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<title>Fehlerfunktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Fehlerfunktion</span></h1>

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<p>Als <b>Fehlerfunktion</b> oder <i>Gaußsche Fehlerfunktion</i> bezeichnet man in der Theorie der <a href="Spezielle_Funktion" title="Spezielle Funktion">speziellen Funktionen</a> die durch das Integral

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,\mathrm {d} t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e659db8ac65279291e40733a160d895de0a9d98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.947ex; height:6.343ex;" alt="{\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,\mathrm {d} t}" loading="lazy"></span></dd></dl>

<p>definierte Funktion.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Damit ist die Fehlerfunktion eine <a href="Stammfunktion" title="Stammfunktion">Stammfunktion</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{\sqrt {\pi }}}e^{-t^{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{\sqrt {\pi }}}e^{-t^{2}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d804f52c7bee6d419f673a751c59423ed22bbeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.166ex; height:4.509ex;" alt="{\displaystyle {\tfrac {2}{\sqrt {\pi }}}e^{-t^{2}}}" loading="lazy"></span>, und zwar die einzige ungerade (gerade Funktionen mit Stammfunktion besitzen genau eine ungerade solche).

</p><p>Für ein reelles Argument <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a8ee8591a5cc9057c9c646d1d07b848c1d68e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.153ex; width:2.809ex; height:2.176ex;" alt="{\displaystyle \operatorname {erf} }" loading="lazy"></span> eine <a href="Reellwertige_Funktion" title="Reellwertige Funktion">reellwertige Funktion</a>; zur Verallgemeinerung auf komplexe Argumente <a href="#Komplexe_Fehlerfunktion">siehe unten</a>.

</p><p>Die Fehlerfunktion ist eine <a href="Sigmoidfunktion" title="Sigmoidfunktion">Sigmoidfunktion</a>, findet Anwendung in der <a href="Statistik" title="Statistik">Statistik</a> und in der Theorie der <a href="Partielle_Differentialgleichung" title="Partielle Differentialgleichung">partiellen Differentialgleichungen</a> und hängt eng mit dem <a href="Fehlerintegral" title="Fehlerintegral">Fehlerintegral</a> zusammen.

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

<p>Die Bezeichnung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textrm {erf}}(x)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab77b37f475b60ee971325debc064b02e2b86596.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.795ex; height:2.843ex;" alt="{\displaystyle {\textrm {erf}}(x)}" loading="lazy"></span> kommt von <i><b>er</b>ror <b>f</b>unction</i>.

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<div class="mw-heading mw-heading3"><h3 id="Komplementäre_Fehlerfunktion"><span id="Komplement.C3.A4re_Fehlerfunktion"></span>Komplementäre Fehlerfunktion</h3></div>

<p>Die <i>komplementäre</i> (bzw. <i>konjugierte</i>) <i>Fehlerfunktion</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfc} (x)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ce4532c4ac939dbe7ef6d2f1fee55f0978847cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.827ex; height:2.843ex;" alt="{\displaystyle \operatorname {erfc} (x)}" loading="lazy"></span> ist gegeben durch:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x)=1-\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-t^{2}}\,\mathrm {d} t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fe0199856c66c50826618a7024b8e5dcc598a8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.578ex; height:6.343ex;" alt="{\displaystyle \operatorname {erfc} (x)=1-\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-t^{2}}\,\mathrm {d} t}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Verallgemeinerte_Fehlerfunktion">Verallgemeinerte Fehlerfunktion</h3></div>

<p>Die <i>verallgemeinerte Fehlerfunktion</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} (a,b)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dba5239aaede16d58a9f951f4907a16f703f4bfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.726ex; height:2.843ex;" alt="{\displaystyle \operatorname {erf} (a,b)}" loading="lazy"></span> wird durch das <a href="Integralrechnung" title="Integralrechnung">Integral</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (a,b)={\frac {2}{\sqrt {\pi }}}\int _{a}^{b}e^{-t^{2}}\,\mathrm {d} t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01fd7252e2388a4bc971ac5ee5e6931c4cc07fe3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.644ex; height:6.843ex;" alt="{\displaystyle \operatorname {erf} (a,b)={\frac {2}{\sqrt {\pi }}}\int _{a}^{b}e^{-t^{2}}\,\mathrm {d} t}" loading="lazy"></span></dd></dl>

<p>definiert.

