-\ln K)T+cT\ln T\\&=(b-\ln K)e^{L}+cLe^{L}\\[5pt]-{\frac {a}{c}}&=\left({\frac {b-\ln K}{c}}+L\right)e^{L}\\[5pt]-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}&=\left(L+{\frac {b-\ln K}{c}}\right)e^{L+{\frac {b-\ln K}{c}}}\\[5pt]L&=W\left(-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}\right)+{\frac {\ln K-b}{c}}\\[5pt]T&=\exp \left(W\left(-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}\right)+{\frac {\ln K-b}{c}}\right).\end{aligned}}}" loading="lazy"></dd></dl>

If a and c have the same sign there will be either two solutions or none (or one if the argument of W is exactly −⁠1/e⁠). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.

<div class="mw-heading mw-heading3"><h3 id="Phase_separation_of_polymer_mixtures">Phase separation of polymer mixtures</h3></div>

In the calculation of the phase diagram of thermodynamically incompatible polymer mixtures according to the <a href="Edmond-Ogston_model" class="mw-redirect" title="Edmond-Ogston model">Edmond-Ogston model</a>, the solutions for binodal and tie-lines are formulated in terms of Lambert W functions.<a href="#cite_note-43">[43]</a>

<div class="mw-heading mw-heading3"><h3 id="Wien's_displacement_law_in_a_D-dimensional_universe">Wien's displacement law in a D-dimensional universe</h3></div>

Wien's displacement law is expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{\max }/T=\alpha =\mathrm {const} }">

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</math><img src="./2ee4f7abba896c7a03bad6d18ba534e05c4f6a20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.231ex; height:2.843ex;" alt="{\displaystyle \nu _{\max }/T=\alpha =\mathrm {const} }" loading="lazy">. With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=h\nu _{\max }/k_{\mathrm {B} }T}">

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</math><img src="./3e273e05ffdd03064b6edd06be27c7d0654bcbd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.613ex; height:2.843ex;" alt="{\displaystyle x=h\nu _{\max }/k_{\mathrm {B} }T}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\rho _{T}\left(x\right)/dx=0}">

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</math><img src="./a7cb722021f45ec48fb803689c2b627ac7333303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.689ex; height:2.843ex;" alt="{\displaystyle d\rho _{T}\left(x\right)/dx=0}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{T}}">

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</math><img src="./1cdc60487130ca7003c98dda06941c49faa1cd00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.396ex; height:4.676ex;" alt="{\displaystyle e^{-x}=1-{\frac {x}{D}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is the number of degrees of freedom for spatial translation. The solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=D+W\left(-De^{-D}\right)}">

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</math><img src="./5d251b924bb5eba7b9aa25c0ec12188cd10e4e02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.832ex; height:3.343ex;" alt="{\displaystyle x=D+W\left(-De^{-D}\right)}" loading="lazy"> shows that the spectral energy density is dependent on the dimensionality of the universe.<a href="#cite_note-44">[44]</a>

<div class="mw-heading mw-heading3"><h3 id="AdS/CFT_correspondence">AdS/CFT correspondence</h3></div>

The classical finite-size corrections to the dispersion relations of giant magnons, single spikes and GKP strings can be expressed in terms of the Lambert W function.<a href="#cite_note-45">[45]</a><a href="#cite_note-46">[46]</a>

<div class="mw-heading mw-heading3"><h3 id="Epidemiology">Epidemiology</h3></div>

In the t → ∞ limit of the <a href="Compartmental_models_in_epidemiology" class="mw-redirect" title="Compartmental models in epidemiology">SIR model</a>, the proportion of susceptible and recovered individuals has a solution in terms of the Lambert W function.<a href="#cite_note-47">[47]</a>

<div class="mw-heading mw-heading3"><h3 id="Determination_of_the_time_of_flight_of_a_projectile">Determination of the time of flight of a projectile</h3></div>

The total time of the journey of a projectile which experiences air resistance proportional to its velocity <a href="Projectile_motion#Time_of_flight_with_air_resistance" title="Projectile motion">can be determined</a> in exact form by using the Lambert W function.<a href="#cite_note-48">[48]</a>

