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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Integralkosinus</span></h1>
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<p>Der <b>Integralkosinus</b> ist eine <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a>, in deren Funktionsvorschrift ein <a href="Integralrechnung" title="Integralrechnung">Integral</a> und die <a href="Kosinus" class="mw-redirect" title="Kosinus">Kosinusfunktion</a> auftreten. Diese Integralfunktion kann mit elementaren Methoden nicht ohne Integral dargestellt werden.
</p><p>Der Integralkosinus ist definiert als:
</p>
<dl><dd><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 {\rm {Ci}}(x):=\gamma +\ln x+\int _{0}^{x}{\frac {\cos t-1}{t}}\,dt\quad =-\int _{x}^{\infty }{\frac {\cos t}{t}}\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">C</mi>
<mi mathvariant="normal">i</mi>
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<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mi>γ<!-- γ --></mi>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>+</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mi>x</mi>
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<mfrac>
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<mi>cos</mi>
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<mn>1</mn>
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<mi>t</mi>
</mfrac>
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<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
<mspace width="1em"></mspace>
<mo>=</mo>
<mo>−<!-- − --></mo>
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<mo>∫<!-- ∫ --></mo>
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<mi>x</mi>
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<mi mathvariant="normal">∞<!-- ∞ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>cos</mi>
<mo><!-- --></mo>
<mi>t</mi>
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<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\rm {Ci}}(x):=\gamma +\ln x+\int _{0}^{x}{\frac {\cos t-1}{t}}\,dt\quad =-\int _{x}^{\infty }{\frac {\cos t}{t}}\,dt}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5724aeebf5c421b5b451810fb42daefc7dbb0dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.411ex; height:5.843ex;" alt="{\displaystyle {\rm {Ci}}(x):=\gamma +\ln x+\int _{0}^{x}{\frac {\cos t-1}{t}}\,dt\quad =-\int _{x}^{\infty }{\frac {\cos t}{t}}\,dt}" loading="lazy"></span></dd></dl>
<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 \gamma =0{,}577215...}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>γ<!-- γ --></mi>
<mo>=</mo>
<mn>0,577</mn>
<mn>215...</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma =0{,}577215...}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c443b5f6a006cb417dd35b932397a40df1a974c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.085ex; height:2.676ex;" alt="{\displaystyle \gamma =0{,}577215...}" loading="lazy"></span> die <a href="Euler-Mascheroni-Konstante" title="Euler-Mascheroni-Konstante">Euler-Mascheroni-Konstante</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<ul><li>Das in der Definition auftretende Integral wird auch mit <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 {Cin} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cin</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cin} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/663844fc0d1201916112614c82dc36cb7a5d200d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.617ex; height:2.176ex;" alt="{\displaystyle \operatorname {Cin} }" loading="lazy"></span> bezeichnet:</li></ul>
<dl><dd><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 {Cin} (x):=\int _{0}^{x}{\frac {1-\cos t}{t}}\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>t</mi>
</mrow>
<mi>t</mi>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cin} (x):=\int _{0}^{x}{\frac {1-\cos t}{t}}\,dt}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0055e8760852f24b8a070db754a34a78bdb3670c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.145ex; height:5.843ex;" alt="{\displaystyle \operatorname {Cin} (x):=\int _{0}^{x}{\frac {1-\cos t}{t}}\,dt}" loading="lazy"></span></dd>
<dd>mit der Beziehung:</dd>
<dd><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 {Cin} (x)=\gamma +\ln x-\operatorname {Ci} (x)\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>γ<!-- γ --></mi>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>Ci</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cin} (x)=\gamma +\ln x-\operatorname {Ci} (x)\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d03571c8b0034cbf8f4d10b4fa1ccd94ebe28dee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.305ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cin} (x)=\gamma +\ln x-\operatorname {Ci} (x)\,}" loading="lazy"></span></dd></dl>
<ul><li>Analog zur Ableitung des <a href="Integralsinus" title="Integralsinus">Integralsinus</a> Si(x):</li></ul>
<dl><dd><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 {Si} '(x)={\frac {\sin x}{x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>Si</mi>
<mo>′</mo>
