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Exponential function

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the function <var style="padding-right: 1px;">f</var>(<var style="padding-right: 1px;">x</var>) = <var style="padding-right: 1px;">e</var><var style="padding-right: 1px;">x</var> and its generalizations. For functions of the form <var style="padding-right: 1px;">f</var>(<var style="padding-right: 1px;">x</var>) = <var style="padding-right: 1px;">x</var><var style="padding-right: 1px;">r</var>, see <a href="Power_function" class="mw-redirect" title="Power function">Power function</a>. For the bivariate function <var style="padding-right: 1px;">f</var>(<var style="padding-right: 1px;">x</var>,<var style="padding-right: 1px;">y</var>) = <var style="padding-right: 1px;">x</var><var style="padding-right: 1px;">y</var>, see <a href="Exponentiation" title="Exponentiation">Exponentiation</a>. For the representation of scientific numbers, see <a href="E_notation" class="mw-redirect" title="E notation">E notation</a>.</div>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e0e0e0;padding:0.15em 0.5em 0.25em;font-weight:bold;">Exponential</th></tr><tr><td colspan="2" class="infobox-image" style="padding-bottom:0.4em;"><div class="infobox-caption">Graph of the exponential function</div></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">General information</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">General definition</th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp z=e^{z}}">

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

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</math><img src="./ffd77a26bc18640fb886ed5e6254d4e8e7da3c81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.211ex; height:2.676ex;" alt="{\displaystyle \exp z=e^{z}}" loading="lazy"></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Domain, codomain and image</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Domain_of_a_function" title="Domain of a function">Domain</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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

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</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Image_(mathematics)" title="Image (mathematics)">Image</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}(0,\infty )&{\text{for }}z\in \mathbb {R} \\\mathbb {C} \setminus \{0\}&{\text{for }}z\in \mathbb {C} \end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}(0,\infty )&{\text{for }}z\in \mathbb {R} \\\mathbb {C} \setminus \{0\}&{\text{for }}z\in \mathbb {C} \end{cases}}}</annotation>

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</math><img src="./472039e098eb4c2fe8921a76baea431719c38297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.151ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}(0,\infty )&{\text{for }}z\in \mathbb {R} \\\mathbb {C} \setminus \{0\}&{\text{for }}z\in \mathbb {C} \end{cases}}}" loading="lazy"></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Specific values</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Y-intercept" title="Y-intercept">At zero</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Value at 1</th><td class="infobox-data"><a href="Euler's_number" class="mw-redirect" title="Euler's number">e</a></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Specific features</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">Fixed point</a></th><td class="infobox-data"><a href="Lambert_W_function" title="Lambert W function">−Wn(−1)</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {Z} }">

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

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</math><img src="./3c1cf6a513f2062531d95dbb198944936f312982.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.786ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {Z} }" loading="lazy"></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Related functions</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Multiplicative_inverse" title="Multiplicative inverse">Reciprocal</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(-z)}">

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

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</math><img src="./de355e33ba6e640879fa7dfc10423df47132ba2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.258ex; height:2.843ex;" alt="{\displaystyle \exp(-z)}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Inverse_function" title="Inverse function">Inverse</a></th><td class="infobox-data"><a href="Natural_logarithm" title="Natural logarithm">Natural logarithm</a>, <a href="Complex_logarithm" title="Complex logarithm">Complex logarithm</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Derivative" title="Derivative">Derivative</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp '\!z=\exp z}">

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</math><img src="./397934ef368e40bcc20eccbef391c4f3a99dda48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.451ex; height:2.843ex;" alt="{\displaystyle \exp '\!z=\exp z}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Antiderivative" title="Antiderivative">Antiderivative</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \exp z\,dz=\exp z+C}">

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<annotation encoding="application/x-tex">{\displaystyle \int \exp z\,dz=\exp z+C}</annotation>

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</math><img src="./3e738aa8b045ee86f64398e634444bb305f3b6c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.032ex; height:5.676ex;" alt="{\displaystyle \int \exp z\,dz=\exp z+C}" loading="lazy"></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Series definition</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Taylor_series" title="Taylor series">Taylor series</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp z=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}">

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<annotation encoding="application/x-tex">{\displaystyle \exp z=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}</annotation>

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</math><img src="./eb3547bfb30b6050f2fb7a59e4b14a458f48cb3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.013ex; height:6.843ex;" alt="{\displaystyle \exp z=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}" loading="lazy"></td></tr></tbody></table>

In <a href="Mathematics" title="Mathematics">mathematics</a>, the exponential function is the unique <a href="Real_function" class="mw-redirect" title="Real function">real function</a> which maps <a href="0" title="0">zero</a> to <a href="1" title="1">one</a> and has a <a href="Derivative_(mathematics)" class="mw-redirect" title="Derivative (mathematics)">derivative</a> everywhere equal to its value. The exponential of a variable ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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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">⁠ is denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp x}">

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

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</math><img src="./eb48cbff4a1363dbf36792710839f53bd5e02ff9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle \exp x}" loading="lazy">⁠ or ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}">

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

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</math><img src="./841c0d168e64191c45a45e54c7e447defd17ec6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}" loading="lazy">⁠, with the two notations used interchangeably. It is called exponential because its argument can be seen as an <a href="Exponent_(mathematics)" class="mw-redirect" title="Exponent (mathematics)">exponent</a> to which a constant <a href="E_(mathematical_constant)" title="E (mathematical constant)">number e ≈ 2.718</a>, the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.

The exponential function converts sums to products: it maps the <a href="Additive_identity" title="Additive identity">additive identity</a> 0 to the <a href="Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a> 1, and the exponential of a sum is equal to the product of separate exponentials, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x+y)=\exp x\cdot \exp y}">

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

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</math><img src="./3fd1e4a39c32987806b9b49e042c794d7cda9efd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.829ex; height:2.843ex;" alt="{\displaystyle \exp(x+y)=\exp x\cdot \exp y}" loading="lazy">⁠. Its <a href="Inverse_function" title="Inverse function">inverse function</a>, the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a>, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln }">

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

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</math><img src="./c0de5ba4f372ede555d00035e70c50ed0b9625d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \ln }" loading="lazy">⁠ or ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log }">

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

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</math><img src="./79e4debd0ab1c6ce342d0172a7643733305c37bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }" loading="lazy">⁠, converts products to sums: ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x\cdot y)=\ln x+\ln y}">

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

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</math><img src="./025cf7a1f2fd3a943a03a819e8c7e6741855e9c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.99ex; height:2.843ex;" alt="{\displaystyle \ln(x\cdot y)=\ln x+\ln y}" loading="lazy">⁠.

The exponential function is occasionally called the natural exponential function, matching the name natural logarithm, for distinguishing it from some other functions that are also commonly called exponential functions. These functions include the functions of the form ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=b^{x}}">

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</math><img src="./766492bec9604d590253dbbe840d712874283a69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle f(x)=b^{x}}" loading="lazy">⁠, which is <a href="Exponentiation" title="Exponentiation">exponentiation</a> with a fixed base ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">⁠. More generally, and especially in applications, functions of the general form ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ab^{x}}">

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

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</math><img src="./5fbbc9abebe92f1aa9f54bdb6650375be8818b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.916ex; height:2.843ex;" alt="{\displaystyle f(x)=ab^{x}}" loading="lazy">⁠ are also called exponential functions. They <a href="Exponential_growth" title="Exponential growth">grow</a> or <a href="Exponential_decay" title="Exponential decay">decay</a> exponentially in that the rate that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">⁠ changes when ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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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">⁠ is increased is proportional to the current value of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">⁠.