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

<p>Es gilt:

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0297c7f9a8514e7885ad07b340ded13929c8f234.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.823ex; height:2.843ex;" alt="{\displaystyle \operatorname {erf} (a,b)=\operatorname {erf} (b)-\operatorname {erf} (a)}" loading="lazy"></span></dd></dl>

<p>Die Fehlerfunktion ist <a href="Gerade_und_ungerade_Funktionen" title="Gerade und ungerade Funktionen">ungerade</a>:

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a966a5b6e70a5d47638563ea227f028d7ccb87bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.692ex; height:2.843ex;" alt="{\displaystyle \operatorname {erf} (-x)=-\operatorname {erf} (x)}" loading="lazy"></span></dd></dl>

<p>Das uneigentliche Integral von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"></span> bis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy"></span> ist

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\sqrt {\pi }}}\int _{-\infty }^{+\infty }e^{-t^{2}}\,\mathrm {d} t=2}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8fd6d4970324c84f570fb35755d1da7f4dceb78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.296ex; height:6.509ex;" alt="{\displaystyle {\frac {2}{\sqrt {\pi }}}\int _{-\infty }^{+\infty }e^{-t^{2}}\,\mathrm {d} t=2}" loading="lazy"></span></dd></dl>

<p>Außerdem gilt:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)^{2}={\frac {4}{\pi }}\int _{0}^{1}{\frac {1-\exp[-x^{2}(y^{2}+1)]}{y^{2}+1}}\mathrm {d} y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12945ca7903dbb5803328b1c5139cabfc79d9ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.76ex; height:6.509ex;" alt="{\displaystyle \operatorname {erf} (x)^{2}={\frac {4}{\pi }}\int _{0}^{1}{\frac {1-\exp[-x^{2}(y^{2}+1)]}{y^{2}+1}}\mathrm {d} y}" loading="lazy"></span></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Verwandtschaft_mit_der_Normalverteilung">Verwandtschaft mit der Normalverteilung</h3></div>

<p>Die Fehlerfunktion hat eine gewisse Ähnlichkeit mit der <a href="Verteilungsfunktion" title="Verteilungsfunktion">Verteilungsfunktion</a> der <a href="Normalverteilung" title="Normalverteilung">Normalverteilung</a>. Sie hat jedoch eine <a href="Zielmenge" title="Zielmenge">Zielmenge</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,1)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e120a3bd60fc89b495dd7ef6039465b7e6a703b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,1)}" loading="lazy"></span>, während eine Verteilungsfunktion zwingend Werte aus dem Bereich <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"></span> annehmen muss.

</p><p>Es gilt für die <a href="Standardnormalverteilung" class="mw-redirect" title="Standardnormalverteilung">Standardnormalverteilung</a>

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<annotation encoding="application/x-tex">{\displaystyle \Phi (x)={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66c83474a0b4197af63d0fdc51c7a3088390563e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.737ex; height:6.509ex;" alt="{\displaystyle \Phi (x)={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right)}" loading="lazy"></span></dd></dl>

<p>bzw. für die Verteilungsfunktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> einer beliebigen Normalverteilung mit <a href="Standardabweichung_(Stochastik)" class="mw-redirect" title="Standardabweichung (Stochastik)">Standardabweichung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span> und <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle F(x)={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14438beb4ff3c0875b76473d9a4fd68359d80314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.307ex; height:6.509ex;" alt="{\displaystyle F(x)={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right).}" loading="lazy"></span></dd></dl>

<p>Falls die Abweichungen der einzelnen Ergebnisse einer <a href="Messreihe" title="Messreihe">Messreihe</a> vom gemeinsamen <a href="Mittelwert" title="Mittelwert">Mittelwert</a> durch eine Normalverteilung mit Standardabweichung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span> und Erwartungswert 0 beschrieben werden können, dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \operatorname {erf} \left({\frac {a}{\sigma {\sqrt {2}}}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \operatorname {erf} \left({\frac {a}{\sigma {\sqrt {2}}}}\right)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b92401dbc1d06afb89bbe733544e04b7bfc426cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.399ex; height:4.843ex;" alt="{\displaystyle \textstyle \operatorname {erf} \left({\frac {a}{\sigma {\sqrt {2}}}}\right)}" loading="lazy"></span> die Wahrscheinlichkeit, mit der der <a href="Messfehler" class="mw-redirect" title="Messfehler">Messfehler</a> einer einzelnen Messung zwischen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +a}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23558d20acf58179a3e9a08ea7c80ba017095873.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle +a}" loading="lazy"></span> liegt (für positives <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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"></span>).

