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<title>Integrallogarithmus</title>

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

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<p>Der <b>Integrallogarithmus</b> ist eine <a href="Analytische_Funktion" title="Analytische Funktion">analytische Funktion</a> auf den <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</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 x\geq 0,\;x\neq 1}">

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<annotation encoding="application/x-tex">{\displaystyle x\geq 0,\;x\neq 1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8bbef9790bd626a8ecab3528de2d00475a650d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.86ex; height:2.676ex;" alt="{\displaystyle x\geq 0,\;x\neq 1}" loading="lazy"></span> (oder <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>1}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0549e1fb7ee2023519833093c6e3b60236e7d09f.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>1}" loading="lazy"></span>) in die reellen Zahlen.

Sie hat praktische Relevanz in einigen Gebieten der Physik wie der <a href="Quantenfeldtheorie" title="Quantenfeldtheorie">Quantenfeldtheorie</a> und bei der Lösung der <a href="Laplace-Gleichung" title="Laplace-Gleichung">Laplace-Gleichung</a> in <a href="Halbleiter" title="Halbleiter">Halbleitern</a> sowie in der <a href="Zahlentheorie" title="Zahlentheorie">Zahlentheorie</a>, da sie eng mit der <a href="Dichte" title="Dichte">Dichte</a> der <a href="Primzahl" title="Primzahl">Primzahlen</a> verknüpft ist.

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

<p>Es sind zwei Definitionen üblich, die sich um eine Konstante unterscheiden. Für eine der wichtigsten Anwendungen – als <a href="Asymptote" title="Asymptote">asymptotische</a> Vergleichsgröße für die Primzahlfunktion im <a href="Primzahlsatz" title="Primzahlsatz">Primzahlsatz</a> – spielt der Unterschied zwischen den beiden Definitionen keine Rolle.

</p><p>Eine Definition im 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 x\geq 0}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2608e2b392b079f5b763f27bf52883dbee3b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}" loading="lazy"></span> lautet

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {\mathrm {d} t}{\ln t}}\ ,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25fa4cf6d885f8e793e8d3b3bdecb000a3117080.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.785ex; height:5.843ex;" alt="{\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {\mathrm {d} t}{\ln t}}\ ,}" loading="lazy"></span></dd></dl>

<p>dabei muss <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 {li} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5d59f5dd40d4a2be9d55f0d6b3334fbafd16775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.294ex; height:2.176ex;" alt="{\displaystyle \operatorname {li} }" loading="lazy"></span> wegen der <a href="Isolierte_Singularit%C3%A4t#Nichtisolierte_Singularitäten" title="Isolierte Singularität">Singularität</a> 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 x=1}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee42176e76ae6b56d68c42ced807e08b962a2b54.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=1}" loading="lazy"></span> für <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>1}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0549e1fb7ee2023519833093c6e3b60236e7d09f.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>1}" loading="lazy"></span> über einen <a href="Grenzwert_(Funktion)" title="Grenzwert (Funktion)">Grenzwert</a> definiert werden (<a href="Cauchyscher_Hauptwert" title="Cauchyscher Hauptwert">cauchyscher Hauptwert</a>):

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {li} (x)=\lim _{\varepsilon \to 0^{+}}\left(\int _{0}^{1-\varepsilon }{\frac {\mathrm {d} t}{\ln t}}+\int _{1+\varepsilon }^{x}{\frac {\mathrm {d} t}{\ln t}}\right)\ .}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17a672a97cffd100dfbf16a329ac81122726ca25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.961ex; height:6.343ex;" alt="{\displaystyle \operatorname {li} (x)=\lim _{\varepsilon \to 0^{+}}\left(\int _{0}^{1-\varepsilon }{\frac {\mathrm {d} t}{\ln t}}+\int _{1+\varepsilon }^{x}{\frac {\mathrm {d} t}{\ln t}}\right)\ .}" loading="lazy"></span></dd></dl>

<p>Eine andere Definition für <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>1}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0549e1fb7ee2023519833093c6e3b60236e7d09f.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>1}" loading="lazy"></span> ist

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} (x)=\operatorname {li} (x)-\operatorname {li} (2)=\int _{2}^{x}{\frac {\mathrm {d} t}{\ln t}}\ .}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb583fc3c9f6c6d435a266bf9d8f3206578a1b03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.227ex; height:5.843ex;" alt="{\displaystyle \operatorname {Li} (x)=\operatorname {li} (x)-\operatorname {li} (2)=\int _{2}^{x}{\frac {\mathrm {d} t}{\ln t}}\ .}" loading="lazy"></span></dd></dl>

<p>Dabei liegt 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 x=1}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee42176e76ae6b56d68c42ced807e08b962a2b54.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=1}" loading="lazy"></span> ein <a href="Verzweigungspunkt" class="mw-redirect" title="Verzweigungspunkt">Verzweigungspunkt</a> vor.