The exponential function can be generalized to accept <a href="Complex_number" title="Complex number">complex numbers</a> as arguments. This reveals relations between multiplication of complex numbers, rotations in the <a href="Complex_plane" title="Complex plane">complex plane</a>, and <a href="Trigonometry" title="Trigonometry">trigonometry</a>. <a href="Euler's_formula" title="Euler's formula">Euler's formula</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp i\theta =\cos \theta +i\sin \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \exp i\theta =\cos \theta +i\sin \theta }</annotation>

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</math><img src="./8caaae4788bfdaa4808a92df269ab1fb25070772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.883ex; height:2.509ex;" alt="{\displaystyle \exp i\theta =\cos \theta +i\sin \theta }" loading="lazy">⁠ expresses and summarizes these relations.

The exponential function can be even further generalized to accept other types of arguments, such as <a href="Matrix_exponentiation" class="mw-redirect" title="Matrix exponentiation">matrices</a> and elements of <a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">Lie algebras</a>.

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

The <a href="Graph_of_a_function" title="Graph of a function">graph</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=e^{x}}">

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

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</math><img src="./e87885ed8d6af200a283fdf841cbea480eb99dd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.51ex; height:2.676ex;" alt="{\displaystyle y=e^{x}}" loading="lazy"> is upward-sloping, and increases faster than every power of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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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">⁠.<a href="#cite_note-1">[1]</a> The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal <a href="Asymptote" title="Asymptote">asymptote</a>. The equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}}">

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

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</math><img src="./3b5adbf2271ec8ca8c004516e8553b085699a448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.247ex; height:3.843ex;" alt="{\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}}" loading="lazy"> means that the <a href="Slope" title="Slope">slope</a> of the <a href="Tangent" title="Tangent">tangent</a> to the graph at each point is equal to its height (its y-coordinate) at that point.

<div class="mw-heading mw-heading2"><h2 id="Definitions_and_fundamental_properties">Definitions and fundamental properties</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Characterizations_of_the_exponential_function" title="Characterizations of the exponential function">Characterizations of the exponential function</a></div>

There are several equivalent definitions of the exponential function, although of very different nature.

<div class="mw-heading mw-heading3"><h3 id="Differential_equation">Differential equation</h3></div>

One of the simplest definitions is: The exponential function is the unique <a href="Differentiable_function" title="Differentiable function">differentiable function</a> that equals its <a href="Derivative" title="Derivative">derivative</a>, and takes the value 1 for the value 0 of its variable.

This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.

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

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

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

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

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy">⁠ are two functions satisfying the above definition, then the derivative of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f/g}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b5c6b7962d3532e248f07cd42b1bdc9e007b137d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.557ex; height:2.843ex;" alt="{\displaystyle f/g}" loading="lazy">⁠ is zero everywhere because of the <a href="Quotient_rule" title="Quotient rule">quotient rule</a>. It follows that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f/g}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b5c6b7962d3532e248f07cd42b1bdc9e007b137d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.557ex; height:2.843ex;" alt="{\displaystyle f/g}" loading="lazy">⁠ is constant; this constant is 1 since ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=g(0)=1}">

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</mrow>

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

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</math><img src="./bd1d3d8496094ac85b995157f8bdeed04ad068a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.697ex; height:2.843ex;" alt="{\displaystyle f(0)=g(0)=1}" loading="lazy">⁠.

Existence is proved in each of the two following sections.

<div class="mw-heading mw-heading3"><h3 id="Inverse_of_natural_logarithm">Inverse of natural logarithm</h3></div>

The exponential function is the <a href="Inverse_function" title="Inverse function">inverse function</a> of the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a>. The <a href="Inverse_function_theorem" title="Inverse function theorem">inverse function theorem</a> implies that the natural logarithm has an inverse function, that satisfies the above definition. This is a first proof of existence. Therefore, one has

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln(\exp x)&=x\\\exp(\ln y)&=y\end{aligned}}}</annotation>

</semantics>

</math><img src="./5722f043cace15ac353f630b1c01379198ed0ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.197ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\ln(\exp x)&=x\\\exp(\ln y)&=y\end{aligned}}}" loading="lazy"></dd></dl>

for every <a href="Real_number" title="Real number">real number</a> <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 every positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y.}">

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

</semantics>

</math><img src="./83f72471aff7c6fbb27df0f971283a068efe091f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Power_series">Power series</h3></div>

The exponential function is the sum of the <a href="Power_series" title="Power series">power series</a><a href="#cite_note-Rudin_1987-2">[2]</a><a href="#cite_note-:0-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\exp(x)&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}},\end{aligned}}}</annotation>

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</math>

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

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

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</math><img src="./bae971720be3cc9b8d82f4cdac89cb89877514a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}" loading="lazy"> is the <a href="Factorial" title="Factorial">factorial</a> of n (the product of the n first positive integers). This series is <a href="Absolutely_convergent" class="mw-redirect" title="Absolutely convergent">absolutely convergent</a> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<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"> per the <a href="Ratio_test" title="Ratio test">ratio test</a>. So, the derivative of the sum can be computed by term-by-term differentiation, and this shows that the sum of the series satisfies the above definition. This is a second existence proof, and shows, as a byproduct, that the exponential function is defined for every ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</mrow>

<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 is everywhere the sum of its <a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a>.

<div class="mw-heading mw-heading3"><h3 id="Functional_equation">Functional equation</h3></div>

The exponential satisfies the functional equation:

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<annotation encoding="application/x-tex">{\displaystyle \exp(x+y)=\exp(x)\cdot \exp(y).}</annotation>

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</math>

This results from the uniqueness and the fact that the function

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

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</math><img src="./be8616c67c5c76f113e9830b7680a386b8684d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.27ex; height:2.843ex;" alt="{\displaystyle f(x)=\exp(x+y)/\exp(y)}" loading="lazy"> satisfies the above definition.

It can be proved that a function that satisfies this functional equation has the form ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \exp(cx)}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto \exp(cx)}</annotation>

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</math><img src="./b306cb4a728e45adaa1bd88d3444ae6dcf750ab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.642ex; height:2.843ex;" alt="{\displaystyle x\mapsto \exp(cx)}" loading="lazy">⁠ if it is either <a href="Continuous_function" title="Continuous function">continuous</a> or <a href="Monotonic_function" title="Monotonic function">monotonic</a>. It is thus <a href="Differentiable_function" title="Differentiable function">differentiable</a>, and equals the exponential function if its derivative at 0 is 1.

<div class="mw-heading mw-heading3"><h3 id="Limit_of_integer_powers">Limit of integer powers</h3></div>

The exponential function is the <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a>, as the integer n goes to infinity,<a href="#cite_note-Maor-4">[4]</a><a href="#cite_note-:0-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle \exp(x)=\lim _{n\to +\infty }\left(1+{\frac {x}{n}}\right)^{n}.}</annotation>

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</math>

By continuity of the logarithm, this can be proved by taking logarithms and proving

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<annotation encoding="application/x-tex">{\displaystyle x=\lim _{n\to \infty }\ln \left(1+{\frac {x}{n}}\right)^{n}=\lim _{n\to \infty }n\ln \left(1+{\frac {x}{n}}\right),}</annotation>

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</math>

for example with <a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a>.

