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<title>Bessel-Funktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Bessel-Funktion</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Als Bessel-Funktionen bezeichnet man Funktionen, welche Lösungen der besselschen Differentialgleichung sind, die eine <a href="Lineare_gew%C3%B6hnliche_Differentialgleichung" title="Lineare gewöhnliche Differentialgleichung">lineare gewöhnliche Differentialgleichung</a> zweiter Ordnung ist. Benannt sind die Funktionen und die Gleichung nach dem deutschen Mathematiker <a href="Friedrich_Wilhelm_Bessel" title="Friedrich Wilhelm Bessel">Friedrich Wilhelm Bessel</a>. Die Bessel-Funktionen werden auch Zylinderfunktionen genannt.

<div class="mw-heading mw-heading2"><h2 id="Besselsche_Differentialgleichung">Besselsche Differentialgleichung</h2></div>

Die Besselsche Differentialgleichung ist eine gewöhnliche lineare Differentialgleichung zweiter Ordnung, die durch

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<annotation encoding="application/x-tex">{\displaystyle x^{2}{\frac {\mathrm {d} ^{2}f}{\mathrm {d} x^{2}}}+x{\frac {\mathrm {d} f}{\mathrm {d} x}}+\left(x^{2}-\nu ^{2}\right)f=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/220343c855e053c3e1f1cfdb333c2e810bb2bdcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.958ex; height:6.176ex;" alt="{\displaystyle x^{2}{\frac {\mathrm {d} ^{2}f}{\mathrm {d} x^{2}}}+x{\frac {\mathrm {d} f}{\mathrm {d} x}}+\left(x^{2}-\nu ^{2}\right)f=0}" loading="lazy"></dd></dl>

definiert ist, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./_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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> reelle oder komplexe <a href="Zahl" title="Zahl">Zahlen</a> sind. Die Lösungen heißen Bessel-Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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

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Entsprechend ist der Bessel-Operator ein <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a> zweiter Ordnung. Er ist definiert durch

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<annotation encoding="application/x-tex">{\displaystyle B_{\nu }:=x^{2}{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}+x{\frac {\mathrm {d} }{\mathrm {d} x}}+\left(x^{2}-\nu ^{2}\right)\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e9b60376f0375cc6b92b9208430d059dfb15036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.065ex; height:6.009ex;" alt="{\displaystyle B_{\nu }:=x^{2}{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}+x{\frac {\mathrm {d} }{\mathrm {d} x}}+\left(x^{2}-\nu ^{2}\right)\,.}" loading="lazy"></dd></dl>

Mit ihm kann man die Besselsche <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichung</a> kurz ausdrücken durch<a href="#cite_note-1">[1]</a>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e0afa60027e215be3692bccd8e7fd54e7692ae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.407ex; height:2.509ex;" alt="{\displaystyle B_{\nu }f=0}" loading="lazy"></dd></dl>

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

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

Die Lösungen der Besselschen Differentialgleichung heißen Bessel-Funktionen. Sie spielen eine wichtige Rolle in der Physik, da die Besselsche Differentialgleichung den radialen Anteil der <a href="Laplace-Gleichung" title="Laplace-Gleichung">Laplace-Gleichung</a> bei zylindrischer <a href="Symmetrie_(Geometrie)" title="Symmetrie (Geometrie)">Symmetrie</a> darstellt. Auf die Bessel-Funktionen trifft man unter anderem bei der Untersuchung von <a href="Eigenschwingung" class="mw-redirect" title="Eigenschwingung">Eigenschwingungen</a> einer kreisförmigen <a href="Schwingungsmembran" title="Schwingungsmembran">Membran</a> oder einer Orgelpfeife, der Ausbreitung von Wasserwellen in runden Behältern, der Wärmeleitung in Stäben, der Analyse des Frequenzspektrums <a href="Frequenzmodulation" title="Frequenzmodulation">frequenzmodulierter</a> Signale, der Feldverteilung im Querschnitt von <a href="Hohlleiter#Rundhohlleiter" title="Hohlleiter">Rundhohlleitern</a>, den stationären Zuständen von Kastenpotentialen, der Leistungsverteilung in <a href="Kernreaktor" title="Kernreaktor">Kernreaktoren</a>, der Intensität von <a href="Beugungsscheibchen" title="Beugungsscheibchen">Lichtbeugung an kreisförmigen Löchern</a> sowie bei <a href="Filter_(Elektrotechnik)" title="Filter (Elektrotechnik)">Filtern</a> in der Elektrotechnik (<a href="Bessel-Filter" title="Bessel-Filter">Bessel-Filter</a>). Man zählt die Bessel-Funktionen wegen ihrer vielfältigen Anwendungen in der <a href="Mathematische_Physik" title="Mathematische Physik">mathematischen Physik</a> zu den <a href="Spezielle_Funktionen" class="mw-redirect" title="Spezielle Funktionen">speziellen Funktionen</a>.

