SherbyNet - /page/wikibook/entry.mu

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Sign function

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</style><div role="note" class="hatnote navigation-not-searchable">"Sgn" redirects here. For other uses, see <a href="SGN_(disambiguation)" class="mw-redirect mw-disambig" title="SGN (disambiguation)">SGN (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Sign_relation" title="Sign relation">Sign relation</a> or <a href="Sine_function" class="mw-redirect" title="Sine function">Sine function</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, the sign function or signum function (from <a href="https://en.wiktionary.org/wiki/signum#Latin" class="extiw external" title="wikt:signum">signum</a>, <a href="Latin_language" class="mw-redirect" title="Latin language">Latin</a> for "sign") is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that has the value −1, +1 or 0 according to whether the <a href="Sign_(mathematics)" title="Sign (mathematics)">sign</a> of a given <a href="Real_number" title="Real number">real number</a> is positive or negative, or the given number is itself zero. In <a href="Mathematical_notation" title="Mathematical notation">mathematical notation</a> the sign function is often represented as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} x}">

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

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</math><img src="./9fd35ed6a7df481b4d410de3073c392e9ea46048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.088ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} x}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(x)}">

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

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</math><img src="./69139aa98adb4695acf691e7a04060fcd40b255c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(x)}" loading="lazy">.<a href="#cite_note-:0-1">[1]</a>

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

The signum function of a real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is a <a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> function which is defined as follows:<a href="#cite_note-:0-1">[1]</a>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} x:={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} x:={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}</annotation>

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The <a href="Law_of_trichotomy" title="Law of trichotomy">law of trichotomy</a> states that every real number must be positive, negative or zero.

The signum function denotes which unique category a number falls into by mapping it to one of the values −1, +1 or 0, which can then be used in mathematical expressions or further calculations.

For example:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcr}\operatorname {sgn}(2)&=&+1\,,\\\operatorname {sgn}(\pi )&=&+1\,,\\\operatorname {sgn}(-8)&=&-1\,,\\\operatorname {sgn}(-{\frac {1}{2}})&=&-1\,,\\\operatorname {sgn}(0)&=&0\,.\end{array}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Basic_properties">Basic properties</h2></div>

Any real number can be expressed as the product of its <a href="Absolute_value" title="Absolute value">absolute value</a> and its sign:

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<annotation encoding="application/x-tex">{\displaystyle x=|x|\operatorname {sgn} x\,.}</annotation>

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It follows that whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is not equal to 0 we have

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} x={\frac {x}{|x|}}={\frac {|x|}{x}}\,.}</annotation>

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Similarly, for any real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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

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<annotation encoding="application/x-tex">{\displaystyle |x|=x\operatorname {sgn} x\,.}</annotation>

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We can also be certain that:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(xy)=(\operatorname {sgn} x)(\operatorname {sgn} y)\,,}</annotation>

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and so

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x^{n})=(\operatorname {sgn} x)^{n}\,.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Some_algebraic_identities">Some algebraic identities</h2></div>

The signum can also be written using the <a href="Iverson_bracket" title="Iverson bracket">Iverson bracket</a> notation:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} x=-[x<0]+[x>0]\,.}</annotation>

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The signum can also be written using the <a href="Floor_and_ceiling_functions" title="Floor and ceiling functions">floor</a> and the absolute value functions:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} x={\Biggl \lfloor }{\frac {x}{|x|+1}}{\Biggr \rfloor }-{\Biggl \lfloor }{\frac {-x}{|x|+1}}{\Biggr \rfloor }\,.}</annotation>

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

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</math><img src="./106f0c4e1cbccbfcbb61001a8c17b8427c65366d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 0^{0}}" loading="lazy"> is accepted to be equal to 1, the signum can also be written for all real numbers as

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} x=0^{\left(-x+\left\vert x\right\vert \right)}-0^{\left(x+\left\vert x\right\vert \right)}\,.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Properties_in_mathematical_analysis">Properties in mathematical analysis</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Discontinuity_at_zero">Discontinuity at zero</h3></div>

Although the sign function takes the value −1 when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is negative, the ringed point (0, −1) in the plot of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} x}">