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

<p>Einige Werte:

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/208707da3f1a23cb5cb820f5c604de2af05f0113.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle \operatorname {li} (0)=0}" loading="lazy"></span></li>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d5d3f5eb75b5b0c981327d438e241443a7624a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.496ex; height:2.843ex;" alt="{\displaystyle \operatorname {li} (1)=-\infty }" loading="lazy"></span></li>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e95c7831acc769ac4b706c6b8da2c92a45cb9640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.766ex; height:2.843ex;" alt="{\displaystyle \operatorname {li} (\mu )=0}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {li} (2)=1{,}04516\;37801\;17492\;78484\ldots }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4593c43d662fa936dad53382f4be54d2a1910caa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.468ex; height:2.843ex;" alt="{\displaystyle \operatorname {li} (2)=1{,}04516\;37801\;17492\;78484\ldots }" loading="lazy"></span> (Folge <a href="https://oeis.org/A069284" class="extiw external" title="oeis:A069284">A069284</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</li></ul>

<p>Dabei 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 \mu =1{,}45136\;92348\;83381\;05028\ldots }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =1{,}45136\;92348\;83381\;05028\ldots }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40b0268658dec5c18ce806e80bde293f5f981750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.604ex; height:2.676ex;" alt="{\displaystyle \mu =1{,}45136\;92348\;83381\;05028\ldots }" loading="lazy"></span> (Folge <a href="https://oeis.org/A070769" class="extiw external" title="oeis:A070769">A070769</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) die <a href="Ramanujan-Soldner-Konstante" title="Ramanujan-Soldner-Konstante">Ramanujan-Soldner-Konstante</a>.

</p><p>Es gilt <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 {li} (x)=\operatorname {Ei} (\ln x)}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63c90a43a7cf1a58117eb2eae76b65de3f9943b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.226ex; height:2.843ex;" alt="{\displaystyle \operatorname {li} (x)=\operatorname {Ei} (\ln x)}" loading="lazy"></span> mit der <a href="Integralexponentialfunktion" title="Integralexponentialfunktion">Integralexponentialfunktion</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 \operatorname {Ei} }">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e9ef2bc3c11559409c3d53f635c610594e899a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.23ex; height:2.176ex;" alt="{\displaystyle \operatorname {Ei} }" loading="lazy"></span>, daraus erhält man die Reihendarstellung

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {li} (x)=\gamma +\ln \left|\ln x\right|+\sum _{k=1}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}\ ,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f46b2b464d09a079a2ad5b9927b7ee60abb5a1d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.723ex; height:7.009ex;" alt="{\displaystyle \operatorname {li} (x)=\gamma +\ln \left|\ln x\right|+\sum _{k=1}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}\ ,}" loading="lazy"></span></dd></dl>

<p>wobei <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 \gamma =0{,}57721\;56649\;01532\;86060\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma =0{,}57721\;56649\;01532\;86060\ldots }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7aae15a6da41ebde84df7e2c7824709a13981682.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.465ex; height:2.676ex;" alt="{\displaystyle \gamma =0{,}57721\;56649\;01532\;86060\ldots }" loading="lazy"></span> (Folge <a href="https://oeis.org/A001620" class="extiw external" title="oeis:A001620">A001620</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) die <a href="Euler-Mascheroni-Konstante" title="Euler-Mascheroni-Konstante">Euler-Mascheroni-Konstante</a> ist.

</p><p>Aus der Definition 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 \operatorname {li} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5d59f5dd40d4a2be9d55f0d6b3334fbafd16775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.294ex; height:2.176ex;" alt="{\displaystyle \operatorname {li} }" loading="lazy"></span> erhält man durch <a href="Integration_durch_Substitution#Lineare_Substitution" title="Integration durch Substitution">lineare Substitution</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {li} (x)=x\int _{0}^{1}{\frac {\mathrm {d} t}{\ln(x\,t)}}\ ,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cf90016cf939f300038899693e99eff95c8ac7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.522ex; height:6.509ex;" alt="{\displaystyle \operatorname {li} (x)=x\int _{0}^{1}{\frac {\mathrm {d} t}{\ln(x\,t)}}\ ,}" loading="lazy"></span></dd></dl>