Als Differentialgleichung zweiter Ableitungsordnung besitzt die Besselsche Differentialgleichung zwei linear unabhängige <a href="L%C3%B6sung_(Mathematik)" title="Lösung (Mathematik)">Lösungen</a>. Sie lassen sich in verschiedenen Varianten beschreiben.

<div class="mw-heading mw-heading3"><h3 id="Bessel-Funktionen_erster_Gattung:_Jν">Bessel-Funktionen erster Gattung: Jν</h3></div>

Die Bessel-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\nu }}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93f5eaf64d9da9c637c9e42042719a0e47bdca28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.394ex; height:2.509ex;" alt="{\displaystyle J_{\nu }}" loading="lazy"> erster Gattung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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<annotation encoding="application/x-tex">{\displaystyle J_{\nu }(x)=\sum _{r=0}^{\infty }{\frac {(-1)^{r}\left({\frac {x}{2}}\right)^{2r+\nu }}{\Gamma (\nu +r+1)r!}}\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c2f629889c8a1eba264dca9609a669f1d489c8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.678ex; height:7.843ex;" alt="{\displaystyle J_{\nu }(x)=\sum _{r=0}^{\infty }{\frac {(-1)^{r}\left({\frac {x}{2}}\right)^{2r+\nu }}{\Gamma (\nu +r+1)r!}}\,}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\cdot )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb56054f7834e05862ff50930fc46b4f8e3f8280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.909ex; height:2.843ex;" alt="{\displaystyle \Gamma (\cdot )}" loading="lazy"> die <a href="Gammafunktion" title="Gammafunktion">Gammafunktion</a> ist. Im Ursprung (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy">) sind diese Funktionen für ganzzahlige <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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Für nicht-ganzzahlige <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\nu }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93f5eaf64d9da9c637c9e42042719a0e47bdca28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.394ex; height:2.509ex;" alt="{\displaystyle J_{\nu }}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{-\nu }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74be9e43f2f5c8cd3efb7932259e1b90dd7eab8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.672ex; height:2.509ex;" alt="{\displaystyle J_{-\nu }}" loading="lazy"> linear unabhängige Lösungen.

Für <a href="Ganzzahlig" class="mw-redirect" title="Ganzzahlig">ganzzahlige</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a97d2387465aaadcc3f166dfb26db0bd871ab67b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.152ex; height:2.843ex;" alt="{\displaystyle J_{-\nu }(x)=(-1)^{\nu }J_{\nu }(x)=J_{\nu }(-x)\,}" loading="lazy">.</dd></dl>

In diesem Fall ist die zweite unabhängige Lösung die Bessel-Funktion zweiter Gattung, die weiter unten diskutiert wird.

<div class="mw-heading mw-heading4"><h4 id="Integraldarstellungen">Integraldarstellungen</h4></div>

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

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<annotation encoding="application/x-tex">{\displaystyle J_{\nu }(x)={\frac {1}{\pi }}\int _{0}^{\pi }{\bigl (}\cos(\nu \,\varphi )\cos {\bigl [}x\sin(\varphi ){\bigr ]}+\sin(\nu \,\varphi )\sin {\bigl [}x\sin(\varphi ){\bigr ]}{\bigr )}\,\mathrm {d} \varphi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3c6f3a4d182a131667772c8a27e3dee1df827e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:66.83ex; height:5.843ex;" alt="{\displaystyle J_{\nu }(x)={\frac {1}{\pi }}\int _{0}^{\pi }{\bigl (}\cos(\nu \,\varphi )\cos {\bigl [}x\sin(\varphi ){\bigr ]}+\sin(\nu \,\varphi )\sin {\bigl [}x\sin(\varphi ){\bigr ]}{\bigr )}\,\mathrm {d} \varphi }" loading="lazy"></dd></dl>