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

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</math><img src="./9fd35ed6a7df481b4d410de3073c392e9ea46048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.088ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} x}" loading="lazy"> indicates that this is not the case when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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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">. Instead, the value jumps abruptly to the solid point at (0, 0) where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(0)=0}">

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

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</math><img src="./c095a0ab1cc3d18b9f13200051ce38eed164e439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.604ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(0)=0}" loading="lazy">. There is then a similar jump to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(x)=+1}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x)=+1}</annotation>

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</math><img src="./82540e8753e72defeac3434e3cde6f9ca24d7366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.579ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(x)=+1}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is positive. Either jump demonstrates visually that the sign function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} x}">

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

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</math><img src="./9fd35ed6a7df481b4d410de3073c392e9ea46048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.088ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} x}" loading="lazy"> is discontinuous at zero, even though it is continuous at any point where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is either positive or negative.

These observations are confirmed by any of the various equivalent formal definitions of <a href="Continuous_function" title="Continuous function">continuity</a> in <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>. A function <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">, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(x),}">

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

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</math><img src="./faba2389093097f6e742f60e2d32b07a82adfb57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.157ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(x),}" loading="lazy"> is continuous at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

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</math><img src="./aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy"> if the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)}">

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</math><img src="./368cb4b81ba5754d7a354a4ce49c2f1084bdaace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.318ex; height:2.843ex;" alt="{\displaystyle f(a)}" loading="lazy"> can be approximated arbitrarily closely by the <a href="Sequence" title="Sequence">sequence</a> of values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a_{1}),f(a_{2}),f(a_{3}),\dots ,}">

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> becomes sufficiently large. In the notation of mathematical <a href="Limit_of_a_sequence" title="Limit of a sequence">limits</a>, continuity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</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"> requires that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a_{n})\to f(a)}">

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

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</math><img src="./d4eb6f33d26476a2dedee0f4e0fad510eb494053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.468ex; height:2.843ex;" alt="{\displaystyle f(a_{n})\to f(a)}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }">

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</math><img src="./a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }" loading="lazy"> for any sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a_{n}\right)_{n=1}^{\infty }}">

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

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</math><img src="./a314effdc3e5893a25f6af5c599359734f71e576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.577ex; height:3.176ex;" alt="{\displaystyle \left(a_{n}\right)_{n=1}^{\infty }}" loading="lazy"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\to a.}">

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

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</math><img src="./ef89cc659ec8d3161bfc97ac9becb0f97b3929b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.939ex; height:2.176ex;" alt="{\displaystyle a_{n}\to a.}" loading="lazy"> The arrow symbol can be read to mean approaches, or tends to, and it applies to the sequence as a whole.

This criterion fails for the sign function at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}">

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

</semantics>

</math><img src="./90d476e5e765a5d77bbcff32e4584579207ec7d8.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=0}" loading="lazy">. For example, we can choose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./790f9209748c2dca7ed7b81932c37c02af1dbc31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" loading="lazy"> to be the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\dots ,}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\dots ,}</annotation>

</semantics>

</math><img src="./56cdbf7902519eff7846fc5da75ade5108a2b608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.03ex; height:3.676ex;" alt="{\displaystyle 1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\dots ,}" loading="lazy"> which tends towards zero as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> increases towards infinity. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\to a}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1215cc19c3c7ee14a4e419478d7ac21cea8c2f2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.292ex; height:2.176ex;" alt="{\displaystyle a_{n}\to a}" loading="lazy"> as required, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(a)=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b98191399187bf434fadd85418a88ef681bf28b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.671ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(a)=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(a_{n})=+1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(a_{n})=+1}</annotation>

</semantics>

</math><img src="./a75f806fc4f27b0051bff6e96ba425cd48f625fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.698ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(a_{n})=+1}" loading="lazy"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./397bfafc701afdf14c2743278a097f6f2957eabb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}" loading="lazy"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(a_{n})\to 1\neq \operatorname {sgn}(a)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(a_{n})\to 1\neq \operatorname {sgn}(a)}</annotation>

</semantics>

</math><img src="./742af6733fe02413b89133462f4491bfebb5ff72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.914ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(a_{n})\to 1\neq \operatorname {sgn}(a)}" loading="lazy">. This counterexample confirms more formally the discontinuity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fd35ed6a7df481b4d410de3073c392e9ea46048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.088ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} x}" loading="lazy"> at zero that is visible in the plot.