</msup>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Si} '(x)={\frac {\sin x}{x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bdea4c8d3c2f5d7d3d5487906208b643f01c792.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.27ex; height:5.176ex;" alt="{\displaystyle \operatorname {Si} '(x)={\frac {\sin x}{x}}}" loading="lazy"></span></dd>
<dd>gilt:</dd>
<dd><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 {Ci} '(x)={\frac {\cos(x)}{x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>Ci</mi>
<mo>′</mo>
</msup>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ci} '(x)={\frac {\cos(x)}{x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74dd2a048c320ec22a89cb697d70a06f26293328.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.333ex; height:5.676ex;" alt="{\displaystyle \operatorname {Ci} '(x)={\frac {\cos(x)}{x}}}" loading="lazy"></span></dd></dl>
<ul><li>Analog der komplexen <a href="Eulersche_Formel" title="Eulersche Formel">Eulerformel</a>-Definition des Cosinus</li></ul>
<dl><dd><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 \cos x={\frac {1}{2}}\left(\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
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<mo>+</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
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<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
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<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos x={\frac {1}{2}}\left(\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86979475fe7bcd50a4574954a1b687d8603aa1e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.885ex; height:5.176ex;" alt="{\displaystyle \cos x={\frac {1}{2}}\left(\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}\right)}" loading="lazy"></span></dd>
<dd>gilt 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} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Ei</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ei} }</annotation>
</semantics>
</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></dd>
<dd><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 {Ci} (x)={\frac {1}{2}}\left(\operatorname {Ei} (\mathrm {i} \ x)+\operatorname {Ei} (-\mathrm {i} \ x)\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Ci</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Ei</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mtext> </mtext>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>Ei</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mtext> </mtext>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ci} (x)={\frac {1}{2}}\left(\operatorname {Ei} (\mathrm {i} \ x)+\operatorname {Ei} (-\mathrm {i} \ x)\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ed792d3030af95387f267777f8d2d7649cabce4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.598ex; height:5.176ex;" alt="{\displaystyle \operatorname {Ci} (x)={\frac {1}{2}}\left(\operatorname {Ei} (\mathrm {i} \ x)+\operatorname {Ei} (-\mathrm {i} \ x)\right)}" loading="lazy"></span></dd></dl>
<ul><li>Es lässt sich eine überall konvergente <a href="Reihe_(Mathematik)" title="Reihe (Mathematik)">Reihe</a> angeben:</li></ul>
<dl><dd><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 {Ci} \left(x\right)=\gamma +\ln x-{\frac {x^{2}}{2!\cdot 2}}+{\frac {x^{4}}{4!\cdot 4}}-\cdots \quad =\gamma +\ln x+\sum _{k=1}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)!\cdot 2k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Ci</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>γ<!-- γ --></mi>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mn>2</mn>
<mo>!</mo>
<mo>⋅<!-- ⋅ --></mo>
<mn>2</mn>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mrow>
<mn>4</mn>
<mo>!</mo>
<mo>⋅<!-- ⋅ --></mo>
<mn>4</mn>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mspace width="1em"></mspace>
<mo>=</mo>
<mi>γ<!-- γ --></mi>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>+</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
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<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
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<mo stretchy="false">)</mo>
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<msup>
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<mo stretchy="false">)</mo>
<mo>!</mo>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ci} \left(x\right)=\gamma +\ln x-{\frac {x^{2}}{2!\cdot 2}}+{\frac {x^{4}}{4!\cdot 4}}-\cdots \quad =\gamma +\ln x+\sum _{k=1}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)!\cdot 2k}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/114ecbf6240c4d2a3991418241c43cd63967c2e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:68.247ex; height:7.009ex;" alt="{\displaystyle \operatorname {Ci} \left(x\right)=\gamma +\ln x-{\frac {x^{2}}{2!\cdot 2}}+{\frac {x^{4}}{4!\cdot 4}}-\cdots \quad =\gamma +\ln x+\sum _{k=1}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)!\cdot 2k}}}" loading="lazy"></span></dd></dl>
<ul><li>Folgende unendliche Summe mit Integralkosinuswerten als Summanden ergibt diesen Wert:</li></ul>