Mit dem Additionstheorem der Kosinusfunktion kann dann noch folgende Vereinfachung vorgenommen werden:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{\nu }(x)&={\frac {1}{\pi }}\int _{0}^{\pi }\cos(x\sin \varphi -\nu \varphi )\,\mathrm {d} \varphi \\&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{\mathrm {i} \,(x\sin \varphi -\nu \varphi )}\,\mathrm {d} \varphi \,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f0260ff07786405e0a453f5da28d29185e518c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:36.156ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}J_{\nu }(x)&={\frac {1}{\pi }}\int _{0}^{\pi }\cos(x\sin \varphi -\nu \varphi )\,\mathrm {d} \varphi \\&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{\mathrm {i} \,(x\sin \varphi -\nu \varphi )}\,\mathrm {d} \varphi \,.\end{aligned}}}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b979c35146145ea840105d88dd98656b9d301e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.533ex; height:2.843ex;" alt="{\displaystyle J_{\nu }(x)}" loading="lazy"> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">-te <a href="Fourier-Koeffizient" class="mw-redirect" title="Fourier-Koeffizient">Fourier-Koeffizient</a> der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \mapsto e^{ix\sin \varphi }}">

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Exemplarisch wird im Folgenden die Bessel-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{0}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/750fead02065cb001fe30d98d309652b642f8e8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{0}}" loading="lazy"> dargestellt:

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<annotation encoding="application/x-tex">{\displaystyle J_{0}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{4^{n}(n!)^{2}}}=\int _{0}^{\pi }{\frac {1}{\pi }}\cos {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y=\int _{0}^{1}{\frac {2\cos(xz)}{\pi {\sqrt {1-z^{2}}}}}\,\mathrm {d} z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e1d8e3a3520c0701e3510647779549bfa1b44a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.429ex; height:7.176ex;" alt="{\displaystyle J_{0}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{4^{n}(n!)^{2}}}=\int _{0}^{\pi }{\frac {1}{\pi }}\cos {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y=\int _{0}^{1}{\frac {2\cos(xz)}{\pi {\sqrt {1-z^{2}}}}}\,\mathrm {d} z}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Hypergeometrische_Funktion">Hypergeometrische Funktion</h4></div>

Die Bessel-Funktion erster Gattung kann durch die <a href="Verallgemeinerte_hypergeometrische_Funktion" title="Verallgemeinerte hypergeometrische Funktion">verallgemeinerte hypergeometrische Funktion</a> ausgedrückt werden:

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<annotation encoding="application/x-tex">{\displaystyle J_{\nu }(x)={\frac {(x/2)^{\nu }}{\Gamma (\nu +1)}}\;_{0}F_{1}(;\nu +1;-x^{2}/4).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7822f869262ce67e244270be85da8620ecf0d37f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.489ex; height:6.509ex;" alt="{\displaystyle J_{\nu }(x)={\frac {(x/2)^{\nu }}{\Gamma (\nu +1)}}\;_{0}F_{1}(;\nu +1;-x^{2}/4).}" loading="lazy"></dd></dl>

Dieser Ausdruck hängt mit der Entwicklung der Bessel-Funktion in Abhängigkeit zur Bessel-Clifford-Funktion zusammen.

<div class="mw-heading mw-heading3"><h3 id="Bessel-Funktionen_zweiter_Gattung:_Yν">Bessel-Funktionen zweiter Gattung: Yν</h3></div>

Auch die Bessel-Funktionen zweiter Gattung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\nu }(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/395f27707de74035520a0649a163b103a006012b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.593ex; height:2.843ex;" alt="{\displaystyle Y_{\nu }(x)}" loading="lazy"> (auch <a href="Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a>-Funktionen oder <a href="Carl_Gottfried_Neumann" title="Carl Gottfried Neumann">Neumann</a>-Funktionen genannt) lösen die Besselsche Differentialgleichung. Eine alternative Bezeichnung ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{\nu }(x)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/642a209f3ac9ca0fd3440440f33f08befb99d1e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.109ex; height:2.843ex;" alt="{\displaystyle N_{\nu }(x)}" loading="lazy">. Für nicht-ganzzahlige <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> kann man die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\nu }(x)}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{\nu }(x)={\frac {J_{\nu }(x)\cos(\nu \pi )-J_{-\nu }(x)}{\sin(\nu \pi )}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb6b80b5afca4a6ca27707d13e43efa76aa89138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.23ex; height:6.509ex;" alt="{\displaystyle Y_{\nu }(x)={\frac {J_{\nu }(x)\cos(\nu \pi )-J_{-\nu }(x)}{\sin(\nu \pi )}}.}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> ist die durch den Grenzübergang <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \rightarrow n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d376f9e86281f654f7a40680f21d17c6be2c0b92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.241ex; height:1.843ex;" alt="{\displaystyle \nu \rightarrow n}" loading="lazy"> gebildete <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a>