Despite the sign function having a very simple form, the step change at zero causes difficulties for traditional <a href="Calculus" title="Calculus">calculus</a> techniques, which are quite stringent in their requirements. Continuity is a frequent constraint. One solution can be to approximate the sign function by a smooth continuous function; others might involve less stringent approaches that build on classical methods to accommodate larger classes of function.

<div class="mw-heading mw-heading3"><h3 id="Smooth_approximations_and_limits">Smooth approximations and limits</h3></div>

The signum function can be given as a number of different (pointwise) limits:

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</munder>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</munder>

<mfrac>

</mfrac>

</mrow>

<mi>arctan</mi>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</munder>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</munder>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {sgn} x&=\lim _{n\to \infty }{\frac {1-2^{-nx}}{1+2^{-nx}}}\\&=\lim _{n\to \infty }{\frac {2}{\pi }}\operatorname {arctan} (nx)\\&=\lim _{n\to \infty }\tanh(nx)\\&=\lim _{\varepsilon \to 0}{\frac {x}{\sqrt {x^{2}+\varepsilon ^{2}}}}.\end{aligned}}}</annotation>

</semantics>

</math>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b1494d5d222bc80846ce45e804a38b0ea37e846b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.652ex; height:2.176ex;" alt="{\displaystyle \tanh }" loading="lazy"> is the <a href="Hyperbolic_tangent" class="mw-redirect" title="Hyperbolic tangent">hyperbolic tangent</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arctan} }">

<mi>arctan</mi>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b08e2b273fb2a8b4a4da5993fccca51f994cf80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.466ex; height:2.009ex;" alt="{\displaystyle \operatorname {arctan} }" loading="lazy"> is the <a href="Arctan" class="mw-redirect" title="Arctan">inverse tangent</a>. The last of these is the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x^{2}+\varepsilon ^{2}}}}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aeeae0ce0dc2cdc63910513da86aac9da4795441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:3.509ex;" alt="{\displaystyle {\sqrt {x^{2}+\varepsilon ^{2}}}}" loading="lazy">. This is inspired from the fact that the above is exactly equal for all nonzero <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"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0cbe653d369abd71fc0be98efbf59e9edc1bdfb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon =0}" loading="lazy">, and has the advantage of simple generalization to higher-dimensional analogues of the sign function (for example, the partial derivatives of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x^{2}+y^{2}}}}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bde512f387a0677a158fd76d3f558ed8b55b3ba1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.763ex; height:4.843ex;" alt="{\displaystyle {\sqrt {x^{2}+y^{2}}}}" loading="lazy">).

See <a href="Heaviside_step_function#Analytic_approximations" title="Heaviside step function">Heaviside step function § Analytic approximations</a>.

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

The signum function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fd35ed6a7df481b4d410de3073c392e9ea46048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.088ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} x}" loading="lazy"> is <a href="Differentiable_function" title="Differentiable function">differentiable</a> everywhere except when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./71320d636c7a43546bf4e94edb94649c5b2e82b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.237ex; height:2.176ex;" alt="{\displaystyle x=0.}" loading="lazy"> Its <a href="Derivative" title="Derivative">derivative</a> is zero when <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"> is non-zero:

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {{\text{d}}\,(\operatorname {sgn} x)}{{\text{d}}x}}=0\qquad {\text{for }}x\neq 0\,.}</annotation>

</semantics>

</math>

This follows from the differentiability of any <a href="Constant_function" title="Constant function">constant function</a>, for which the derivative is always zero on its domain of definition. The signum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fd35ed6a7df481b4d410de3073c392e9ea46048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.088ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} x}" loading="lazy"> acts as a constant function when it is restricted to the negative <a href="Interval_(mathematics)#Definitions_and_terminology" title="Interval (mathematics)">open region</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./878f7bcba151597923cfab9a9f13c880cbae59cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x<0,}" loading="lazy"> where it equals −1. It can similarly be regarded as a constant function within the positive open region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b4c8d8607cfd12cb95feef5a2517f4d8aa82ab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x>0,}" loading="lazy"> where the corresponding constant is +1. Although these are two different constant functions, their derivative is equal to zero in each case.