</p><p>Die Fehlerfunktion kann verwendet werden, um mit Hilfe der <a href="Inversionsmethode" title="Inversionsmethode">Inversionsmethode</a> normalverteilte <a href="Pseudozufallszahl" class="mw-redirect" title="Pseudozufallszahl">Pseudozufallszahlen</a> zu generieren.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Wärmeleitungsgleichung"><span id="W.C3.A4rmeleitungsgleichung"></span>Wärmeleitungsgleichung</h3></div>

<p>Die Fehlerfunktion und die komplementäre Fehlerfunktion kommen beispielsweise in Lösungen der <a href="W%C3%A4rmeleitungsgleichung" title="Wärmeleitungsgleichung">Wärmeleitungsgleichung</a> vor, wenn <a href="Randwertproblem" title="Randwertproblem">Randwertbedingungen</a> durch die <a href="Heaviside-Funktion" title="Heaviside-Funktion">Heaviside-Funktion</a> vorgegeben sind.

</p>

<div class="mw-heading mw-heading2"><h2 id="Numerische_Berechnung">Numerische Berechnung</h2></div>

<p>Die Fehlerfunktion ist wie die Verteilungsfunktion der Normalverteilung nicht durch eine geschlossene Funktion darstellbar und muss <a href="Numerische_Mathematik" title="Numerische Mathematik">numerisch</a> bestimmt werden.

</p><p>Für kleine reelle Werte erfolgt die Berechnung mit der <a href="Taylorreihe" title="Taylorreihe">Reihenentwicklung</a>

</p>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)n!}}={\frac {2}{\sqrt {\pi }}}\left(x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{10}}-{\frac {x^{7}}{42}}+{\frac {x^{9}}{216}}-\dotsb \right),}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/969c81b156842901845410dd662fc173dcd07db5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:73.715ex; height:7.009ex;" alt="{\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)n!}}={\frac {2}{\sqrt {\pi }}}\left(x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{10}}-{\frac {x^{7}}{42}}+{\frac {x^{9}}{216}}-\dotsb \right),}" loading="lazy"></span></dd></dl>

<p>für große reelle Werte mit der <a href="Kettenbruch" title="Kettenbruch">Kettenbruch</a>entwicklung

</p>

<mfrac>

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<mfrac>

<msup>

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</msup>

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<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)=1-{\frac {1}{\sqrt {\pi }}}\cdot {\frac {e^{-x^{2}}}{x+{\frac {1}{2x+{\frac {2}{x+{\frac {3}{2x+{\frac {4}{x+\dotsb }}}}}}}}}}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ca20c5cb286c3292487ad4815a5f97c67a5852c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:38.352ex; height:12.843ex;" alt="{\displaystyle \operatorname {erf} (x)=1-{\frac {1}{\sqrt {\pi }}}\cdot {\frac {e^{-x^{2}}}{x+{\frac {1}{2x+{\frac {2}{x+{\frac {3}{2x+{\frac {4}{x+\dotsb }}}}}}}}}}.}" loading="lazy"></span></dd></dl>

<p>Für den kompletten Wertebereich gibt es folgende Approximation mit einem maximalen Fehler von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1{,}2\cdot 10^{-7}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1{,}2\cdot 10^{-7}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1987b207a1be873044b5ec3fb8b43ed5391271de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.309ex; height:3.009ex;" alt="{\displaystyle 1{,}2\cdot 10^{-7}}" loading="lazy"></span>:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

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<mtr>

<mtd>

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<mtd>

<mtext>falls </mtext>

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</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>sonst,</mtext>

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</mtd>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)\approx {\begin{cases}1-\tau (x){\text{,}}&{\text{falls }}x\geq 0{\text{,}}\\\tau (-x)-1&{\text{sonst,}}\end{cases}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/089e67d31253c83ff4215a35344d457b3b8815c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.765ex; height:6.176ex;" alt="{\displaystyle \operatorname {erf} (x)\approx {\begin{cases}1-\tau (x){\text{,}}&{\text{falls }}x\geq 0{\text{,}}\\\tau (-x)-1&{\text{sonst,}}\end{cases}}}" loading="lazy"></span></dd></dl>