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<annotation encoding="application/x-tex">{\displaystyle Y_{n}(x)=\lim _{\nu \to n}Y_{\nu }(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7cf9b57e3728fddaf7cdd7f814017f797131b1f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.287ex; height:3.843ex;" alt="{\displaystyle Y_{n}(x)=\lim _{\nu \to n}Y_{\nu }(x)}" loading="lazy"></dd></dl>

weiterhin eine Lösung der Besselschen Differentialgleichung.

Wie für die Bessel-Funktionen erster Gattung gilt auch für die Besselfunktionen zweiter Gattung folgende Beziehung:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dbb1a260f37cd23563454cfd068d0a73fac8794.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.791ex; height:2.843ex;" alt="{\displaystyle Y_{-n}(x)=(-1)^{n}Y_{n}(x)}" loading="lazy">.</dd></dl>

Nach Ausführung des Grenzüberganges mit der <a href="Regel_von_de_L%E2%80%99Hospital" title="Regel von de L’Hospital">Regel von de L’Hospital</a> ergibt sich

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<annotation encoding="application/x-tex">{\displaystyle Y_{n}(x)={\frac {1}{\pi }}\left[{\operatorname {d} \over \operatorname {d} \!\nu }J_{\nu }(x){\Big |}_{\nu =n}+(-1)^{n}{\operatorname {d} \over \operatorname {d} \!\nu }J_{\nu }(x){\Big |}_{\nu =-n}\right].}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f00fde02e624dace0b3040a83280ecc7bd287ee8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.785ex; height:6.176ex;" alt="{\displaystyle Y_{n}(x)={\frac {1}{\pi }}\left[{\operatorname {d} \over \operatorname {d} \!\nu }J_{\nu }(x){\Big |}_{\nu =n}+(-1)^{n}{\operatorname {d} \over \operatorname {d} \!\nu }J_{\nu }(x){\Big |}_{\nu =-n}\right].}" loading="lazy"></dd></dl>

Explizit findet man

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<annotation encoding="application/x-tex">{\displaystyle Y_{n}(x)=\,{\frac {2}{\pi }}\left(\gamma +\ln {\frac {x}{2}}\right)J_{n}(x)-{\frac {1}{\pi }}\sum _{k=0}^{n-1}{\frac {(n-k-1)!}{k!}}\left({\frac {x}{2}}\right)^{2k-n}-{\frac {1}{\pi }}\sum _{k=0}^{\infty }(-1)^{k}{\frac {H_{k}+H_{k+n}}{k!(n+k)!}}\left({\frac {x}{2}}\right)^{2k+n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cc94e8dc507a909037874c9e9e71f3bfb016673.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:95.867ex; height:7.509ex;" alt="{\displaystyle Y_{n}(x)=\,{\frac {2}{\pi }}\left(\gamma +\ln {\frac {x}{2}}\right)J_{n}(x)-{\frac {1}{\pi }}\sum _{k=0}^{n-1}{\frac {(n-k-1)!}{k!}}\left({\frac {x}{2}}\right)^{2k-n}-{\frac {1}{\pi }}\sum _{k=0}^{\infty }(-1)^{k}{\frac {H_{k}+H_{k+n}}{k!(n+k)!}}\left({\frac {x}{2}}\right)^{2k+n}}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} _{0}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cf5c8ad993619a325ca57a25c22cdc75a460f88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.968ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} _{0}}" loading="lazy">. Hierbei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> die <a href="Euler-Mascheroni-Konstante" title="Euler-Mascheroni-Konstante">Euler-Mascheroni-Konstante</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e63458b04288bbe116a9a8037dfae0b36b2c639a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.15ex; height:2.509ex;" alt="{\displaystyle H_{n}}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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Die Bessel-Funktionen zweiter Gattung haben also bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"> eine logarithmische Singularität und einen Pol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> ist neben der Bessel-Funktion erster Gattung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\nu }}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93f5eaf64d9da9c637c9e42042719a0e47bdca28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.394ex; height:2.509ex;" alt="{\displaystyle J_{\nu }}" loading="lazy"> die Bessel-Funktion zweiter Gattung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\nu }}">