<div class="mw-heading mw-heading3"><h3 id="Electromagnetic_surface_wave_propagation">Electromagnetic surface wave propagation</h3></div>

The transcendental equation that appears in the determination of the propagation wave number of an electromagnetic axially symmetric surface wave (a low-attenuation single TM01 mode) propagating in a cylindrical metallic wire gives rise to an equation like u ln u = v (where u and v clump together the geometrical and physical factors of the problem), which is solved by the Lambert W function. The first solution to this problem, due to Sommerfeld circa 1898, already contained an iterative method to determine the value of the Lambert W function.<a href="#cite_note-49">[49]</a>

<div class="mw-heading mw-heading3"><h3 id="Orthogonal_trajectories_of_real_ellipses">Orthogonal trajectories of real ellipses</h3></div>

The family of ellipses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+(1-\varepsilon ^{2})y^{2}=\varepsilon ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+(1-\varepsilon ^{2})y^{2}=\varepsilon ^{2}}</annotation>

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</math><img src="./7812e5bbe8824c2892b77186f2c737fe4e4ae736.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.625ex; height:3.176ex;" alt="{\displaystyle x^{2}+(1-\varepsilon ^{2})y^{2}=\varepsilon ^{2}}" loading="lazy"> centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}">

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</math><img src="./5d630d3e781a53b0a3559ae7e5b45f9479a3141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" loading="lazy"> is parameterized by eccentricity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

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</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">. The orthogonal trajectories of this family are given by the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {1}{y}}+y\right)dy=\left({\frac {1}{x}}-x\right)dx}">

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {1}{y}}+y\right)dy=\left({\frac {1}{x}}-x\right)dx}</annotation>

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</math><img src="./d661cfb622311f0b30abd36f91f6b398dcb2b91b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.962ex; height:6.176ex;" alt="{\displaystyle \left({\frac {1}{y}}+y\right)dy=\left({\frac {1}{x}}-x\right)dx}" loading="lazy"> whose general solution is the family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=}">

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<annotation encoding="application/x-tex">{\displaystyle y^{2}=}</annotation>

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</math><img src="./199efd744370706e88d3daca732e2cd17cc47238.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.668ex; height:3.009ex;" alt="{\displaystyle y^{2}=}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}(x^{2}\exp(-2C-x^{2}))}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(x^{2}\exp(-2C-x^{2}))}</annotation>

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</math><img src="./dc088b340fc429960692cbd485ac1929d612c430.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.151ex; height:3.176ex;" alt="{\displaystyle W_{0}(x^{2}\exp(-2C-x^{2}))}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

The standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form:

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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-cx}=a_{0}(x-r)}">

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</math></td> <td></td> <td class="nowrap">1</td></tr></tbody></table>

where a0, c and r are real constants. The solution is

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<annotation encoding="application/x-tex">{\displaystyle x=r+{\frac {1}{c}}W\left({\frac {c\,e^{-cr}}{a_{0}}}\right).}</annotation>

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Generalizations of the Lambert W function<a href="#cite_note-50">[50]</a><a href="#cite_note-51">[51]</a><a href="#cite_note-52">[52]</a> include:

<ul>

<li>An application to <a href="General_relativity" title="General relativity">general relativity</a> and <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> (<a href="Quantum_gravity#The_dilaton" title="Quantum gravity">quantum gravity</a>) in lower dimensions, in fact a link (unknown prior to 2007<a href="#cite_note-53">[53]</a>) between these two areas, where the right-hand side of (<a href="#math_1">1</a>) is replaced by a quadratic polynomial in x:

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<annotation encoding="application/x-tex">{\displaystyle e^{-cx}=a_{0}\left(x-r_{1}\right)\left(x-r_{2}\right),}</annotation>