It is not possible to define a classical derivative at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

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

</semantics>

</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">, because there is a discontinuity there.

Although it is not differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

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

</semantics>

</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"> in the ordinary sense, under the generalized notion of differentiation in <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distribution theory</a>,

the derivative of the signum function is two times the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. This can be demonstrated using the identity <a href="#cite_note-2">[2]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} x=2H(x)-1\,,}</annotation>

</semantics>

</math>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./11f241aa7195bebab9d0a3c248ea97ef0c78b1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle H(x)}" loading="lazy"> is the <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> using the standard <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(0)={\frac {1}{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(0)={\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./9e9e52bd1c68460c797efcb120e5d3822c9d856d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.132ex; height:5.176ex;" alt="{\displaystyle H(0)={\frac {1}{2}}}" loading="lazy"> formalism.

Using this identity, it is easy to derive the distributional derivative:<a href="#cite_note-3">[3]</a>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {{\text{d}}\operatorname {sgn} x}{{\text{d}}x}}=2{\frac {{\text{d}}H(x)}{{\text{d}}x}}=2\delta (x)\,.}</annotation>

</semantics>

</math>

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

The signum function has a <a href="Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a> between any pair of finite values a and b, even when the interval of integration includes zero. The resulting integral for a and b is then equal to the difference between their absolute values:

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}(\operatorname {sgn} x)\,{\text{d}}x=|b|-|a|\,.}</annotation>

</semantics>

</math>

In fact, the signum function is the derivative of the absolute value function, except where there is an abrupt change in <a href="Slope" title="Slope">gradient</a> at zero:

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {{\text{d}}|x|}{{\text{d}}x}}=\operatorname {sgn} x\qquad {\text{for }}x\neq 0\,.}</annotation>

</semantics>

</math>

We can understand this as before by considering the definition of the absolute value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4eb41e5fd5dc37eaa1718dfbf4bc082edb991936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle |x|}" loading="lazy"> on the separate regions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./80d24be5f0eb4a9173da6038badc8659546021d0.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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eae2426ff09e1cd53f1658b8949e8dbd0fd8ad04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.237ex; height:2.176ex;" alt="{\displaystyle x<0.}" loading="lazy"> For example, the absolute value function is identical to <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"> in the region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b4c8d8607cfd12cb95feef5a2517f4d8aa82ab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x>0,}" loading="lazy"> whose derivative is the constant value +1, which equals the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fd35ed6a7df481b4d410de3073c392e9ea46048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.088ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} x}" loading="lazy"> there.

Because the absolute value is a <a href="Convex_function" title="Convex function">convex function</a>, there is at least one <a href="Subderivative" title="Subderivative">subderivative</a> at every point, including at the origin. Everywhere except zero, the resulting <a href="Subdifferential" class="mw-redirect" title="Subdifferential">subdifferential</a> consists of a single value, equal to the value of the sign function. In contrast, there are many subderivatives at zero, with just one of them taking the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(0)=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c095a0ab1cc3d18b9f13200051ce38eed164e439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.604ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(0)=0}" loading="lazy">. A subderivative value 0 occurs here because the absolute value function is at a minimum. The full family of valid subderivatives at zero constitutes the subdifferential interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./51e3b7f14a6f70e614728c583409a0b9a8b9de01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}" loading="lazy">, which might be thought of informally as "filling in" the graph of the sign function with a vertical line through the origin, making it continuous as a two dimensional curve.

In integration theory, the signum function is a <a href="Weak_derivative" title="Weak derivative">weak derivative</a> of the absolute value function. Weak derivatives are equivalent if they are equal <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>, making them impervious to isolated anomalies at a single point. This includes the change in gradient of the absolute value function at zero, which prohibits there being a classical derivative.