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

<a href="Multiplicative_inverse" title="Multiplicative inverse">Reciprocal</a>: The functional equation implies ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}e^{-x}=1}">

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

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</math><img src="./5667378aaa63a09a1c046123ce802bd18b2f239a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.051ex; height:2.509ex;" alt="{\displaystyle e^{x}e^{-x}=1}" loading="lazy">⁠. Therefore ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}\neq 0}">

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</msup>

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

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</math><img src="./f054a82acf7fc9801ae1eead462f62ad5278d7d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.517ex; height:2.843ex;" alt="{\displaystyle e^{x}\neq 0}" loading="lazy">⁠ for every ⁠<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

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

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</math>

Positiveness: ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}>0}">

<msup>

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</mrow>

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

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</math><img src="./f61fd417a5ce808e3a571bfd89ec1c2d65aa17c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.517ex; height:2.343ex;" alt="{\displaystyle e^{x}>0}" loading="lazy">⁠ for every real number ⁠<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">⁠. This results from the <a href="Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a>, since ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{0}=1}">

<msup>

</mrow>

</msup>

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</mrow>

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

</semantics>

</math><img src="./ea9335287ddbe5eb6cf606a44f58d2acf782a979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.399ex; height:2.676ex;" alt="{\displaystyle e^{0}=1}" loading="lazy">⁠ and, if one would have ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}<0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math><img src="./866e05d2fe323ef7edeb41ca0dd19c02658ddc52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.517ex; height:2.343ex;" alt="{\displaystyle e^{x}<0}" loading="lazy">⁠ for some ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<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">⁠, there would be an ⁠<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">⁠ such that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{y}=0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math><img src="./04f1d1c225fee3c29b1f870d3ee5e1f1b3bd2328.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.394ex; height:2.343ex;" alt="{\displaystyle e^{y}=0}" loading="lazy">⁠ between ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">⁠ and ⁠<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">⁠. Since the exponential function equals its derivative, this implies that the exponential function is <a href="Monotonically_increasing" class="mw-redirect" title="Monotonically increasing">monotonically increasing</a>.

Extension of <a href="Exponentiation" title="Exponentiation">exponentiation</a> to positive real bases: Let b be a positive real number. The exponential function and the natural logarithm being the inverse each of the other, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\exp(\ln b).}">

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

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</math><img src="./72d688f1018f1f513d8216f915f2ad7532bad8dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.429ex; height:2.843ex;" alt="{\displaystyle b=\exp(\ln b).}" loading="lazy"> If n is an integer, the functional equation of the logarithm implies

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle b^{n}=\exp(\ln b^{n})=\exp(n\ln b).}</annotation>

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</math>

Since the right-most expression is defined if n is any real number, this allows defining ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{x}}">

<msup>

</mrow>

</msup>

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</mrow>

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

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</math><img src="./2bb7406a338fb530330582bc63420d091897c709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.343ex;" alt="{\displaystyle b^{x}}" loading="lazy">⁠ for every positive real number b and every real number x:

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

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In particular, if b is the <a href="Euler's_number" class="mw-redirect" title="Euler's number">Euler's number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\exp(1),}">

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

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</math><img src="./5788ee8a9b7d2f155f4c635142cf74879daea13f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.353ex; height:2.843ex;" alt="{\displaystyle e=\exp(1),}" loading="lazy"> one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln e=1}">

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

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</math><img src="./0ac39a99258627c071061334973932ca14bdf0a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.671ex; height:2.176ex;" alt="{\displaystyle \ln e=1}" loading="lazy"> (inverse function) and thus <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}=\exp(x).}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{x}=\exp(x).}</annotation>

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</math> This shows the equivalence of the two notations for the exponential function.

<div class="mw-heading mw-heading2"><h2 id="General_exponential_functions">General exponential functions</h2></div>

A function is commonly called an exponential function—with an indefinite article—if it has the form ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto b^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./873987f9618fe2c30ce4e72cbd1a967ff759c1d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.114ex; height:2.343ex;" alt="{\displaystyle x\mapsto b^{x}}" loading="lazy">⁠, that is, if it is obtained from <a href="Exponentiation" title="Exponentiation">exponentiation</a> by fixing the base and letting the exponent vary.

More generally and especially in applied contexts, the term exponential function is commonly used for functions of the form ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ab^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5fbbc9abebe92f1aa9f54bdb6650375be8818b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.916ex; height:2.843ex;" alt="{\displaystyle f(x)=ab^{x}}" loading="lazy">⁠. This may be motivated by the fact that, if the values of the function represent <a href="Quantities" class="mw-redirect" title="Quantities">quantities</a>, a change of <a href="Measurement_unit" class="mw-redirect" title="Measurement unit">measurement unit</a> changes the value of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">⁠, and so, it is nonsensical to impose ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1}">

</mstyle>

</mrow>

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

</semantics>

</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">⁠.

These most general exponential functions are the <a href="Differentiable_function" title="Differentiable function">differentiable functions</a> that satisfy the following equivalent characterizations.

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5fbbc9abebe92f1aa9f54bdb6650375be8818b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.916ex; height:2.843ex;" alt="{\displaystyle f(x)=ab^{x}}" loading="lazy">⁠ for every ⁠<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 some constants ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./94436473a90bd55191a79c59474cb5456dcbec00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>0}" loading="lazy">⁠.</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./df9cb5b25b423f3be01d26239c13b188f3a294c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.858ex; height:3.176ex;" alt="{\displaystyle f(x)=ae^{kx}}" loading="lazy">⁠ for every ⁠<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 some constants ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</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">⁠.</li>

<li>The value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)/f(x)}">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./04a877b047709a3c67f88129674b3894ad553a37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.724ex; height:3.009ex;" alt="{\displaystyle f'(x)/f(x)}" loading="lazy"> is independent of <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">.</li>

<li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1b42a115d30706dde56ddece71bb4248da2115d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.863ex; height:2.509ex;" alt="{\displaystyle d,}" loading="lazy"> the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+d)/f(x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+d)/f(x)}</annotation>

</semantics>

</math><img src="./a5e940ab370e9c952b35bf44430d706fa7e5efde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.054ex; height:2.843ex;" alt="{\displaystyle f(x+d)/f(x)}" loading="lazy"> is independent of <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="./05a950b252371110b85a784de7babc2448d28cc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x;}" loading="lazy"> that is, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f(x+d)}{f(x)}}={\frac {f(y+d)}{f(y)}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {f(x+d)}{f(x)}}={\frac {f(y+d)}{f(y)}}}</annotation>

</semantics>

</math> for every x, y.<a href="#cite_note-5">[5]</a></li></ul>

The base of an exponential function is the base of the <a href="Exponentiation" title="Exponentiation">exponentiation</a> that appears in it when written as ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to ab^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f8cc897f5ac4c8465c2a733c066f6fe94caeca31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.344ex; height:2.343ex;" alt="{\displaystyle x\to ab^{x}}" loading="lazy">⁠, namely ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">⁠.<a href="#cite_note-6">[6]</a> The base is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5eedc905f8f6e254ae23c7d7e10ca32731449ad0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.172ex; height:2.676ex;" alt="{\displaystyle e^{k}}" loading="lazy">⁠ in the second characterization, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \exp {\frac {f'(x)}{f(x)}}}">

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \exp {\frac {f'(x)}{f(x)}}}</annotation>

</semantics>

</math><img src="./22862534e0211e2898259df507b3cf32e7dfd979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.461ex; height:5.009ex;" alt="{\textstyle \exp {\frac {f'(x)}{f(x)}}}" loading="lazy"> in the third one, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left({\frac {f(x+d)}{f(x)}}\right)^{1/d}}">

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left({\frac {f(x+d)}{f(x)}}\right)^{1/d}}</annotation>

</semantics>

</math><img src="./850a3f5f878a524bcbaa9fd3cde1ca86c5f35cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.61ex; height:5.343ex;" alt="{\textstyle \left({\frac {f(x+d)}{f(x)}}\right)^{1/d}}" loading="lazy"> in the last one.

<div class="mw-heading mw-heading3"><h3 id="In_applications">In applications</h3></div>

The last characterization is important in <a href="Empirical_science" class="mw-redirect" title="Empirical science">empirical sciences</a>, as allowing a direct <a href="Experimental" class="mw-redirect" title="Experimental">experimental</a> test whether a function is an exponential function.