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</math></td> <td></td> <td class="nowrap">2</td></tr></tbody></table>

where r1 and r2 are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function which has a single argument x but the terms like ri and a0 are parameters of that function. In this respect, the generalization resembles the <a href="Hypergeometric" class="mw-disambig" title="Hypergeometric">hypergeometric</a> function and the <a href="Meijer_G-function" title="Meijer G-function">Meijer G function</a> but it belongs to a different class of functions. When r1 = r2, both sides of (<a href="#math_2">2</a>) can be factored and reduced to (<a href="#math_1">1</a>) and thus the solution reduces to that of the standard W function. Equation (<a href="#math_2">2</a>) expresses the equation governing the <a href="Dilaton" title="Dilaton">dilaton</a> field, from which is derived the metric of the <a href="R_%3D_T_model" class="mw-redirect" title="R = T model">R = T</a> or lineal two-body gravity problem in 1 + 1 dimensions (one spatial dimension and one time dimension) for the case of unequal rest masses, as well as the eigenenergies of the quantum-mechanical <a href="Delta_potential#Double_Delta_Potential" title="Delta potential">double-well Dirac delta function model</a> for unequal charges in one dimension.

</li>

<li>Analytical solutions of the eigenenergies of a special case of the quantum mechanical <a href="Euler's_three-body_problem" title="Euler's three-body problem">three-body problem</a>, namely the (three-dimensional) <a href="Hydrogen_molecule-ion" class="mw-redirect" title="Hydrogen molecule-ion">hydrogen molecule-ion</a>.<a href="#cite_note-54">[54]</a> Here the right-hand side of (<a href="#math_1">1</a>) is replaced by a ratio of infinite order polynomials in x:

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<annotation encoding="application/x-tex">{\displaystyle e^{-cx}=a_{0}{\frac {\displaystyle \prod _{i=1}^{\infty }(x-r_{i})}{\displaystyle \prod _{i=1}^{\infty }(x-s_{i})}}}</annotation>

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</math></td> <td></td> <td class="nowrap">3</td></tr></tbody></table>

where ri and si are distinct real constants and x is a function of the eigenenergy and the internuclear distance R. Equation (<a href="#math_3">3</a>) with its specialized cases expressed in (<a href="#math_1">1</a>) and (<a href="#math_2">2</a>) is related to a large class of <a href="Delay_differential_equation" title="Delay differential equation">delay differential equations</a>. <a href="G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a>'s notion of a "false derivative" provides exact multiple roots to special cases of (<a href="#math_3">3</a>).<a href="#cite_note-55">[55]</a>

</li>

</ul>

Applications of the Lambert W function in fundamental physical problems are not exhausted even for the standard case expressed in (<a href="#math_1">1</a>) as seen recently in the area of <a href="Atomic%2C_molecular%2C_and_optical_physics" title="Atomic, molecular, and optical physics">atomic, molecular, and optical physics</a>.<a href="#cite_note-56">[56]</a>

<div class="mw-heading mw-heading2"><h2 id="Plots">Plots</h2></div>

<li class="gallerycaption">Plots of the Lambert W function on the complex plane</li>

</li>

</li>

</li>

<div class="gallerytext">Superimposition of the previous three plots</div>

</li>

</ul>

<div class="mw-heading mw-heading2"><h2 id="Numerical_evaluation">Numerical evaluation</h2></div>

The W function may be approximated using <a href="Newton's_method" title="Newton's method">Newton's method</a>, with successive approximations to w = W(z) (so z = wew) being

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<annotation encoding="application/x-tex">{\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}+w_{j}e^{w_{j}}}}.}</annotation>

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</math><img src="./59ff8e39515daac4a18bd674b1f3bbddcfabab2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.498ex; height:6.176ex;" alt="{\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}+w_{j}e^{w_{j}}}}.}" loading="lazy"></dd></dl>

The W function may also be approximated using <a href="Halley's_method" title="Halley's method">Halley's method</a>,

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<annotation encoding="application/x-tex">{\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}\left(w_{j}+1\right)-{\dfrac {\left(w_{j}+2\right)\left(w_{j}e^{w_{j}}-z\right)}{2w_{j}+2}}}}}</annotation>

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</math><img src="./9e19096e29c25687b5b83db01a0714f8af312bf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:49.971ex; height:9.843ex;" alt="{\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}\left(w_{j}+1\right)-{\dfrac {\left(w_{j}+2\right)\left(w_{j}e^{w_{j}}-z\right)}{2w_{j}+2}}}}}" loading="lazy"></dd></dl>

given in Corless et al.<a href="#cite_note-Corless-4">[4]</a> to compute W.