<div class="mw-heading mw-heading3"><h3 id="Fourier_transform">Fourier transform</h3></div>

The <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the signum function is<a href="#cite_note-4">[4]</a>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PV\int _{-\infty }^{\infty }(\operatorname {sgn} x)e^{-ikx}{\text{d}}x={\frac {2}{ik}}\qquad {\text{for }}k\neq 0,}</annotation>

</semantics>

</math>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8b6ee9fce2e1b90f60d71f454bbfcee9ea3bf0d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.533ex; height:2.176ex;" alt="{\displaystyle PV}" loading="lazy"> means taking the <a href="Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a>.

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

<div class="mw-heading mw-heading3"><h3 id="Complex_signum">Complex signum</h3></div>

The signum function can be generalized to <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> as:

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math>

for any 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"> except <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 signum of a given 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"> is the <a href="Point_(geometry)" title="Point (geometry)">point</a> on the <a href="Unit_circle" title="Unit circle">unit circle</a> of the <a href="Complex_plane" title="Complex plane">complex plane</a> that is nearest to <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">. Then, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3b8b7eb2d2a30057811a7835502717d3d6ece962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.676ex;" alt="{\displaystyle z\neq 0}" loading="lazy">,

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} z=e^{i\arg z}\,,}</annotation>

</semantics>

</math>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec03a9c123925f400a40064ca491d268f9312956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.237ex; height:2.009ex;" alt="{\displaystyle \arg }" loading="lazy"> is the <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">complex argument function</a>.

For reasons of symmetry, and to keep this a proper generalization of the signum function on the reals, also in the complex domain one usually defines, 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">:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(0+0i)=0}</annotation>

</semantics>

</math>

Another generalization of the sign function for real and complex expressions is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{csgn}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d883d826b37498543c6939b0ef861fef3de88492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.404ex; height:2.009ex;" alt="{\displaystyle {\text{csgn}}}" loading="lazy">,<a href="#cite_note-5">[5]</a> which is defined as:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {csgn} z={\begin{cases}1&{\text{if }}\mathrm {Re} (z)>0,\\-1&{\text{if }}\mathrm {Re} (z)<0,\\\operatorname {sgn} \mathrm {Im} (z)&{\text{if }}\mathrm {Re} (z)=0\end{cases}}}">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {csgn} z={\begin{cases}1&{\text{if }}\mathrm {Re} (z)>0,\\-1&{\text{if }}\mathrm {Re} (z)<0,\\\operatorname {sgn} \mathrm {Im} (z)&{\text{if }}\mathrm {Re} (z)=0\end{cases}}}</annotation>

</semantics>

</math>

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

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5c8f911257677a06c782dc7fc57e52d5f59ea41a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.64ex; height:2.843ex;" alt="{\displaystyle {\text{Re}}(z)}" loading="lazy"> is 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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Im}}(z)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b3b0880160d0c3862617cea3a506a50f26b367ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.673ex; height:2.843ex;" alt="{\displaystyle {\text{Im}}(z)}" loading="lazy"> is the imaginary 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">.

We then have (for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3b8b7eb2d2a30057811a7835502717d3d6ece962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.676ex;" alt="{\displaystyle z\neq 0}" loading="lazy">):

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {csgn} z={\frac {z}{\sqrt {z^{2}}}}={\frac {\sqrt {z^{2}}}{z}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Polar_decomposition_of_matrices">Polar decomposition of matrices</h3></div>

Thanks to the <a href="Polar_decomposition" title="Polar decomposition">Polar decomposition</a> theorem, a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}\in \mathbb {K} ^{n\times n}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}\in \mathbb {K} ^{n\times n}}</annotation>