<p>wobei für <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>1}">

</mstyle>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0549e1fb7ee2023519833093c6e3b60236e7d09f.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>1}" loading="lazy"></span> wegen der Singularität 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 t=1/x}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de420bf971fa73e5d0b6d7a39133e921017716d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.593ex; height:2.843ex;" alt="{\displaystyle t=1/x}" loading="lazy"></span> der cauchysche Hauptwert eingesetzt werden muss.<br>

Ferner haben wir für <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\geq 0,x\neq 1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x\geq 0,x\neq 1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d150770a4e6fe866200667765f330dee32500c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.215ex; height:2.676ex;" alt="{\displaystyle x\geq 0,x\neq 1}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{x}\operatorname {li} (t)\,{\mathrm {d} t}=x\,\operatorname {li} (x)-\operatorname {li} (x^{2}).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/112d3d46d0b42ff8df6fb730b7fb46715c7c6fef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.095ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{x}\operatorname {li} (t)\,{\mathrm {d} t}=x\,\operatorname {li} (x)-\operatorname {li} (x^{2}).}" loading="lazy"></span></dd></dl>

<p>Außerdem gilt für <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 p>-1,p\not =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p>-1,p\not =0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5841c2bd5af4140388c190bfe96f175c489fb3eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.792ex; height:2.676ex;" alt="{\displaystyle p>-1,p\not =0}" loading="lazy"></span>

</p>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}\operatorname {li} (t)\,t^{p-1}\,\mathrm {d} t=-{\tfrac {1}{p}}\ln(p+1),}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59304cb2fda2b76e391c03d4d3aff948aa636f0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.278ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}\operatorname {li} (t)\,t^{p-1}\,\mathrm {d} t=-{\tfrac {1}{p}}\ln(p+1),}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c29a2f2fb3f642618036ed7a79712202e7ada924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}" loading="lazy"></span> erhält man <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 \int _{0}^{1}\operatorname {li} (t)\,\mathrm {d} t=-\ln 2.}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{0}^{1}\operatorname {li} (t)\,\mathrm {d} t=-\ln 2.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb5ef8191b5933be02dca66f71fd9e172243c1de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.847ex; height:3.676ex;" alt="{\displaystyle \textstyle \int _{0}^{1}\operatorname {li} (t)\,\mathrm {d} t=-\ln 2.}" loading="lazy"></span><br>

Im Grenzfall <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 p=0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3e6ac10fa45fb984d886065f959a6bdd467b5e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=0}" loading="lazy"></span> 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 \int _{0}^{1}\operatorname {li} (t)\,t^{-1}\,\mathrm {d} t=-1.}">

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{0}^{1}\operatorname {li} (t)\,t^{-1}\,\mathrm {d} t=-1.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b38090faab28d4d1bfa949cb098405646bef32d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.693ex; height:3.676ex;" alt="{\displaystyle \textstyle \int _{0}^{1}\operatorname {li} (t)\,t^{-1}\,\mathrm {d} t=-1.}" loading="lazy"></span>

</p><p>Eine weitere Formel 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 \int _{0}^{1}\operatorname {li} (t^{-1})\,t\,\mathrm {d} t=\textstyle \int _{1}^{\infty }\operatorname {li} (t)\,t^{-3}\,\mathrm {d} t=0.}">

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{0}^{1}\operatorname {li} (t^{-1})\,t\,\mathrm {d} t=\textstyle \int _{1}^{\infty }\operatorname {li} (t)\,t^{-3}\,\mathrm {d} t=0.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1eaf49ee16c13ccc4a3073baf51ae7a8469eeb42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.781ex; height:3.676ex;" alt="{\displaystyle \textstyle \int _{0}^{1}\operatorname {li} (t^{-1})\,t\,\mathrm {d} t=\textstyle \int _{1}^{\infty }\operatorname {li} (t)\,t^{-3}\,\mathrm {d} t=0.}" loading="lazy"></span>

</p><p>Die <a href="Golomb-Dickman-Konstante" title="Golomb-Dickman-Konstante">Golomb-Dickman-Konstante</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 \lambda =\textstyle \int _{0}^{1}\mathrm {e} ^{\operatorname {li} (x)}\mathrm {d} x=0{,}62432\;99885\;43550\;87099\ldots }">