For real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq -1/e}">

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<annotation encoding="application/x-tex">{\displaystyle w_{n+1}(x)={\frac {w_{n}(x)}{1+w_{n}(x)}}\left(1+\log \left({\frac {x}{w_{n}(x)}}\right)\right).}</annotation>

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</math><img src="./b727c9c91ed562d258c5f83aa2750b2b27e6f268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.177ex; height:6.509ex;" alt="{\displaystyle w_{n+1}(x)={\frac {w_{n}(x)}{1+w_{n}(x)}}\left(1+\log \left({\frac {x}{w_{n}(x)}}\right)\right).}" loading="lazy"></dd></dl>

Lajos Lóczi proves<a href="#cite_note-57">[57]</a> that by using this iteration with an appropriate starting value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x)}">

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<ul><li>For the principal branch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}:}">

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</math><img src="./cc544773e077602c01d5a22b4287147898ceb461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.54ex; height:2.509ex;" alt="{\displaystyle W_{0}:}" loading="lazy">

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</math><img src="./f3a4ddf7eb9ccb9950da6caab48ca86c4686805f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.421ex; height:2.843ex;" alt="{\displaystyle x\in (e,\infty )}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x)=\log(x)-\log(\log(x)),}">

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</math><img src="./17802be09f847ebe729ef2a4982ad9061cce7ac4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.446ex; height:2.843ex;" alt="{\displaystyle w_{0}(x)=\log(x)-\log(\log(x)),}" loading="lazy"></li>

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</math><img src="./a78ac7b7ff50bae6712afb0e6df7475502f71417.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.552ex; height:2.843ex;" alt="{\displaystyle x\in (0,e):}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x)=x/e,}">

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</math><img src="./d60a0825d2e633caed6cbc3897105e3a810a12c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.685ex; height:2.843ex;" alt="{\displaystyle x\in (-1/e,0):}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x)={\frac {ex\log(1+{\sqrt {1+ex}})}{1+ex+{\sqrt {1+ex}}}},}">

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<li>For the branch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{-1}:}">

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<annotation encoding="application/x-tex">{\displaystyle x\in (-1/e,-1/4]:}</annotation>

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</math><img src="./24b94a22ce7703c01fad21e3dc02064c914fb7aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.56ex; height:2.843ex;" alt="{\displaystyle x\in (-1/e,-1/4]:}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x)=-1-{\sqrt {2}}{\sqrt {1+ex}},}">

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{0}(x)=-1-{\sqrt {2}}{\sqrt {1+ex}},}</annotation>

</semantics>

</math><img src="./06133d87eb3dedaa5a0962c0b895c2ab76a438a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.864ex; height:3.176ex;" alt="{\displaystyle w_{0}(x)=-1-{\sqrt {2}}{\sqrt {1+ex}},}" loading="lazy"></li></ul></li></ul>

one can determine the maximum number of iteration steps in advance for any precision:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in (e,\infty )}</annotation>

</semantics>

</math><img src="./f3a4ddf7eb9ccb9950da6caab48ca86c4686805f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.421ex; height:2.843ex;" alt="{\displaystyle x\in (e,\infty )}" loading="lazy"> (Theorem 2.4): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<W_{0}(x)-w_{n}(x)<\left(\log(1+1/e)\right)^{2^{n}},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<W_{0}(x)-w_{n}(x)<\left(\log(1+1/e)\right)^{2^{n}},}</annotation>

</semantics>

</math><img src="./45a96dffaf1a2cd67362bf4adad44c1bdda3dda7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.276ex; height:3.343ex;" alt="{\displaystyle 0<W_{0}(x)-w_{n}(x)<\left(\log(1+1/e)\right)^{2^{n}},}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in (0,e)}</annotation>

</semantics>

</math><img src="./e09b92a8df6dba38d25e82ffaa55a59b01d0bb68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.26ex; height:2.843ex;" alt="{\displaystyle x\in (0,e)}" loading="lazy"> (Theorem 2.9): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<W_{0}(x)-w_{n}(x)<{\frac {\left(1-1/e\right)^{2^{n}-1}}{5}},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<W_{0}(x)-w_{n}(x)<{\frac {\left(1-1/e\right)^{2^{n}-1}}{5}},}</annotation>