</semantics>

</math><img src="./9aa13dd87fb63ac0e9fe714a881f5369255e78b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.152ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {A}}\in \mathbb {K} ^{n\times n}}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4d1dfbdbc9f282a890a4539ce701c2029c1e1820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.364ex; height:2.843ex;" alt="{\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}" loading="lazy">) can be decomposed as a product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {Q}}{\boldsymbol {P}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7d24a3d8d0074eaeb9ea8fe103c6ff32ad97a159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.988ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {Q}}{\boldsymbol {P}}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {Q}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./94cf8183e844214cfa05d23bf2c0403a4820855e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.019ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {Q}}}" loading="lazy"> is a unitary matrix and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {P}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5d6351e321b7d5cb9e071bb996fd1838060725b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.968ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {P}}}" loading="lazy"> is a self-adjoint, or Hermitian, positive definite matrix, both in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ^{n\times n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./883a3248d77195b40cee4d8c5a3b0c9cd9a0c361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.291ex; height:2.343ex;" alt="{\displaystyle \mathbb {K} ^{n\times n}}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5a8b5a6d1dbadead8b1dc48719c888e6cac5f861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {A}}}" loading="lazy"> is invertible then such a decomposition is unique and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {Q}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./94cf8183e844214cfa05d23bf2c0403a4820855e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.019ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {Q}}}" loading="lazy"> plays the role of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5a8b5a6d1dbadead8b1dc48719c888e6cac5f861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {A}}}" loading="lazy">'s signum. A dual construction is given by the decomposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}={\boldsymbol {S}}{\boldsymbol {R}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}={\boldsymbol {S}}{\boldsymbol {R}}}</annotation>

</semantics>

</math><img src="./907cca71445db0876a2d86850c4b45924621674a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.825ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {A}}={\boldsymbol {S}}{\boldsymbol {R}}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7945da372695d49cd4229e2a84ac6dc8ae6c99b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.047ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {R}}}" loading="lazy"> is unitary, but generally different than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {Q}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./94cf8183e844214cfa05d23bf2c0403a4820855e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.019ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {Q}}}" loading="lazy">. This leads to each invertible matrix having a unique left-signum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {Q}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./94cf8183e844214cfa05d23bf2c0403a4820855e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.019ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {Q}}}" loading="lazy"> and right-signum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7945da372695d49cd4229e2a84ac6dc8ae6c99b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.047ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {R}}}" loading="lazy">.

In the special case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} =\mathbb {R} ,\ n=2,}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {K} =\mathbb {R} ,\ n=2,}</annotation>

</semantics>

</math><img src="./8dbaa06973977b8a103d7906c472c141d35279e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.502ex; height:2.509ex;" alt="{\displaystyle \mathbb {K} =\mathbb {R} ,\ n=2,}" loading="lazy"> and the (invertible) matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}=\left[{\begin{array}{rr}a&-b\\b&a\end{array}}\right]}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}=\left[{\begin{array}{rr}a&-b\\b&a\end{array}}\right]}</annotation>

</semantics>

</math><img src="./11d1907b0fe28351347d9ba7b63a33549a72a636.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.682ex; height:6.176ex;" alt="{\displaystyle {\boldsymbol {A}}=\left[{\begin{array}{rr}a&-b\\b&a\end{array}}\right]}" loading="lazy">, which identifies with the (nonzero) complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+\mathrm {i} b=c}">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./2e615e264b56315d0b8d10eeb57c8a60eae18cf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.82ex; height:2.343ex;" alt="{\displaystyle a+\mathrm {i} b=c}" loading="lazy">, then the signum matrices satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {Q}}={\boldsymbol {P}}=\left[{\begin{array}{rr}a&-b\\b&a\end{array}}\right]/|c|}">

</mrow>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {Q}}={\boldsymbol {P}}=\left[{\begin{array}{rr}a&-b\\b&a\end{array}}\right]/|c|}</annotation>

</semantics>

</math><img src="./d491d90e057f67de0f0b15d29bc01ff4fa58d73d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.599ex; height:6.176ex;" alt="{\displaystyle {\boldsymbol {Q}}={\boldsymbol {P}}=\left[{\begin{array}{rr}a&-b\\b&a\end{array}}\right]/|c|}" loading="lazy"> and identify with the complex signum of <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">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} c=c/|c|}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./710cc334f9e283fd9a56fdfe130bbf98181292e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.333ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn} c=c/|c|}" loading="lazy">. In this sense, polar decomposition generalizes to matrices the signum-modulus decomposition of complex numbers.