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda =\textstyle \int _{0}^{1}\mathrm {e} ^{\operatorname {li} (x)}\mathrm {d} x=0{,}62432\;99885\;43550\;87099\ldots }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e304c1920f20e8ff782b56816542fdaaa733176e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.633ex; height:3.676ex;" alt="{\displaystyle \lambda =\textstyle \int _{0}^{1}\mathrm {e} ^{\operatorname {li} (x)}\mathrm {d} x=0{,}62432\;99885\;43550\;87099\ldots }" loading="lazy"></span> (Folge <a href="https://oeis.org/A084945" class="extiw external" title="oeis:A084945">A084945</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) tritt in der Theorie <a href="Zuf%C3%A4llige_Permutation" title="Zufällige Permutation">zufälliger Permutationen</a> bei der Abschätzung der Länge des längsten <a href="Zyklische_Permutation" title="Zyklische Permutation">Zykels</a> einer <a href="Permutation" title="Permutation">Permutation</a> und in der Zahlentheorie bei der Abschätzung der Größe des größten <a href="Primfaktorzerlegung" title="Primfaktorzerlegung">Primfaktors</a> einer Zahl auf.

</p>

<div class="mw-heading mw-heading2"><h2 id="Asymptotisches_Verhalten">Asymptotisches Verhalten</h2></div>

<p>Für große <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> lässt sich <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 {li} (x)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0f7c22225b6931ab65c480fc209b2dfc4b71f9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.433ex; height:2.843ex;" alt="{\displaystyle \operatorname {li} (x)}" loading="lazy"></span> durch

</p>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {li} (x)=0!\,{\frac {x}{\ln x}}+1!\,{\frac {x}{\ln ^{2}x}}+2!\,{\frac {x}{\ln ^{3}x}}+3!\,{\frac {x}{\ln ^{4}x}}+\dotsb }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4725fc8e1a65cf678c8eaa368b8a9f0651b2bbe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.533ex; height:5.343ex;" alt="{\displaystyle \operatorname {li} (x)=0!\,{\frac {x}{\ln x}}+1!\,{\frac {x}{\ln ^{2}x}}+2!\,{\frac {x}{\ln ^{3}x}}+3!\,{\frac {x}{\ln ^{4}x}}+\dotsb }" loading="lazy"></span></dd></dl>

<p><a href="Approximation" title="Approximation">approximieren</a>. Die <a href="Reihe_(Mathematik)" title="Reihe (Mathematik)">Reihe</a> ist eine <a href="Asymptotische_Entwicklung" class="mw-redirect" title="Asymptotische Entwicklung">asymptotische Entwicklung</a>; sie <a href="Grenzwert_(Folge)" title="Grenzwert (Folge)">konvergiert</a> nicht, sondern nähert sich dem <a href="Wahrer_Wert" title="Wahrer Wert">wahren Wert</a> an, um sich dann wieder zu entfernen.

Die beste Approximation wird nach etwa <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 \ln x}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed172b0f5195382a3500c713941f945ad4db3898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \ln x}" loading="lazy"></span> Gliedern erreicht, dann werden die Summanden größer durch die stärker werdende Wirkung der <a href="Fakult%C3%A4t_(Mathematik)" title="Fakultät (Mathematik)">Fakultät</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

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

<li><a href="Integralcosinus" class="mw-redirect" title="Integralcosinus">Integralcosinus</a></li></ul>

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

<ul><li><a href="Johann_Georg_von_Soldner" title="Johann Georg von Soldner">Johann Georg Soldner</a>: <i>Théorie et tables d’une nouvelle fonction transcendante</i>. Lindauer, München 1809 (französisch; <a rel="nofollow" class="external text" href="https://books.google.de/books?id=g4Q_AAAAcAAJ&hl=de">bei Google Books</a>)</li>

<li><a href="Niels_Nielsen_(Mathematiker)" title="Niels Nielsen (Mathematiker)">Niels Nielsen</a>: <i>Theorie des Integrallogarithmus und verwandter Transzendenten</i>, B. G. Teubner, Leipzig 1906 (<a rel="nofollow" class="external text" href="https://archive.org/details/theoriedesinteg00nielgoog/mode/1up">archive.org</a>)</li></ul>

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

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LogarithmicIntegral.html"><i>Logarithmic Integral</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>

<li><a rel="nofollow" class="external text" href="https://functions.wolfram.com/GammaBetaErf/LogIntegral/"><i>Logarithmic integral</i>.</a> bei <i>The Wolfram Functions Site</i> (englisch; mit Berechnungsmöglichkeit)</li></ul>

<div>

Normdaten (Sachbegriff): <a href="Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="-print"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4212683-6">4212683-6</a></span> </div>

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