</semantics>

</math><img src="./a1a36f9843a499511b6a1bd4a2742170919e06a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.431ex; height:6.176ex;" alt="{\displaystyle 0<W_{0}(x)-w_{n}(x)<{\frac {\left(1-1/e\right)^{2^{n}-1}}{5}},}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in (-1/e,0):}</annotation>

</semantics>

</math><img src="./d60a0825d2e633caed6cbc3897105e3a810a12c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.685ex; height:2.843ex;" alt="{\displaystyle x\in (-1/e,0):}" loading="lazy">

<ul><li>for the principal branch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{0}}</annotation>

</semantics>

</math><img src="./7f541f57fd799ba5137a2e50a1a728dde4306c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{0}}" loading="lazy"> (Theorem 2.17): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<w_{n}(x)-W_{0}(x)<\left(1/10\right)^{2^{n}},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<w_{n}(x)-W_{0}(x)<\left(1/10\right)^{2^{n}},}</annotation>

</semantics>

</math><img src="./91df0b7451050c994ae014e4a90839363fe3b254.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.733ex; height:3.343ex;" alt="{\displaystyle 0<w_{n}(x)-W_{0}(x)<\left(1/10\right)^{2^{n}},}" loading="lazy"></li>

<li>for the branch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{-1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{-1}}</annotation>

</semantics>

</math><img src="./bdf91805ba18586ff58b3035ff4b8fe48d39dd67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.526ex; height:2.509ex;" alt="{\displaystyle W_{-1}}" loading="lazy">(Theorem 2.23): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<W_{-1}(x)-w_{n}(x)<\left(1/2\right)^{2^{n}}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<W_{-1}(x)-w_{n}(x)<\left(1/2\right)^{2^{n}}.}</annotation>

</semantics>

</math><img src="./57cadf2b56d581a1eb81ddf495dd06c998db4c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.849ex; height:3.343ex;" alt="{\displaystyle 0<W_{-1}(x)-w_{n}(x)<\left(1/2\right)^{2^{n}}.}" loading="lazy"></li></ul></li></ul>

Toshio Fukushima has presented a fast method for approximating the real valued parts of the principal and secondary branches of the W function without using any iteration.<a href="#cite_note-58">[58]</a> In this method the W function is evaluated as a conditional switch of <a href="Rational_functions" class="mw-redirect" title="Rational functions">rational functions</a> on transformed variables:

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{0}(z)={\begin{cases}X_{k}(x),&(z_{k-1}<=z<z_{k},\quad k=1,2,\ldots ,17),\\U_{k}(u),&(z_{k-1}<=z<z_{k},\quad k=18,19),\end{cases}}}</annotation>

</semantics>

</math>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{-1}(z)={\begin{cases}Y_{k}(y),&(z_{k-1}<=z<z_{k},\quad k=-1,-2,\ldots ,-7),\\V_{k}(u),&(z_{k-1}<=z<z_{k},\quad k=-8,-9,-10),\end{cases}}}</annotation>

</semantics>

</math>

where x, u, y and v are transformations of z:

<msqrt>

</mrow>

</msqrt>

</mrow>

</mrow>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x={\sqrt {z+1/e}},\quad u=\ln {z},\quad y=-z/(x+1/{\sqrt {e}}),\quad v=\ln(-z)}</annotation>

</semantics>

</math><img src="./b01e270caec340dd344f0a7e28712f0c90cd201b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:62.508ex; height:4.843ex;" alt="{\displaystyle x={\sqrt {z+1/e}},\quad u=\ln {z},\quad y=-z/(x+1/{\sqrt {e}}),\quad v=\ln(-z)}" loading="lazy">.</dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{k}(x)}</annotation>

</semantics>

</math><img src="./73e7a1b06cc14d348301ba42d0f9afedc708deec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.152ex; height:2.843ex;" alt="{\displaystyle X_{k}(x)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{k}(u)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{k}(u)}</annotation>

</semantics>

</math><img src="./6044aad79a1d89f89f9490047f0ca12130daf491.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.815ex; height:2.843ex;" alt="{\displaystyle U_{k}(u)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{k}(y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{k}(y)}</annotation>