<dl><dd><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 \sum _{n=1}^{\infty }\operatorname {Ci} (2\pi n)={\frac {1}{4}}-{\frac {1}{2}}\gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
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<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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<mi>Ci</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
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<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
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<mi>γ<!-- γ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }\operatorname {Ci} (2\pi n)={\frac {1}{4}}-{\frac {1}{2}}\gamma }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/165d15e71fa0b9ccfa278f0de0af23e6205b7a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.964ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }\operatorname {Ci} (2\pi n)={\frac {1}{4}}-{\frac {1}{2}}\gamma }" loading="lazy"></span></dd></dl>
<ul><li>Denn es gelten folgende Integrale:</li></ul>
<dl><dd><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 \int _{0}^{\infty }{\frac {x\exp(-wx)}{x^{2}+1}}dx={\frac {1}{2}}\pi \sin(w)-\operatorname {Si} (w)\sin(w)-\operatorname {Ci} (w)\cos(w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mo>∫<!-- ∫ --></mo>
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<mn>0</mn>
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<mi>exp</mi>
<mo><!-- --></mo>
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<mi>w</mi>
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<msup>
<mi>x</mi>
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<mi>π<!-- π --></mi>
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<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>w</mi>
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<mo>−<!-- − --></mo>
<mi>Si</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>Ci</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>w</mi>
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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }{\frac {x\exp(-wx)}{x^{2}+1}}dx={\frac {1}{2}}\pi \sin(w)-\operatorname {Si} (w)\sin(w)-\operatorname {Ci} (w)\cos(w)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/572d3bf8a9e15eb2ff5be335a38360ae6aec2ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:63.714ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{\infty }{\frac {x\exp(-wx)}{x^{2}+1}}dx={\frac {1}{2}}\pi \sin(w)-\operatorname {Si} (w)\sin(w)-\operatorname {Ci} (w)\cos(w)}" loading="lazy"></span></dd>
<dd><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 \int _{0}^{\infty }{\frac {x}{(x^{2}+1)[\exp(2\pi x)-1]}}dx={\frac {1}{2}}\gamma -{\frac {1}{4}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">[</mo>
<mi>exp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
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<mo stretchy="false">)</mo>
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<mi>d</mi>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }{\frac {x}{(x^{2}+1)[\exp(2\pi x)-1]}}dx={\frac {1}{2}}\gamma -{\frac {1}{4}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0241efdd91972c7478be1af98eb19de3a56803e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.985ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{\infty }{\frac {x}{(x^{2}+1)[\exp(2\pi x)-1]}}dx={\frac {1}{2}}\gamma -{\frac {1}{4}}}" loading="lazy"></span></dd></dl>
<p><i>Anmerkung: In verschiedenen Formelsammlungen wird der Integralkosinus mit umgekehrten <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> definiert.</i>
</p><p>Eng verwandt ist der <a href="Integralsinus" title="Integralsinus">Integralsinus</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 {Si} (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Si</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Si} (x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaae5702ee43d96453e58588e2fac127cf9acf38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \operatorname {Si} (x)}" loading="lazy"></span>, der zusammen mit dem Integralcosinus <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 {Ci} (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Ci</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ci} (x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4558b3d0592d9d46565019eb2005cb0c4850290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.464ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ci} (x)}" loading="lazy"></span> in parametrischer Darstellung eine <a href="Klothoide" title="Klothoide">Klothoide</a> bildet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Integralexponentialfunktion" title="Integralexponentialfunktion">Integralexponentialfunktion</a></li>
<li><a href="Integralsinus" title="Integralsinus">Integralsinus</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/CosineIntegral.html"><i>Cosine Integral</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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