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<div class="mw-heading mw-heading3"><h3 id="Bessel-Funktionen_dritter_Gattung:_Hν(1),_Hν(2)">Bessel-Funktionen dritter Gattung: Hν(1), Hν(2)</h3></div>

Die Bessel-Funktionen dritter Gattung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }^{(1)}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8370d4930b673d5821205a6ffb442cb4706eed27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.437ex; height:3.343ex;" alt="{\displaystyle H_{\nu }^{(1)}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }^{(2)}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e8834bc3836b620d3c6f1ff072151fc1ebcea6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.437ex; height:3.343ex;" alt="{\displaystyle H_{\nu }^{(2)}}" loading="lazy"> (auch bekannt als <a href="Hermann_Hankel" title="Hermann Hankel">Hankel</a>-Funktionen) sind Linearkombinationen der Bessel-Funktionen erster und zweiter Gattung

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{\nu }^{(1)}(x)&=J_{\nu }(x)+\mathrm {i} \cdot Y_{\nu }(x)\,,\\H_{\nu }^{(2)}(x)&=J_{\nu }(x)-\mathrm {i} \cdot Y_{\nu }(x)\,,\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c244e1d93be746c4d96b662fd88d71e9a9546998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.752ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}H_{\nu }^{(1)}(x)&=J_{\nu }(x)+\mathrm {i} \cdot Y_{\nu }(x)\,,\\H_{\nu }^{(2)}(x)&=J_{\nu }(x)-\mathrm {i} \cdot Y_{\nu }(x)\,,\end{aligned}}}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18f0f09f6fc40e634d34aed6e205ac0f7a40e062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }" loading="lazy"> die <a href="Imagin%C3%A4re_Einheit" class="mw-redirect" title="Imaginäre Einheit">imaginäre Einheit</a> bezeichnet. Auch diese beiden Funktionen sind linear unabhängige Lösungen der Besselschen Differentialgleichung.

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

<div class="mw-heading mw-heading4"><h4 id="Beziehungen_von_Ordnungen_einer_Gattung">Beziehungen von Ordnungen einer Gattung</h4></div>

<ul><li>Für die Bessel-Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\nu }}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93f5eaf64d9da9c637c9e42042719a0e47bdca28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.394ex; height:2.509ex;" alt="{\displaystyle J_{\nu }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\nu }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f519c94ab4d76342a667c23d5c54d181bbf6eca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.509ex;" alt="{\displaystyle Y_{\nu }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }^{(1)}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8370d4930b673d5821205a6ffb442cb4706eed27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.437ex; height:3.343ex;" alt="{\displaystyle H_{\nu }^{(1)}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }^{(2)}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e8834bc3836b620d3c6f1ff072151fc1ebcea6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.437ex; height:3.343ex;" alt="{\displaystyle H_{\nu }^{(2)}}" loading="lazy"> gelten die <a href="Rekursion" title="Rekursion">Rekursionsbeziehungen</a>:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\nu }{x}}\Omega _{\nu }={\frac {1}{2}}(\Omega _{\nu -1}+\Omega _{\nu +1})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}\Omega _{\nu }={\frac {1}{2}}(\Omega _{\nu -1}-\Omega _{\nu +1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-{\frac {1}{x}}{\frac {\rm {d}}{{\rm {d}}x}}\right)^{n}J_{0}(x)={\frac {J_{n}(x)}{x^{n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \left(-{\frac {1}{x}}{\frac {\rm {d}}{{\rm {d}}x}}\right)^{n}J_{0}(x)={\frac {J_{n}(x)}{x^{n}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceab203eb62c06afeffeb812d5a71c0d94f71997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.138ex; height:6.343ex;" alt="{\displaystyle \left(-{\frac {1}{x}}{\frac {\rm {d}}{{\rm {d}}x}}\right)^{n}J_{0}(x)={\frac {J_{n}(x)}{x^{n}}}}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Asymptotisches_Verhalten">Asymptotisches Verhalten</h4></div>