</semantics>

</math><img src="./540d701e4cbb89964267aa9a53b7ad2e3d36c124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.404ex; height:2.843ex;" alt="{\displaystyle Y_{k}(y)}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{k}(v)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{k}(v)}</annotation>

</semantics>

</math><img src="./c1459ab96304a8294e623682ad71068860a51f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.381ex; height:2.843ex;" alt="{\displaystyle V_{k}(v)}" loading="lazy"> are rational functions whose coefficients for different k-values are listed in the referenced paper together with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{k}}</annotation>

</semantics>

</math><img src="./a51cdfac24f8b95ed711f11ea9502da4087b6a24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.17ex; height:2.009ex;" alt="{\displaystyle z_{k}}" loading="lazy"> values that determine the subdomains. With higher degree polynomials in these rational functions the method can approximate the W function more accurately.

For example, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1/e\leq z\leq 2.0082178115844727}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1/e\leq z\leq 2.0082178115844727}</annotation>

</semantics>

</math><img src="./0f98bf911a1b054aed9a4c9130a582362ec87940.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.91ex; height:2.843ex;" alt="{\displaystyle -1/e\leq z\leq 2.0082178115844727}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{0}(z)}</annotation>

</semantics>

</math><img src="./783994969d7b6885909dc863299915dacc86f3a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.145ex; height:2.843ex;" alt="{\displaystyle W_{0}(z)}" loading="lazy"> can be approximated to 24 bits of accuracy on 64-bit floating point values as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}(z)\approx X_{1}(x)={\frac {\sum _{i}^{4}P_{i}x^{i}}{\sum _{i}^{3}Q_{i}x^{i}}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{0}(z)\approx X_{1}(x)={\frac {\sum _{i}^{4}P_{i}x^{i}}{\sum _{i}^{3}Q_{i}x^{i}}}}</annotation>

</semantics>

</math><img src="./21564bbdd107b3d0f9e5f9b137afdc8fb444f5b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.958ex; height:7.509ex;" alt="{\displaystyle W_{0}(z)\approx X_{1}(x)={\frac {\sum _{i}^{4}P_{i}x^{i}}{\sum _{i}^{3}Q_{i}x^{i}}}}" loading="lazy"> where x is defined with the transformation above and the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{i}}</annotation>

</semantics>

</math><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{i}}</annotation>

</semantics>

</math><img src="./b9f7193081d440425e522698e80817b5d558df03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}" loading="lazy"> are given in the table below.

<caption>Coefficients

</caption>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{i}}</annotation>

</semantics>

</math><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{i}}</annotation>

</semantics>

</math><img src="./b9f7193081d440425e522698e80817b5d558df03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}" loading="lazy">

</th></tr>

<tr>

<td>1

</td></tr>

<tr>

<td>2.275906559863465

</td></tr>

<tr>

<td>1.367597013868904

</td></tr>

<tr>

<td>0.18615823452831623

</td></tr>

<tr>

<td>0

</td></tr></tbody></table>

Fukushima also offers an approximation with 50 bits of accuracy on 64-bit floats that uses 8th- and 7th-degree polynomials.

<div class="mw-heading mw-heading2"><h2 id="Software">Software</h2></div>

The Lambert W function is implemented in many programming languages. Some of them are listed below:

<th>Language</th>

<th>Function name</th>

<th>Required library

</th></tr>

<tr>

<td><code>gsl_sf_lambert_W0</code> and <code>gsl_sf_lambert_Wm1</code></td>

<td>Special functions section of the GNU Scientific Library (GSL)<a href="#cite_note-59">[59]</a>

</td></tr>

<tr>

<td><code>lambert_w0</code>, <code>lambert_wm1</code>, <code>lambert_w0_prime</code>, and <code>lambert_wm1_prime</code></td>

<td>Boost C++ libraries<a href="#cite_note-60">[60]</a>

</td></tr>

<tr>

<td><code>LambertW</code></td>

<td>LambertW-function<a href="#cite_note-61">[61]</a>

</td></tr>

<tr>

<td><code>lambertw</code></td>

<td>

</td></tr>

<tr>

<td><a href="Julia_(programming_language)" title="Julia (programming language)">Julia</a></td>