Exponential <a href="Exponential_growth" title="Exponential growth">growth</a> or <a href="Exponential_decay" title="Exponential decay">exponential decay</a>—where the variable change is <a href="Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the variable value—are thus modeled with exponential functions. Examples are unlimited population growth leading to <a href="Malthusian_catastrophe" class="mw-redirect" title="Malthusian catastrophe">Malthusian catastrophe</a>, <a href="Compound_interest#Continuous_compounding" title="Compound interest">continuously compounded interest</a>, and <a href="Radioactive_decay" title="Radioactive decay">radioactive decay</a>.

If the modeling function has the form ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto ae^{kx},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto ae^{kx},}</annotation>

</semantics>

</math><img src="./9ecfc30924245b3cf1fe1840e533dd7e50061365.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.933ex; height:3.009ex;" alt="{\displaystyle x\mapsto ae^{kx},}" loading="lazy">⁠ or, equivalently, is a solution of the differential equation ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=ky}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b10d87f9a74551a2e20a57eba7546792d1d529ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.31ex; height:2.843ex;" alt="{\displaystyle y'=ky}" loading="lazy">⁠, the constant ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</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">⁠ is called, depending on the context, the decay constant, disintegration constant,<a href="#cite_note-Serway-Moses-Moyer_1989-7">[7]</a> rate constant,<a href="#cite_note-Simmons_1972-8">[8]</a> or transformation constant.<a href="#cite_note-McGrawHill_2007-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Equivalence_proof">Equivalence proof</h3></div>

For proving the equivalence of the above properties, one can proceed as follows.

The two first characterizations are equivalent, since, if ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=e^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3a29cea59b64b388ad51eacd9124639a6b99071f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.268ex; height:2.676ex;" alt="{\displaystyle b=e^{k}}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\ln b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1ee1f542f6444e3ab3063dc9385d60e31c70fd10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.634ex; height:2.176ex;" alt="{\displaystyle k=\ln b}" loading="lazy">⁠, one has

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{kx}=(e^{k})^{x}=b^{x}.}</annotation>

</semantics>

</math>

The basic properties of the exponential function (derivative and functional equation) implies immediately the third and the last condition.

Suppose that the third condition is verified, and let ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</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">⁠ be the constant value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)/f(x).}">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(x)/f(x).}</annotation>

</semantics>

</math><img src="./31491a11d8c63801a7a9272951e901b9e1d8857e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.371ex; height:3.009ex;" alt="{\displaystyle f'(x)/f(x).}" loading="lazy"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\partial e^{kx}}{\partial x}}=ke^{kx},}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {\partial e^{kx}}{\partial x}}=ke^{kx},}</annotation>

</semantics>

</math><img src="./e861fb7b33e50b342769d2841696ddd80cc38511.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.226ex; height:4.176ex;" alt="{\textstyle {\frac {\partial e^{kx}}{\partial x}}=ke^{kx},}" loading="lazy"> the <a href="Quotient_rule" title="Quotient rule">quotient rule</a> for derivation

implies that

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial x}}\,{\frac {f(x)}{e^{kx}}}=0,}</annotation>

</semantics>

</math> and thus that there is a constant ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">⁠ such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ae^{kx}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=ae^{kx}.}</annotation>

</semantics>

</math><img src="./14adad7acdd2791955db6aab38b831f8d9d5fc94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.505ex; height:3.176ex;" alt="{\displaystyle f(x)=ae^{kx}.}" loading="lazy">

If the last condition is verified, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi (d)=f(x+d)/f(x),}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi (d)=f(x+d)/f(x),}</annotation>

</semantics>

</math><img src="./529ccdb41ac972483394ab9ff189937905096aa5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.344ex; height:2.843ex;" alt="{\textstyle \varphi (d)=f(x+d)/f(x),}" loading="lazy"> which is independent of ⁠<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">⁠. Using ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (0)=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4d30cbf14b09de7224c796a19162320dd34fa086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.753ex; height:2.843ex;" alt="{\displaystyle \varphi (0)=1}" loading="lazy">⁠, one gets

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {f(x+d)-f(x)}{d}}=f(x)\,{\frac {\varphi (d)-\varphi (0)}{d}}.}</annotation>

</semantics>

</math>

Taking the limit when ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">⁠ tends to zero, one gets that the third condition is verified with ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\varphi '(0)}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./78295f28eef6b1cdd74fb6bad77e8aa52c1e52ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.486ex; height:3.009ex;" alt="{\displaystyle k=\varphi '(0)}" loading="lazy">⁠. It follows therefore that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ae^{kx}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./df9cb5b25b423f3be01d26239c13b188f3a294c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.858ex; height:3.176ex;" alt="{\displaystyle f(x)=ae^{kx}}" loading="lazy">⁠ for some ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f059f053fcf9f421b7c74362cf3bd5ed024e19d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (d)=e^{kd}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (d)=e^{kd}.}</annotation>

</semantics>

</math><img src="./13265c7c41f9f586e66c87ce00c23b1c88484733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.323ex; height:3.176ex;" alt="{\displaystyle \varphi (d)=e^{kd}.}" loading="lazy">⁠ As a byproduct, one gets that

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

is independent of both ⁠<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 d}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">⁠.

<div class="mw-heading mw-heading2"><h2 id="Compound_interest">Compound interest</h2></div>

The earliest occurrence of the exponential function was in <a href="Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a>'s study of <a href="Compound_interest" title="Compound interest">compound interests</a> in 1683.<a href="#cite_note-O'Connor_2001-10">[10]</a>

This is this study that led Bernoulli to consider the number

</mrow>

</munder>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}</annotation>

</semantics>

</math>

now known as <a href="Euler's_number" class="mw-redirect" title="Euler's number">Euler's number</a> and denoted ⁠<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">⁠.

The exponential function is involved as follows in the computation of <a href="Compound_interest#Continuous_compounding" title="Compound interest">continuously compounded interests</a>.

If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is <style data-mw-deduplicate="TemplateStyles:r1214402035">

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

.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}

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</style>⁠x/12⁠ times the current value, so each month the total value is multiplied by (1 + ⁠x/12⁠), and the value at the end of the year is (1 + ⁠x/12⁠)12. If instead interest is compounded daily, this becomes (1 + ⁠x/365⁠)365. Letting the number of time intervals per year grow without bound leads to the <a href="Limit_of_a_function" title="Limit of a function">limit</a> definition of the exponential function,

</mrow>

</munder>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}}</annotation>

</semantics>

</math>

first given by <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>.<a href="#cite_note-Maor-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Differential_equations">Differential equations</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Linear_differential_equation" title="Linear differential equation">Linear differential equation</a></div>

Exponential functions occur very often in solutions of <a href="Differential_equation" title="Differential equation">differential equations</a>.

The exponential functions can be defined as solutions of <a href="Differential_equation" title="Differential equation">differential equations</a>. Indeed, the exponential function is a solution of the simplest possible differential equation, namely ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=y}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e6239f12a70a7f715303934acf9dbae208fceb80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.099ex; height:2.843ex;" alt="{\displaystyle y'=y}" loading="lazy">⁠. Every other exponential function, of the form ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=ab^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./83da71210200ba688ab4dcb3bf5ce469b04a64b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.654ex; height:2.676ex;" alt="{\displaystyle y=ab^{x}}" loading="lazy">⁠, is a solution of the differential equation ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=ky}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b10d87f9a74551a2e20a57eba7546792d1d529ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.31ex; height:2.843ex;" alt="{\displaystyle y'=ky}" loading="lazy">⁠, and every solution of this differential equation has this form.

The solutions of an equation of the form

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y'+ky=f(x)}</annotation>

</semantics>

</math>

involve exponential functions in a more sophisticated way, since they have the form

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=ce^{-kx}+e^{-kx}\int f(x)e^{kx}dx,}</annotation>

</semantics>

</math>

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

</mstyle>

</mrow>

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

</semantics>

</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">⁠ is an arbitrary constant and the integral denotes any <a href="Antiderivative" title="Antiderivative">antiderivative</a> of its argument.