<div class="mw-heading mw-heading3"><h3 id="Signum_as_a_generalized_function">Signum as a generalized function</h3></div>

At real values 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">, it is possible to define a <a href="Generalized_function" title="Generalized function">generalized function</a>–version of the signum function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b34a99af9d60f2d9770f5fe5f997b7bb11531ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.222ex; height:2.843ex;" alt="{\displaystyle \varepsilon (x)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (x)^{2}=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4567b4cdbab8cae4c28b584cabdd5f60da813fac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.538ex; height:3.176ex;" alt="{\displaystyle \varepsilon (x)^{2}=1}" loading="lazy"> everywhere, including at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

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

</semantics>

</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">, unlike <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec838dfd8a4a659b2877f93a6b53f22fc7777d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} }" loading="lazy">, for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {sgn} 0)^{2}=0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dbd5920c76506b6748662397e44437941e13230d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.045ex; height:3.176ex;" alt="{\displaystyle (\operatorname {sgn} 0)^{2}=0}" loading="lazy">. This generalized signum allows construction of the <a href="Algebra_of_generalized_functions" class="mw-redirect" title="Algebra of generalized functions">algebra of generalized functions</a>, but the price of such generalization is the loss of <a href="Commutativity" class="mw-redirect" title="Commutativity">commutativity</a>. In particular, the generalized signum anticommutes with the Dirac delta function<a href="#cite_note-Algebra-6">[6]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon (x)\delta (x)+\delta (x)\varepsilon (x)=0\,;}</annotation>

</semantics>

</math>

in addition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b34a99af9d60f2d9770f5fe5f997b7bb11531ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.222ex; height:2.843ex;" alt="{\displaystyle \varepsilon (x)}" loading="lazy"> cannot be evaluated at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

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

</semantics>

</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">; and the special name, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> is necessary to distinguish it from the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec838dfd8a4a659b2877f93a6b53f22fc7777d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} }" loading="lazy">. (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (0)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3df6963ea793fa56e699a9f1e0157d34803b8143.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.055ex; height:2.843ex;" alt="{\displaystyle \varepsilon (0)}" loading="lazy"> is not defined, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} 0=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e2c2101121dfaef50858718a6c4a214975e2ffb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.182ex; height:2.509ex;" alt="{\displaystyle \operatorname {sgn} 0=0}" loading="lazy">.)

<ul><li><a href="Absolute_value" title="Absolute value">Absolute value</a></li>

<li><a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a></li>

<li><a href="Negative_number" title="Negative number">Negative number</a></li>

<li><a href="Rectangular_function" title="Rectangular function">Rectangular function</a></li>

<li><a href="Sigmoid_function" title="Sigmoid function">Sigmoid function</a> (<a href="Hard_sigmoid" title="Hard sigmoid">Hard sigmoid</a>)</li>

<li><a href="Step_function" title="Step function">Step function</a> (<a href="Piecewise_constant_function" class="mw-redirect" title="Piecewise constant function">Piecewise constant function</a>)</li>

<li><a href="Three-way_comparison" title="Three-way comparison">Three-way comparison</a></li>

<li><a href="Zero_crossing" title="Zero crossing">Zero crossing</a></li>

<li><a href="Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li></ul>

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.maeckes.nl/Signum%20functie%20GB.html">"Signum function - Maeckes"</a>. www.maeckes.nl.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Sign.html">"Sign"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HeavisideStepFunction.html">"Heaviside Step Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFBurrowsColwell1990" class="citation journal cs1">Burrows, B. L.; Colwell, D. J. (1990). "The Fourier transform of the unit step function". International Journal of Mathematical Education in Science and Technology. 21 (4): 629–635. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0020739900210418">10.1080/0020739900210418</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Maple V documentation. May 21, 1998

</li>

<cite id="CITEREFYu.M.Shirokov1979" class="citation journal cs1">Yu.M.Shirokov (1979). <a rel="nofollow" class="external text" href="https://archive.today/20121208232109/http://springerlink.metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3">"Algebra of one-dimensional generalized functions"</a>. <a href="Theoretical_and_Mathematical_Physics" title="Theoretical and Mathematical Physics">Theoretical and Mathematical Physics</a>. 39 (3): 471–477. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979TMP....39..471S">1979TMP....39..471S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01017992">10.1007/BF01017992</a>. Archived from <a rel="nofollow" class="external text" href="http://springerlink.metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3">the original</a> on 2012-12-08.</cite>

</li>

</ol></div></div><div><div class="zim-footer">

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