<td><code>lambertw</code></td>

<td><code>LambertW</code><a href="#cite_note-62">[62]</a></td>

<td>

</td></tr>

<tr>

<td><a href="Maple_(software)" title="Maple (software)">Maple</a></td>

<td><code>LambertW</code><a href="#cite_note-63">[63]</a></td>

<td>

</td></tr>

<tr>

<td><a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></td>

<td><code>ProductLog</code> (with <code>LambertW</code> as a silent alias)<a href="#cite_note-64">[64]</a></td>

<td>

</td></tr>

<tr>

<td><a href="Matlab" class="mw-redirect" title="Matlab">Matlab</a></td>

<td><code>lambertw</code><a href="#cite_note-65">[65]</a></td>

<td>

</td></tr>

<tr>

<td><a href="Maxima_(software)" title="Maxima (software)">Maxima</a></td>

<td><code>lambert_w</code><a href="#cite_note-66">[66]</a></td>

<td>

</td></tr>

<tr>

<td><a href="GNU_Octave" title="GNU Octave">Octave</a></td>

<td><code>lambertw</code></td>

<td><code>specfun</code><a href="#cite_note-67">[67]</a>

</td></tr>

<tr>

<td><code>glambertW, lambertWC, glambertW_i, mplambertW, lambertW</code></td>

<td>

</td></tr>

<tr>

<td><code>LambertW</code></td>

<td><code>ntheory</code><a href="#cite_note-68">[68]</a>

</td></tr>

<tr>

<td><a href="Python_(programming_language)" title="Python (programming language)">Python</a></td>

<td><code>lambertw</code></td>

<td><code><a href="Scipy" class="mw-redirect" title="Scipy">scipy</a></code><a href="#cite_note-69">[69]</a>

</td></tr>

<tr>

<td><code>lambertW0</code> and <code>lambertWm1</code></td>

</td></tr>

<tr>

<td><code>lambert_w</code>, <code>lambert_w0</code> and <code>lambert_wm1</code></td>

<td><code>lambert_w</code><a href="#cite_note-71">[71]</a>

</td></tr></tbody></table>

<ul><li><a href="Wright_omega_function" title="Wright omega function">Wright omega function</a></li>

<li>Lambert's <a href="Trinomial" title="Trinomial">trinomial equation</a></li>

<li><a href="Lagrange_inversion_theorem#Lambert_W_function" title="Lagrange inversion theorem">Lagrange inversion theorem</a></li>

<li><a href="Experimental_mathematics" title="Experimental mathematics">Experimental mathematics</a></li>

<li><a href="Holstein%E2%80%93Herring_method" title="Holstein–Herring method">Holstein–Herring method</a></li>

<li><a href="R_%3D_T_model" class="mw-redirect" title="R = T model">R = T model</a></li>

<li><a href="Ross'_%CF%80_lemma" title="Ross' π lemma">Ross' π lemma</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<cite id="CITEREFCorlessGonnetHareJeffrey1996" class="citation journal cs1">Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf">"On the Lambert W function"</a> (PDF). Advances in Computational Mathematics. 5: 329–359. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02124750">10.1007/BF02124750</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:29028411">29028411</a>.</cite>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">⁠ function in Maple". The Maple Technical Newsletter. 9: 12–22. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.2556">10.1.1.33.2556</a>.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFBronsteinCorlessDavenportJeffrey2008" class="citation journal cs1">Bronstein, Manuel; Corless, Robert M.; Davenport, James H.; Jeffrey, D. J. (2008). <a rel="nofollow" class="external text" href="http://opus.bath.ac.uk/27004/1/Davenport_ITSF_19_10_709.pdf">"Algebraic properties of the Lambert ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">⁠ function from a result of Rosenlicht and of Liouville"</a> (PDF). Integral Transforms and Special Functions. 19 (10): 709–712. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10652460802332342">10.1080/10652460802332342</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120069437">120069437</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151211132056/http://opus.bath.ac.uk/27004/1/Davenport_ITSF_19_10_709.pdf">Archived</a> (PDF) from the original on 2015-12-11.</cite>

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<li id="cite_note-:0-11">^ <a href="#cite_ref-:0_11-0">a</a> <a href="#cite_ref-:0_11-1">b</a> A. Hoorfar, M. Hassani, <a rel="nofollow" class="external text" href="https://www.emis.de/journals/JIPAM/article983.html?sid=983">Inequalities on the Lambert W Function and Hyperpower Function</a>, JIPAM, Theorem 2.7, page 7, volume 9, issue 2, article 51. 2008.