More generally, the solutions of every linear differential equation with constant coefficients can be expressed in terms of exponential functions and, when they are not homogeneous, antiderivatives. This holds true also for systems of linear differential equations with constant coefficients.

<div class="mw-heading mw-heading2"><h2 id="Complex_exponential">Complex exponential</h2></div>

The exponential function can be naturally extended to a <a href="Complex_function" class="mw-redirect" title="Complex function">complex function</a>, which is a function with the <a href="Complex_number" title="Complex number">complex numbers</a> as <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Codomain" title="Codomain">codomain</a>, such that its <a href="Restriction_(mathematics)" title="Restriction (mathematics)">restriction</a> to the reals is the above-defined exponential function, called real exponential function in what follows. This function is also called the exponential function, and also denoted ⁠<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">⁠ or ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fc14dcff41f956163726d55369923012da262b3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.45ex; height:2.843ex;" alt="{\displaystyle \exp(z)}" loading="lazy">⁠. For distinguishing the complex case from the real one, the extended function is also called complex exponential function or simply complex exponential.

Most of the definitions of the exponential function can be used verbatim for definiting the complex exponential function, and the proof of their equivalence is the same as in the real case.

The complex exponential function can be defined in several equivalent ways that are the same as in the real case.

The complex exponential is the unique complex function that equals its <a href="Complex_derivative" class="mw-redirect" title="Complex derivative">complex derivative</a> and takes the value ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">⁠ for the argument ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">⁠:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {de^{z}}{dz}}=e^{z}\quad {\text{and}}\quad e^{0}=1.}</annotation>

</semantics>

</math>

The complex exponential function is the sum of the <a href="Series_(mathematics)" title="Series (mathematics)">series</a>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{z}=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}.}</annotation>

</semantics>

</math>

This series is <a href="Absolutely_convergent" class="mw-redirect" title="Absolutely convergent">absolutely convergent</a> for every complex number ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<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">⁠. So, the complex differential is an <a href="Entire_function" title="Entire function">entire function</a>.

The complex exponential function is the <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{z}=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}}</annotation>

</semantics>

</math>

The functional equation

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

holds for every complex numbers ⁠<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="./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">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<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 complex exponential is the unique <a href="Continuous_function" title="Continuous function">continuous function</a> that satisfies this functional equation and has the value ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">⁠ for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b92bfc06485cc90286474b14a516a68d8bfdd7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}" loading="lazy">⁠.

The <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> is a <a href="Left_inverse_function" class="mw-redirect" title="Left inverse function">right-inverse function </a> of the complex exponential:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

However, since the complex logarithm is a <a href="Multivalued_function" title="Multivalued function">multivalued function</a>, one has

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log e^{z}=\{z+2ik\pi \mid k\in \mathbb {Z} \},}</annotation>

</semantics>

</math>

and it is difficult to define the complex exponential from the complex logarithm. On the opposite, this is the complex logarithm that is often defined from the complex exponential.

The complex exponential has the following properties:

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

and

<msup>

</mrow>

</msup>

<mtext>for every </mtext>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{z}\neq 0\quad {\text{for every }}z\in \mathbb {C} .}</annotation>

</semantics>

</math>

It is <a href="Periodic_function" title="Periodic function">periodic function</a> of period ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2i\pi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c94604258102818cfe51a8048b00ab0e2c96829b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.297ex; height:2.176ex;" alt="{\displaystyle 2i\pi }" loading="lazy">⁠; that is

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mtext>for every </mtext>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{z+2ik\pi }=e^{z}\quad {\text{for every }}k\in \mathbb {Z} .}</annotation>

</semantics>

</math>

This results from <a href="Euler's_identity" title="Euler's identity">Euler's identity</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\pi }=-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./00ba628c5431f37a0be483c26111b1078c036f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.894ex; height:2.843ex;" alt="{\displaystyle e^{i\pi }=-1}" loading="lazy">⁠ and the functional identity.

The <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the complex exponential is

<mover>

<msup>

</mrow>

</msup>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

Its modulus is

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |e^{z}|=e^{|\Re (z)|},}</annotation>

</semantics>

</math>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2bec794fff1c9fd60700215775ff27f35fb0742a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle \Re (z)}" loading="lazy">⁠ denotes the real part of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<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">⁠.

<div class="mw-heading mw-heading3"><h3 id="Relationship_with_trigonometry">Relationship with trigonometry</h3></div>

Complex exponential and <a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> are strongly related by <a href="Euler's_formula" title="Euler's formula">Euler's formula</a>:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{it}=\cos(t)+i\sin(t).}</annotation>

</semantics>

</math>

This formula provides the decomposition of complex exponentials into <a href="Real_and_imaginary_parts" class="mw-redirect" title="Real and imaginary parts">real and imaginary parts</a>:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{x+iy}=e^{x}e^{iy}=e^{x}\,\cos y+ie^{x}\,\sin y.}</annotation>

</semantics>

</math>

The trigonometric functions can be expressed in terms of complex exponentials:

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos x&={\frac {e^{ix}+e^{-ix}}{2}}\\\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\\tan x&=i\,{\frac {1-e^{2ix}}{1+e^{2ix}}}\end{aligned}}}</annotation>

</semantics>

</math>

In these formulas, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,t}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./77a04fe169d6625eedb4e4db82339863f01f5ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.393ex; height:2.343ex;" alt="{\displaystyle x,y,t}" loading="lazy">⁠ are commonly interpreted as real variables, but the formulas remain valid if the variables are interpreted as complex variables. These formulas may be used to define trigonometric functions of a complex variable.<a href="#cite_note-Apostol_1974-11">[11]</a>

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

<li class="gallerycaption">3D plots of real part, imaginary part, and modulus of the exponential function</li>

</li>

</li>

</li>

</ul>

Considering the complex exponential function as a function involving four real variables:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v+iw=\exp(x+iy)}</annotation>

</semantics>

</math>

the graph of the exponential function is a two-dimensional surface curving through four dimensions.

Starting with a color-coded portion of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy"> domain, the following are depictions of the graph as variously projected into two or three dimensions.

<li class="gallerycaption">Graphs of the complex exponential function</li>

<div class="gallerytext">Checker board key: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0:\;{\text{green}}}">

<mtext>green</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x>0:\;{\text{green}}}</annotation>

</semantics>

</math><img src="./83eb3ab5abbf71592bc261cc64aa49e1c62f9f44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.604ex; height:2.509ex;" alt="{\displaystyle x>0:\;{\text{green}}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0:\;{\text{red}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<0:\;{\text{red}}}</annotation>

</semantics>

</math><img src="./a07c4dba54cfc32e9d541c8f1eb62e6c4fe0ea10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.409ex; height:2.176ex;" alt="{\displaystyle x<0:\;{\text{red}}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y>0:\;{\text{yellow}}}">

<mtext>yellow</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y>0:\;{\text{yellow}}}</annotation>

</semantics>

</math><img src="./62e9a44fdeda2fd5a9a0a2aeacd574fece140fb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.393ex; height:2.509ex;" alt="{\displaystyle y>0:\;{\text{yellow}}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<0:\;{\text{blue}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y<0:\;{\text{blue}}}</annotation>

</semantics>

</math><img src="./c629031aedfa69a5d3342154a1606503e4d1a26b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.263ex; height:2.509ex;" alt="{\displaystyle y<0:\;{\text{blue}}}" loading="lazy"></div>

</li>

<div class="gallerytext">Projection onto the range complex plane (V/W). Compare to the next, perspective picture.</div>

</li>

<div class="gallerytext">Projection into the <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">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">, and <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="./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"> dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image)</div>

</li>

<div class="gallerytext">Projection into the <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">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">, and <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="./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"> dimensions, producing a spiral shape (<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"> range extended to ±2π, again as 2-D perspective image)</div>

</li>

</ul>

The second image shows how the domain complex plane is mapped into the range complex plane:

<ul><li>zero is mapped to 1</li>

</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"> axis is mapped to the positive real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> axis</li>

<li>the imaginary <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"> axis is wrapped around the unit circle at a constant angular rate</li>

<li>values with negative real parts are mapped inside the unit circle</li>

<li>values with positive real parts are mapped outside of the unit circle</li>

<li>values with a constant real part are mapped to circles centered at zero</li>

<li>values with a constant imaginary part are mapped to rays extending from zero</li></ul>

The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image.