<p>mit

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}\tau (x)&=&t\cdot \exp {\bigl (}-x^{2}-1{,}26551223+1{,}00002368\cdot t+0{,}37409196\cdot t^{2}+0{,}09678418\cdot t^{3}\\&&\qquad -0{,}18628806\cdot t^{4}+0{,}27886807\cdot t^{5}-1{,}13520398\cdot t^{6}+1{,}48851587\cdot t^{7}\\&&\qquad -0{,}82215223\cdot t^{8}+0{,}17087277\cdot t^{9}{\bigr )}\end{array}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1ac6314b96fbb3602182cbc63209af2e80f9c9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.348ex; margin-bottom: -0.323ex; width:89.755ex; height:10.509ex;" alt="{\displaystyle {\begin{array}{rcl}\tau (x)&=&t\cdot \exp {\bigl (}-x^{2}-1{,}26551223+1{,}00002368\cdot t+0{,}37409196\cdot t^{2}+0{,}09678418\cdot t^{3}\\&&\qquad -0{,}18628806\cdot t^{4}+0{,}27886807\cdot t^{5}-1{,}13520398\cdot t^{6}+1{,}48851587\cdot t^{7}\\&&\qquad -0{,}82215223\cdot t^{8}+0{,}17087277\cdot t^{9}{\bigr )}\end{array}}}" loading="lazy"></span></dd></dl>

<p>und

</p>

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<annotation encoding="application/x-tex">{\displaystyle t={\frac {1}{1+0{,}5\,|x|}}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f81e3d40f27a89a5b77524172efe967f1c55130e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.406ex; height:6.009ex;" alt="{\displaystyle t={\frac {1}{1+0{,}5\,|x|}}.}" loading="lazy"></span></dd></dl>

<p>Eine für alle reellen Werte von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.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"></span> schnell konvergierende Entwicklung<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> erhält man unter Verwendung des Theorems von Heinrich H. Bürmann:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<mtr>

<mtd>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {erf} (x)&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn}(x){\sqrt {1-e^{-x^{2}}}}\left(1-{\frac {1}{12}}\left(1-e^{-x^{2}}\right)-{\frac {7}{480}}\left(1-e^{-x^{2}}\right)^{2}-{\frac {5}{896}}\left(1-e^{-x^{2}}\right)^{3}-{\frac {787}{276480}}\left(1-e^{-x^{2}}\right)^{4}-\ \cdots \right)\\&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn}(x){\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+\sum _{k=1}^{\infty }c_{k}e^{-k\,x^{2}}\right)\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a86e253b117e5f61d346123ffd28bce08d2eac4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:121.949ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {erf} (x)&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn} (x){\sqrt {1-e^{-x^{2}}}}\left(1-{\frac {1}{12}}\left(1-e^{-x^{2}}\right)-{\frac {7}{480}}\left(1-e^{-x^{2}}\right)^{2}-{\frac {5}{896}}\left(1-e^{-x^{2}}\right)^{3}-{\frac {787}{276480}}\left(1-e^{-x^{2}}\right)^{4}-\ \cdots \right)\\&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn} (x){\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+\sum _{k=1}^{\infty }c_{k}e^{-k\,x^{2}}\right)\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Durch geeignete Wahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77b7dc6d279091d354e0b90889b463bfa7eb7247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{1}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b30ba1b247fb8d334580cec68561e749d24aff2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{2}}" loading="lazy"></span> ergibt sich daraus eine Näherung, deren größter relativer Fehler bei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle x=\pm 1{,}3796}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle x=\pm 1{,}3796}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/875c1b70023632115c92c3bf1fc4baaeeda637af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.695ex; height:2.509ex;" alt="{\displaystyle \textstyle x=\pm 1{,}3796}" loading="lazy"></span> kleiner als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 3{,}6127\cdot 10^{-3}}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle 3{,}6127\cdot 10^{-3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fddfccba2a9f671d9d38f70432c47acc0fa0911b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.796ex; height:3.009ex;" alt="{\displaystyle \textstyle 3{,}6127\cdot 10^{-3}}" loading="lazy"></span> ist:

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<msqrt>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)\approx {\frac {2}{\sqrt {\pi }}}\operatorname {sgn}(x){\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+{\frac {31}{200}}\,e^{-x^{2}}-{\frac {341}{8000}}\,e^{-2\,x^{2}}\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2da25e89c0834af632ed1ef16cffcf193705c0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:64.705ex; height:6.843ex;" alt="{\displaystyle \operatorname {erf} (x)\approx {\frac {2}{\sqrt {\pi }}}\operatorname {sgn} (x){\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+{\frac {31}{200}}\,e^{-x^{2}}-{\frac {341}{8000}}\,e^{-2\,x^{2}}\right)}" loading="lazy"></span></dd></dl>

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

</mstyle>

</mrow>

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

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</th>

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</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2a96a1b139214ea50c6d6f436fb555e6429134e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.795ex; height:2.843ex;" alt="{\displaystyle \operatorname {erf} (x)}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</th>

</td>

</mstyle>

</mrow>

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

</semantics>

</th>

</mstyle>

</mrow>

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

</semantics>

</th>

</mstyle>

</mrow>

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

</semantics>

</th></tr>

<tr>

<td>0,00

</td>

<td>0,0000000

</td>

<td>1,0000000

</td>

<td>1,30

</td>

<td>0,9340079

</td>

<td>0,0659921

</td></tr>

<tr>

<td>0,05

</td>

<td>0,0563720

</td>

<td>0,9436280

</td>

<td>1,40

</td>

<td>0,9522851

</td>

<td>0,0477149

</td></tr>

<tr>

<td>0,10

</td>

<td>0,1124629

</td>

<td>0,8875371

</td>

<td>1,50

</td>

<td>0,9661051

</td>

<td>0,0338949

</td></tr>

<tr>

<td>0,15

</td>

<td>0,1679960

</td>

<td>0,8320040

</td>

<td>1,60

</td>

<td>0,9763484

</td>

<td>0,0236516

</td></tr>

<tr>

<td>0,20

</td>

<td>0,2227026

</td>

<td>0,7772974

</td>

<td>1,70

</td>

<td>0,9837905

</td>

<td>0,0162095

</td></tr>

<tr>

<td>0,25

</td>

<td>0,2763264

</td>

<td>0,7236736

</td>

<td>1,80

</td>

<td>0,9890905

</td>

<td>0,0109095

</td></tr>

<tr>

<td>0,30

</td>

<td>0,3286268

</td>

<td>0,6713732

</td>

<td>1,90

</td>

<td>0,9927904

</td>

<td>0,0072096

</td></tr>

<tr>

<td>0,35

</td>

<td>0,3793821

</td>

<td>0,6206179

</td>

<td>2,00

</td>

<td>0,9953223

</td>

<td>0,0046777

</td></tr>

<tr>

<td>0,40

</td>

<td>0,4283924

</td>

<td>0,5716076

</td>

<td>2,10

</td>

<td>0,9970205

</td>

<td>0,0029795

</td></tr>

<tr>

<td>0,45

</td>

<td>0,4754817

</td>

<td>0,5245183

</td>

<td>2,20

</td>

<td>0,9981372

</td>

<td>0,0018628

</td></tr>

<tr>

<td>0,50

</td>

<td>0,5204999

</td>

<td>0,4795001

</td>

<td>2,30

</td>

<td>0,9988568

</td>

<td>0,0011432

</td></tr>

<tr>

<td>0,55

</td>

<td>0,5633234

</td>

<td>0,4366766

</td>

<td>2,40

</td>

<td>0,9993115

</td>

<td>0,0006885

</td></tr>

<tr>

<td>0,60

</td>

<td>0,6038561

</td>

<td>0,3961439

</td>

<td>2,50

</td>

<td>0,9995930

</td>

<td>0,0004070

</td></tr>

<tr>

<td>0,65

</td>

<td>0,6420293

</td>

<td>0,3579707

</td>

<td>2,60

</td>

<td>0,9997640

</td>

<td>0,0002360

</td></tr>

<tr>

<td>0,70

</td>

<td>0,6778012

</td>

<td>0,3221988

</td>

<td>2,70

</td>

<td>0,9998657

</td>

<td>0,0001343

</td></tr>

<tr>

<td>0,75

</td>

<td>0,7111556

</td>

<td>0,2888444

</td>

<td>2,80

</td>

<td>0,9999250

</td>

<td>0,0000750

</td></tr>

<tr>

<td>0,80

</td>

<td>0,7421010

</td>

<td>0,2578990

</td>

<td>2,90

</td>

<td>0,9999589

</td>

<td>0,0000411

</td></tr>

<tr>

<td>0,85

</td>

<td>0,7706681

</td>

<td>0,2293319

</td>

<td>3,00

</td>

<td>0,9999779

</td>

<td>0,0000221

</td></tr>