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<li id="cite_note-doi.org-12">^ <a href="#cite_ref-doi.org_12-0">a</a> <a href="#cite_ref-doi.org_12-1">b</a> <cite id="CITEREFIaconoBoyd2017" class="citation journal cs1">Iacono, Roberto; Boyd, John P. (2017-12-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s10444-017-9530-3">"New approximations to the principal real-valued branch of the Lambert W-function"</a>. Advances in Computational Mathematics. 43 (6): 1403–1436. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10444-017-9530-3">10.1007/s10444-017-9530-3</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1572-9044">1572-9044</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:254184098">254184098</a>.</cite>

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFStewart2009" class="citation journal cs1">Stewart, Seán M. (2009). <a rel="nofollow" class="external text" href="https://eudml.org/doc/224588">"On certain inequalities involving the Lambert W function"</a>. JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]. 10 (4): Paper No. 96, 4 p., electronic only–Paper No. 96, 4 p., electronic only. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1443-5756">1443-5756</a>.</cite>

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<cite id="CITEREFChatzigeorgiou2013" class="citation journal cs1">Chatzigeorgiou, I. (2013). "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation". IEEE Communications Letters. 17 (8): 1505–1508. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1601.04895">1601.04895</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FLCOMM.2013.070113.130972">10.1109/LCOMM.2013.070113.130972</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10062685">10062685</a>.</cite>

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<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://functions.wolfram.com/ElementaryFunctions/ProductLog/17/01/0001/">"Lambert function: Identities (formula 01.31.17.0001)"</a>.</cite>

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<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/LambertW-Function.html">"Lambert W-Function"</a>.</cite>

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a rel="nofollow" class="external free" href="https://isa-afp.org/entries/Lambert_W.html">https://isa-afp.org/entries/Lambert_W.html</a> Note: although one of the assumptions of the relevant lemma states that x must be > 1/e, inspection of said lemma reveals that this assumption is unused. The lower bound is in fact x > 0. The reason for the branch switch at e is simple: for x > 1 there are always two solutions, −ln x and another one that you'd get from the x on the other side of e that would feed the same value to W; these must crossover at x = e: <a rel="nofollow" class="external autonumber" href="https://wolframalpha.com/input/?i=plot+LambertW%280%2C-ln%28x%29%2Fx%29%3BLambertW%28-1%2C-ln%28x%29%2Fx%29x%3D0...5">[1]</a> Wn cannot distinguish a value of ln x/x from an x < e from the same value from the other x > e, so it cannot flip the order of its return values.

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<li id="cite_note-Finch-18"><a href="#cite_ref-Finch_18-0">^</a> <cite id="CITEREFFinch2003" class="citation book cs1">Finch, S. R. (2003). Mathematical constants. Cambridge University Press. p. 450.</cite>

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<ul><li><a rel="nofollow" class="external text" href="http://dlmf.nist.gov/4.13">National Institute of Science and Technology Digital Library – Lambert W</a></li>

<li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/LambertW-Function.html">MathWorld – Lambert W-Function</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060824060948/http://www.whim.org/nebula/math/lambertw.html">Computing the Lambert W function</a></li>

<li><a rel="nofollow" class="external text" href="http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/">Corless et al. Notes about Lambert W research</a></li>

<li>GPL <a rel="nofollow" class="external text" href="https://github.com/DarkoVeberic/LambertW">C++ implementation</a> with Halley's and Fritsch's iteration.</li>

<li><a rel="nofollow" class="external text" href="https://www.gnu.org/software/gsl/manual/html_node/Special-Functions.html">Special Functions</a> of the <a rel="nofollow" class="external text" href="https://www.gnu.org/software/gsl/">GNU Scientific Library</a> – GSL</li>

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