The third image shows the graph extended along the real <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"> axis. It shows the graph is a surface of revolution about the <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"> axis of the graph of the real exponential function, producing a horn or funnel shape.

The fourth image shows the graph extended along the imaginary <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"> axis. It shows that the graph's surface for positive and negative <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"> values doesn't really meet along the negative real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> axis, but instead forms a spiral surface about the <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"> axis. Because its <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"> values have been extended to ±2π, this image also better depicts the 2π periodicity in the imaginary <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"> value.

<div class="mw-heading mw-heading2"><h2 id="Matrices_and_Banach_algebras">Matrices and Banach algebras</h2></div>

The power series definition of the exponential function makes sense for square <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> (for which the function is called the <a href="Matrix_exponential" title="Matrix exponential">matrix exponential</a>) and more generally in any unital <a href="Banach_algebra" title="Banach algebra">Banach algebra</a> B. In this setting, e0 = 1, and ex is invertible with inverse e−x for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommuting x and y.

Some alternative definitions lead to the same function. For instance, ex can be defined as

</mrow>

</munder>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}</annotation>

</semantics>

</math>

Or ex can be defined as fx(1), where fx : R → B is the solution to the differential equation ⁠dfx/dt⁠(t) = x fx(t), with initial condition fx(0) = 1; it follows that fx(t) = etx for every t in R.

<div class="mw-heading mw-heading2"><h2 id="Lie_algebras">Lie algebras</h2></div>

Given a <a href="Lie_group" title="Lie group">Lie group</a> G and its associated <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">, the <a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">exponential map</a> is a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> ↦ G satisfying similar properties. In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group GL(n,R) of invertible n × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

The identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x+y)=\exp(x)\exp(y)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b87ab180d2b6339daeb4f3e5499a6f898b03a17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.381ex; height:2.843ex;" alt="{\displaystyle \exp(x+y)=\exp(x)\exp(y)}" loading="lazy"> can fail for Lie algebra elements x and y that do not commute; the <a href="Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff formula</a> supplies the necessary correction terms.

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

The function ez is a <a href="Transcendental_function" title="Transcendental function">transcendental function</a>, which means that it is not a <a href="Polynomial_root" class="mw-redirect" title="Polynomial root">root</a> of a polynomial over the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of the <a href="Rational_fraction" class="mw-redirect" title="Rational fraction">rational fractions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} (z).}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} (z).}</annotation>

</semantics>

</math><img src="./bc7e0d9c1259658cbe845de78e88a5f7876822b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.222ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} (z).}" loading="lazy">

If a1, ..., an are distinct complex numbers, then ea1z, ..., eanz are linearly independent over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} (z)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5917f17c3d7c66273521f830ef30a8950703188c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} (z)}" loading="lazy">, and hence ez is <a href="Transcendental_function" title="Transcendental function">transcendental</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} (z)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5917f17c3d7c66273521f830ef30a8950703188c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} (z)}" loading="lazy">.

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

The Taylor series definition above is generally efficient for computing (an approximation of) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./841c0d168e64191c45a45e54c7e447defd17ec6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}" loading="lazy">. However, when computing near the argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

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

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</math><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy">, the result will be close to 1, and computing the value of the difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}-1}">

<msup>

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

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</math><img src="./9098adf157cfb3697609aaa10d9e3b4ec875d1a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.259ex; height:2.509ex;" alt="{\displaystyle e^{x}-1}" loading="lazy"> with <a href="Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a> may lead to the loss of (possibly all) <a href="Significant_figures" title="Significant figures">significant figures</a>, producing a large relative error, possibly even a meaningless result.

Following a proposal by <a href="William_Kahan" title="William Kahan">William Kahan</a>, it may thus be useful to have a dedicated routine, often called <code>expm1</code>, which computes ex − 1 directly, bypassing computation of ex. For example,

one may use the Taylor series:

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<annotation encoding="application/x-tex">{\displaystyle e^{x}-1=x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+\cdots +{\frac {x^{n}}{n!}}+\cdots .}</annotation>

</semantics>

</math>

This was first implemented in 1979 in the <a href="Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a> <a href="HP-41C" title="HP-41C">HP-41C</a> calculator, and provided by several calculators,<a href="#cite_note-HP48_AUR-12">[12]</a><a href="#cite_note-HP50_AUR-13">[13]</a> <a href="Operating_system" title="Operating system">operating systems</a> (for example <a href="Berkeley_UNIX_4.3BSD" class="mw-redirect" title="Berkeley UNIX 4.3BSD">Berkeley UNIX 4.3BSD</a><a href="#cite_note-Beebe_2017-14">[14]</a>), <a href="Computer_algebra_system" title="Computer algebra system">computer algebra systems</a>, and programming languages (for example <a href="C99" title="C99">C99</a>).<a href="#cite_note-Beebe_2002-15">[15]</a>

In addition to base e, the <a href="IEEE_754-2008" class="mw-redirect" title="IEEE 754-2008">IEEE 754-2008</a> standard defines similar exponential functions near 0 for base 2 and 10: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{x}-1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 2^{x}-1}</annotation>

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</math><img src="./0b48590c4c77174ac7363888b9c434668358813f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.338ex; height:2.509ex;" alt="{\displaystyle 2^{x}-1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{x}-1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 10^{x}-1}</annotation>

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</math><img src="./347b67eb9e0e08c17f6528c275da798b0c2bb7b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.5ex; height:2.509ex;" alt="{\displaystyle 10^{x}-1}" loading="lazy">.

A similar approach has been used for the logarithm; see <a href="Log1p" class="mw-redirect" title="Log1p">log1p</a>.

An identity in terms of the <a href="Hyperbolic_tangent" class="mw-redirect" title="Hyperbolic tangent">hyperbolic tangent</a>,

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {expm1} (x)=e^{x}-1={\frac {2\tanh(x/2)}{1-\tanh(x/2)}},}</annotation>

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</math>

gives a high-precision value for small values of x on systems that do not implement expm1(x).

<div class="mw-heading mw-heading3"><h3 id="Continued_fractions">Continued fractions</h3></div>

The exponential function can also be computed with <a href="Continued_fraction" title="Continued fraction">continued fractions</a>.