<tr>

<td>0,90

</td>

<td>0,7969082

</td>

<td>0,2030918

</td>

<td>3,10

</td>

<td>0,9999884

</td>

<td>0,0000116

</td></tr>

<tr>

<td>0,95

</td>

<td>0,8208908

</td>

<td>0,1791092

</td>

<td>3,20

</td>

<td>0,9999940

</td>

<td>0,0000060

</td></tr>

<tr>

<td>1,00

</td>

<td>0,8427008

</td>

<td>0,1572992

</td>

<td>3,30

</td>

<td>0,9999969

</td>

<td>0,0000031

</td></tr>

<tr>

<td>1,10

</td>

<td>0,8802051

</td>

<td>0,1197949

</td>

<td>3,40

</td>

<td>0,9999985

</td>

<td>0,0000015

</td></tr>

<tr>

<td>1,20

</td>

<td>0,9103140

</td>

<td>0,0896860

</td>

<td>3,50

</td>

<td>0,9999993

</td>

<td>0,0000007

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

<div class="mw-heading mw-heading2"><h2 id="Komplexe_Fehlerfunktion">Komplexe Fehlerfunktion</h2></div>

<p>Die Definitionsgleichung der Fehlerfunktion kann auf komplexe Argumente <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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"></span> ausgeweitet werden:

</p>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-\tau ^{2}}\,\mathrm {d} \tau }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8e200e22c594e1d3c41bfa9c8a6a2fc58eede50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.193ex; height:6.343ex;" alt="{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-\tau ^{2}}\,\mathrm {d} \tau }" loading="lazy"></span></dd></dl>

<p>In diesem Fall ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a8ee8591a5cc9057c9c646d1d07b848c1d68e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.153ex; width:2.809ex; height:2.176ex;" alt="{\displaystyle \operatorname {erf} }" loading="lazy"></span> eine <a href="Komplexwertige_Funktion" title="Komplexwertige Funktion">komplexwertige Funktion</a>. Unter <a href="Komplexe_Konjugation" title="Komplexe Konjugation">komplexer Konjugation</a> gilt

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (z^{*})=\operatorname {erf} (z)^{*}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2f86599e7bab5ae32f9b11d024ad6760af51133.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.316ex; height:2.843ex;" alt="{\displaystyle \operatorname {erf} (z^{*})=\operatorname {erf} (z)^{*}}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Imaginäre_Fehlerfunktion"><span id="Imagin.C3.A4re_Fehlerfunktion"></span>Imaginäre Fehlerfunktion</h3></div>

<p>Die <i>imaginäre Fehlerfunktion</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfi} (x)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6aa89763ce865e0cf3a8f70b2b5f000044a9be1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.442ex; height:2.843ex;" alt="{\displaystyle \operatorname {erfi} (x)}" loading="lazy"></span> ist gegeben durch

</p>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfi} (x)={\frac {\operatorname {erf} (\mathrm {i} x)}{\mathrm {i} }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/344d6ffdb13e3502ff2941ace12ff8b94d0cf1f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.818ex; height:5.676ex;" alt="{\displaystyle \operatorname {erfi} (x)={\frac {\operatorname {erf} (\mathrm {i} x)}{\mathrm {i} }}}" loading="lazy"></span></dd></dl>

<p>mit der Reihenentwicklung

</p>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfi} (x)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{n!(2n+1)}}={\frac {2}{\sqrt {\pi }}}\left(x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{10}}+{\frac {x^{7}}{42}}+{\frac {x^{9}}{216}}+\dotsb \right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c2f65d07d30d4ce601f28168c2b6eada4b871a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:72.27ex; height:6.843ex;" alt="{\displaystyle \operatorname {erfi} (x)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{n!(2n+1)}}={\frac {2}{\sqrt {\pi }}}\left(x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{10}}+{\frac {x^{7}}{42}}+{\frac {x^{9}}{216}}+\dotsb \right)}" loading="lazy"></span>.</dd></dl>