A continued fraction for ex can be obtained via <a href="Euler's_continued_fraction_formula" title="Euler's continued fraction formula">an identity of Euler</a>:

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<annotation encoding="application/x-tex">{\displaystyle e^{x}=1+{\cfrac {x}{1-{\cfrac {x}{x+2-{\cfrac {2x}{x+3-{\cfrac {3x}{x+4-\ddots }}}}}}}}}</annotation>

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</math>

The following <a href="Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">generalized continued fraction</a> for ez converges more quickly:<a href="#cite_note-Lorentzen_2008-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle e^{z}=1+{\cfrac {2z}{2-z+{\cfrac {z^{2}}{6+{\cfrac {z^{2}}{10+{\cfrac {z^{2}}{14+\ddots }}}}}}}}}</annotation>

</semantics>

</math>

or, by applying the substitution z = ⁠x/y⁠:

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<annotation encoding="application/x-tex">{\displaystyle e^{\frac {x}{y}}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+\ddots }}}}}}}}}</annotation>

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with a special case for z = 2:

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<annotation encoding="application/x-tex">{\displaystyle e^{2}=1+{\cfrac {4}{0+{\cfrac {2^{2}}{6+{\cfrac {2^{2}}{10+{\cfrac {2^{2}}{14+\ddots }}}}}}}}=7+{\cfrac {2}{5+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{11+\ddots }}}}}}}}}</annotation>

</semantics>

</math>

This formula also converges, though more slowly, for z > 2. For example:

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<annotation encoding="application/x-tex">{\displaystyle e^{3}=1+{\cfrac {6}{-1+{\cfrac {3^{2}}{6+{\cfrac {3^{2}}{10+{\cfrac {3^{2}}{14+\ddots }}}}}}}}=13+{\cfrac {54}{7+{\cfrac {9}{14+{\cfrac {9}{18+{\cfrac {9}{22+\ddots }}}}}}}}}</annotation>

</semantics>

</math>

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<ul><li><a href="Carlitz_exponential" title="Carlitz exponential">Carlitz exponential</a>, a characteristic p analogue</li>

<li><a href="Double_exponential_function" title="Double exponential function">Double exponential function</a> – Exponential function of an exponential function</li>

<li><a href="Exponential_field" title="Exponential field">Exponential field</a> – Mathematical field with an extra operation</li>

<li><a href="Gaussian_function" title="Gaussian function">Gaussian function</a></li>

<li><a href="Half-exponential_function" title="Half-exponential function">Half-exponential function</a>, a compositional square root of an exponential function</li>

<li><a href="Lambert_W_function#Solving_equations" title="Lambert W function">Lambert W function#Solving equations</a> – Multivalued function in mathematics - Used for solving exponential equations</li>

<li><a href="List_of_exponential_topics" title="List of exponential topics">List of exponential topics</a></li>

<li><a href="List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">List of integrals of exponential functions</a></li>

<li><a href="Mittag-Leffler_function" title="Mittag-Leffler function">Mittag-Leffler function</a>, a generalization of the exponential function</li>

<li><a href="P-adic_exponential_function" title="P-adic exponential function">p-adic exponential function</a></li>

<li><a href="Pad%C3%A9_table_for_exponential_function" class="mw-redirect" title="Padé table for exponential function">Padé table for exponential function</a> – <a href="Pad%C3%A9_approximation" class="mw-redirect" title="Padé approximation">Padé approximation</a> of exponential function by a fraction of polynomial functions</li>

<li><a href="Phase_factor" title="Phase factor">Phase factor</a></li></ul>

</div>

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

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</div>

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

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

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/sets/function-exponential.html">"Exponential Function Reference"</a>. www.mathsisfun.com. Retrieved 2020-08-28.</cite>

</li>

<li id="cite_note-Rudin_1987-2"><a href="#cite_ref-Rudin_1987_2-0">^</a> <cite id="CITEREFRudin1987" class="citation book cs1">Rudin, Walter (1987). <a rel="nofollow" class="external text" href="https://archive.org/details/RudinW.RealAndComplexAnalysis3e1987">Real and complex analysis</a> (3rd ed.). New York: <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. p. 1. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054234-1</bdi>.</cite>

</li>

<li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ExponentialFunction.html">"Exponential Function"</a>. mathworld.wolfram.com. Retrieved 2020-08-28.</cite>

</li>

<li id="cite_note-Maor-4">^ <a href="#cite_ref-Maor_4-0">a</a> <a href="#cite_ref-Maor_4-1">b</a> <cite id="CITEREFMaor" class="citation book cs1"><a href="Eli_Maor" title="Eli Maor">Maor, Eli</a>. e: the Story of a Number. p. 156.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> G. Harnett, Calculus 1, 1998, Functions continued:

"General exponential functions have the property that the ratio of two outputs depends only on the difference of inputs. The ratio of outputs for a unit change in input is the base."

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> G. Harnett, Calculus 1, 1998; Functions continued / Exponentials & logarithms: "The ratio of outputs for a unit change in input is the base of a general exponential function."

</li>

<li id="cite_note-Serway-Moses-Moyer_1989-7"><a href="#cite_ref-Serway-Moses-Moyer_1989_7-0">^</a> <cite id="CITEREFSerwayMosesMoyer1989" class="citation book cs1">Serway, Raymond A.; Moses, Clement J.; Moyer, Curt A. (1989). Modern Physics. Fort Worth: <a href="Harcourt_Brace_Jovanovich" class="mw-redirect" title="Harcourt Brace Jovanovich">Harcourt Brace Jovanovich</a>. p. 384. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-03-004844-3</bdi>.</cite>

</li>

<li id="cite_note-Simmons_1972-8"><a href="#cite_ref-Simmons_1972_8-0">^</a> <cite id="CITEREFSimmons1972" class="citation book cs1"><a href="George_F._Simmons" title="George F. Simmons">Simmons, George F.</a> (1972). Differential Equations with Applications and Historical Notes. New York: <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. p. 15. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/75173716">75173716</a>.</cite>

</li>

<li id="cite_note-McGrawHill_2007-9"><a href="#cite_ref-McGrawHill_2007_9-0">^</a> <cite class="citation encyclopaedia cs1">McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. 2007. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-144143-8</bdi>.</cite>

</li>

<li id="cite_note-O'Connor_2001-10"><a href="#cite_ref-O'Connor_2001_10-0">^</a> <cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/e.html">"Exponential function"</a>, <a href="MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite>

</li>

<li id="cite_note-Apostol_1974-11"><a href="#cite_ref-Apostol_1974_11-0">^</a> <cite id="CITEREFApostol1974" class="citation book cs1">Apostol, Tom M. (1974). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalanal00apos_530">Mathematical Analysis</a> (2nd ed.). Reading, Mass.: <a href="Addison_Wesley" class="mw-redirect" title="Addison Wesley">Addison Wesley</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalanal00apos_530/page/n32">19</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-00288-1</bdi>.</cite>

</li>

<li id="cite_note-HP48_AUR-12"><a href="#cite_ref-HP48_AUR_12-0">^</a> <cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://www.hpcalc.org/details.php?id=6036">HP 48G Series – Advanced User's Reference Manual (AUR)</a> (4 ed.). <a href="Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a>. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2. Retrieved 2015-09-06.</cite>

</li>

<li id="cite_note-HP50_AUR-13"><a href="#cite_ref-HP50_AUR_13-0">^</a> <cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://www.hpcalc.org/details.php?id=7141">HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR)</a> (2 ed.). <a href="Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a>. 2009-07-14 [2005]. HP F2228-90010. Retrieved 2015-10-10.</cite> <a rel="nofollow" class="external autonumber" href="http://holyjoe.net/hp/HP_50g_AUR_v2_English_searchable.pdf">[1]</a>

</li>

<li id="cite_note-Beebe_2017-14"><a href="#cite_ref-Beebe_2017_14-0">^</a> <cite id="CITEREFBeebe2017" class="citation book cs1">Beebe, Nelson H. F. (2017-08-22). "Chapter 10.2. Exponential near zero". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA: <a href="Springer_International_Publishing_AG" class="mw-redirect" title="Springer International Publishing AG">Springer International Publishing AG</a>. pp. 273–282. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-64110-2">10.1007/978-3-319-64110-2</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-64109-6</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2017947446">2017947446</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:30244721">30244721</a>. <q>Berkeley UNIX 4.3BSD introduced the expm1() function in 1987.</q></cite>

</li>

<li id="cite_note-Beebe_2002-15"><a href="#cite_ref-Beebe_2002_15-0">^</a> <cite id="CITEREFBeebe2002" class="citation web cs1">Beebe, Nelson H. F. (2002-07-09). <a rel="nofollow" class="external text" href="http://www.math.utah.edu/~beebe/reports/expm1.pdf">"Computation of expm1 = exp(x)−1"</a> (PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. Retrieved 2015-11-02.</cite>

</li>

<li id="cite_note-Lorentzen_2008-16"><a href="#cite_ref-Lorentzen_2008_16-0">^</a> <cite id="CITEREFLorentzenWaadeland2008" class="citation book cs1"><a href="Lisa_Lorentzen" title="Lisa Lorentzen">Lorentzen, L.</a>; Waadeland, H. (2008). <a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/bbm%3A978-94-91216-37-4%2F1">"A.2.2 The exponential function."</a>. Continued Fractions. Atlantis Studies in Mathematics. Vol. 1. p. 268. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2991%2F978-94-91216-37-4">10.2991/978-94-91216-37-4</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-94-91216-37-4</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Exponential_function">"Exponential function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

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</style><div id="Calculus249" style="font-size:114%;margin:0 4em"><a href="Calculus" title="Calculus">Calculus</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Precalculus" title="Precalculus">Precalculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_theorem" title="Binomial theorem">Binomial theorem</a></li>

<li><a href="Concave_function" title="Concave function">Concave function</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous function</a></li>

<li><a href="Factorial" title="Factorial">Factorial</a></li>

<li><a href="Finite_difference" title="Finite difference">Finite difference</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li>

<li><a href="Graph_of_a_function" title="Graph of a function">Graph of a function</a></li>

<li><a href="Linear_function" title="Linear function">Linear function</a></li>

<li><a href="Radian" title="Radian">Radian</a></li>

<li><a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a></li>

<li><a href="Secant_line" title="Secant line">Secant</a></li>

<li><a href="Slope" title="Slope">Slope</a></li>

<li><a href="Tangent" title="Tangent">Tangent</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li>

<li><a href="Limit_of_a_function" title="Limit of a function">Limit of a function</a>

<ul><li><a href="One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li>

<li><a href="Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li>

<li><a href="Order_of_approximation" title="Order of approximation">Order of approximation</a></li>

<li><a href="(%CE%B5%2C_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Derivative" title="Derivative">Derivative</a></li>

<li><a href="Second_derivative" title="Second derivative">Second derivative</a></li>

<li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li>

<li><a href="Differential_operator" title="Differential operator">Differential operator</a></li>

<li><a href="Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li>

<li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation</a>

<ul><li><a href="Leibniz's_notation" title="Leibniz's notation">Leibniz's notation</a></li>

<li><a href="Newton's_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li>

<li><a href="Differentiation_rules" title="Differentiation rules">Rules of differentiation</a>

<ul><li><a href="Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li>

<li><a href="Power_rule" title="Power rule">Power</a></li>

<li><a href="Chain_rule" title="Chain rule">Chain</a></li>

<li><a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's</a></li>

<li><a href="Product_rule" title="Product rule">Product</a>

<ul><li><a href="General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li>

<li><a href="Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li>

<li>Other techniques

<ul><li><a href="Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li>

<li><a href="Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li>

<li><a href="Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li>

<li><a href="Related_rates" title="Related rates">Related rates</a></li></ul></li>

<li><a href="Stationary_point" title="Stationary point">Stationary points</a>

<ul><li><a href="First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li>

<li><a href="Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li>

<li><a href="Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li>

<li><a href="Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li>

<li>Further applications

<ul><li><a href="Newton's_method" title="Newton's method">Newton's method</a></li>

<li><a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li>

<li><a href="Differential_equation" title="Differential equation">Differential equation</a>

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Antiderivative" title="Antiderivative">Antiderivative</a></li>

<li><a href="Arc_length" title="Arc length">Arc length</a></li>

<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>

<li><a href="Integral#Properties" title="Integral">Basic properties</a></li>

<li><a href="Constant_of_integration" title="Constant of integration">Constant of integration</a></li>

<li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a>

<ul><li><a href="Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li>

<li><a href="Integration_by_parts" title="Integration by parts">Integration by parts</a></li>

<li><a href="Integration_by_substitution" title="Integration by substitution">Integration by substitution</a>

<ul><li><a href="Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li>

<li><a href="Euler_substitution" title="Euler substitution">Euler</a></li>

<li><a href="Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li>

<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a>

<ul><li><a href="Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li>

<li><a href="Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li>

<li>Volumes

<ul><li><a href="Disc_integration" title="Disc integration">Washer method</a></li>

<li><a href="Shell_integration" title="Shell integration">Shell method</a></li></ul></li>

<li><a href="Integral_equation" title="Integral equation">Integral equation</a></li>

<li><a href="Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Derivatives

<li><a href="Directional_derivative" title="Directional derivative">Directional derivative</a></li>

<li><a href="Divergence" title="Divergence">Divergence</a></li>

<li><a href="Gradient" title="Gradient">Gradient</a></li>

<li><a href="Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li>

<li>Basic theorems

<ul><li><a href="Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li>

<li><a href="Green's_theorem" title="Green's theorem">Green's</a></li>

<li><a href="Stokes'_theorem" title="Stokes' theorem">Stokes'</a></li>

<li><a href="Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li>

<li><a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li>

<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li>

<li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Line_integral" title="Line integral">Line integral</a></li>

<li><a href="Matrix_calculus" title="Matrix calculus">Matrix</a></li>

<li><a href="Multiple_integral" title="Multiple integral">Multiple integral</a></li>

<li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>

<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>

<li>Advanced topics

<ul><li><a href="Differential_form" title="Differential form">Differential forms</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Generalized_Stokes'_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li>

<li>Types of series

<ul><li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Binomial_series" title="Binomial series">Binomial</a></li>

<li><a href="Fourier_series" title="Fourier series">Fourier</a></li>

<li><a href="Geometric_series" title="Geometric series">Geometric</a></li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li>

<li><a href="Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li>

<li><a href="Power_series" title="Power series">Power</a>

<ul><li><a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor</a></li></ul></li>

<li><a href="Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li>

<li>Tests of convergence

<li><a href="Alternating_series_test" title="Alternating series test">Alternating series</a></li>

<li><a href="Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li>

<li><a href="Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li>

<li><a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet's</a></li>

<li><a href="Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li>

<li><a href="Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li>

<li><a href="Ratio_test" title="Ratio test">Ratio</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li>

<li><a href="E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li>

<li><a href="Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li>

<li><a href="Stirling's_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adequality" title="Adequality">Adequality</a></li>

<li><a href="Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li>

<li><a href="Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li>

<li><a href="Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li>

<li><a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li>

<li><a href="Infinitesimal" title="Infinitesimal">Infinitesimal</a></li>

<li><a href="Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li>

<li><a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li>

<li><a href="Fluxion" title="Fluxion">Fluxion</a></li>

<li><a href="Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></li>

<li><a href="The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li>

<li><a href="List_of_integrals_of_irrational_algebraic_functions" title="List of integrals of irrational algebraic functions">irrational algebraic functions</a></li>

<li><a href="List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li>

<li><a href="List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li>

<li><a href="List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a>

<ul><li><a href="List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li>

<li><a href="List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a>

<ul><li><a href="List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li>

<li><a href="Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li>

<li><a href="Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_limits" title="List of limits">List of limits</a></li>

<li><a href="Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Complex calculus

<ul><li><a href="Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li>

<li>Differential geometry

<ul><li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Curvature" title="Curvature">Curvature</a></li>

<li><a href="Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li>

<li><a href="Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li></ul></li>

<li><a href="Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li>

<li><a href="Gabriel's_horn" title="Gabriel's horn">Gabriel's horn</a></li>

<li><a href="Integration_Bee" title="Integration Bee">Integration Bee</a></li>

<li><a href="Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li>

<li><a href="Regiomontanus'_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li>

<li><a href="Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul>

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

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This article is issued from <a class="external text" title="Last edited on 2025-07-07" href="https://en.wikipedia.org/wiki/?title=Exponential_function&oldid=1299285929">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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