<p>Zur Berechnung können <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf,erfi,erfc} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf,erfi,erfc} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f91ff2f1e99da2427da13e0d8a5eab3df0394d8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.175ex; height:2.509ex;" alt="{\displaystyle \operatorname {erf,erfi,erfc} }" loading="lazy"></span> und weitere verwandte Funktionen auch durch die <a href="Faddeeva-Funktion" title="Faddeeva-Funktion">Faddeeva-Funktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(z)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13626102fd9be0d15ff137c157c1aed6caf339a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.562ex; height:2.843ex;" alt="{\displaystyle w(z)}" loading="lazy"></span> ausgedrückt werden. Die Faddeeva-Funktion ist eine skalierte komplexe komplementäre Fehlerfunktion und auch als relativistische Plasma-Dispersions-Funktion bekannt. Sie ist mit den <a href="Dawson-Integral" class="mw-redirect" title="Dawson-Integral">Dawson-Integralen</a> und dem <a href="Voigt-Profil" title="Voigt-Profil">Voigt-Profil</a> verwandt. Eine numerische Implementierung von Steven G. Johnson steht als C-Bibliothek <i>libcerf</i> zur Verfügung.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

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

<ul><li><a href="Milton_Abramowitz" title="Milton Abramowitz">Milton Abramowitz</a>, <a href="Irene_Stegun" title="Irene Stegun">Irene A. Stegun</a> (Hrsg.): <i><a href="Abramowitz-Stegun" title="Abramowitz-Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>.</i> Dover, New York 1972, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_297.htm">Chapter 7.</a></li>

<li>William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery: <i>Numerical Recipes in C</i>. 2. Auflage. Cambridge 1992, <a rel="nofollow" class="external text" href="http://www.nrbook.com/a/bookcpdf/c6-2.pdf">S. 220 ff.</a> (PDF; 76 kB)</li>

<li><a href="Ilja_Nikolajewitsch_Bronstein" class="mw-redirect" title="Ilja Nikolajewitsch Bronstein">Bronstein, I.N.</a> und <a href="Konstantin_Adolfowitsch_Semendjajew" title="Konstantin Adolfowitsch Semendjajew">Semendjajew, K.A.</a>, <i><a href="Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a></i>, 6. Auflage, <a href="Verlag_Harri_Deutsch" title="Verlag Harri Deutsch">Verlag Harri Deutsch</a>, Frankfurt am Main 2005, S. 782</li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Bronstein, Semendjajew, <i>Taschenbuch der Mathematik</i>, 6. Auflage, S. 782</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Für eine konkrete Implementierung siehe z. B. Peter John Acklam: <style data-mw-deduplicate="TemplateStyles:r261891140">

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

.mw-parser-output .webarchiv-memento a{color:inherit}

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

</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070505093933/http://home.online.no/~pjacklam/notes/invnorm/"><i>An algorithm for computing the inverse normal cumulative distribution function</i>.</a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 5. Mai 2007 im <i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a></i>)</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><i>Numerical Recipes in Fortran 77: The Art of Scientific Computing</i>. Cambridge University Press, 1992, ISBN 0-521-43064-X, S. 214.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">H. M. Schöpf, P. H. Supancic: <a rel="nofollow" class="external text" href="http://www.mathematica-journal.com/2014/11/on-burmanns-theorem-and-its-application-to-problems-of-linear-and-nonlinear-heat-transfer-and-diffusion/#more-39602/"><i>On Bürmann’s Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion</i>.</a> In: <i>The Mathematica Journal</i>, 2014. <a href="https://doi.org/10.3888/tmj.16-11" class="extiw external" title="doi:10.3888/tmj.16-11">doi:10.3888/tmj.16-11</a>.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Moritz Cantor: <i><a href="https://de.wikisource.org/wiki/ADB:B%C3%BCrmann,_Heinrich" class="extiw external" title="s:ADB:Bürmann, Heinrich">Bürmann, Heinrich</a>.</i> In: <i><a href="Allgemeine_Deutsche_Biographie" title="Allgemeine Deutsche Biographie">Allgemeine Deutsche Biographie</a></i> (ADB). Band 47, Duncker & Humblot, Leipzig 1903, S. 392–394.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">E. W. Weisstein: <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BuermannsTheorem.html"><i>Bürmann’s Theorem</i>.</a> mathworld</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Steven G. Johnson, Joachim Wuttke: <a rel="nofollow" class="external text" href="https://jugit.fz-juelich.de/mlz/libcerf">libcerf.</a></span>

</li>

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

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