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,\nu \in \mathbb {R} ,\nu \geq 0}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/141060adfb7e95a464caceb556a1dc676dfeb660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.641ex; height:2.509ex;" alt="{\displaystyle x,\nu \in \mathbb {R} ,\nu \geq 0}" loading="lazy">, dann gelten für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x\ll {\sqrt {\nu +1}}}">

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<annotation encoding="application/x-tex">{\displaystyle 0<x\ll {\sqrt {\nu +1}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e033d8b67eea017790ab659e7c7451f46a0ce9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.376ex; height:3.009ex;" alt="{\displaystyle 0<x\ll {\sqrt {\nu +1}}}" loading="lazy"> die asymptotischen Darstellungen

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}J_{\nu }(x)&\approx {\frac {1}{\Gamma (\nu +1)}}\left({\frac {x}{2}}\right)^{\nu }\\Y_{\nu }(x)&\approx {\begin{cases}{\frac {2}{\pi }}\left(\ln \left({\frac {x}{2}}\right)+\gamma \right)&{\text{wenn }}\nu =0\\\\-{\frac {\Gamma (\nu )}{\pi }}\left({\frac {2}{x}}\right)^{\nu }&{\text{wenn }}\nu >0.\end{cases}}\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{\nu }(x)&\approx {\frac {1}{\Gamma (\nu +1)}}\left({\frac {x}{2}}\right)^{\nu }\\Y_{\nu }(x)&\approx {\begin{cases}{\frac {2}{\pi }}\left(\ln \left({\frac {x}{2}}\right)+\gamma \right)&{\text{wenn }}\nu =0\\\\-{\frac {\Gamma (\nu )}{\pi }}\left({\frac {2}{x}}\right)^{\nu }&{\text{wenn }}\nu >0.\end{cases}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e471701354b97b75044c04b596afa3e7de4af020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:40.842ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}J_{\nu }(x)&\approx {\frac {1}{\Gamma (\nu +1)}}\left({\frac {x}{2}}\right)^{\nu }\\Y_{\nu }(x)&\approx {\begin{cases}{\frac {2}{\pi }}\left(\ln \left({\frac {x}{2}}\right)+\gamma \right)&{\text{wenn }}\nu =0\\\\-{\frac {\Gamma (\nu )}{\pi }}\left({\frac {2}{x}}\right)^{\nu }&{\text{wenn }}\nu >0.\end{cases}}\end{aligned}}}" loading="lazy"></dd></dl>

Für große Argumente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\gg |\nu ^{2}-1/4|}">

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<annotation encoding="application/x-tex">{\displaystyle x\gg |\nu ^{2}-1/4|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e229a3195d8deafeb8ebbf949689640e7ab0fa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.877ex; height:3.176ex;" alt="{\displaystyle x\gg |\nu ^{2}-1/4|}" loading="lazy"> findet man

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{\nu }(x)&\approx {\sqrt {\frac {2}{\pi x}}}\cos \left(x-{\frac {\nu \pi }{2}}-{\frac {\pi }{4}}\right)\\Y_{\nu }(x)&\approx {\sqrt {\frac {2}{\pi x}}}\sin \left(x-{\frac {\nu \pi }{2}}-{\frac {\pi }{4}}\right).\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d9aeffc2c84d24b0ef65fc45c45468359551937.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:34.509ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}J_{\nu }(x)&\approx {\sqrt {\frac {2}{\pi x}}}\cos \left(x-{\frac {\nu \pi }{2}}-{\frac {\pi }{4}}\right)\\Y_{\nu }(x)&\approx {\sqrt {\frac {2}{\pi x}}}\sin \left(x-{\frac {\nu \pi }{2}}-{\frac {\pi }{4}}\right).\end{aligned}}}" loading="lazy"></dd></dl>

Diese Formeln sind für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =1/2}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f4a1f94e25928532f6f700d868c6c09c3f90ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.818ex; height:2.843ex;" alt="{\displaystyle \nu =1/2}" loading="lazy"> exakt. Vergleiche hierfür mit den sphärischen Besselfunktionen weiter unten.

<div class="mw-heading mw-heading2"><h2 id="Modifizierte_Bessel-Funktionen:_Iν,_Kν"> Modifizierte Bessel-Funktionen: Iν, Kν</h2></div>

Die Differentialgleichung

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<annotation encoding="application/x-tex">{\displaystyle x^{2}{\frac {\mathrm {d} ^{2}f}{\mathrm {d} x^{2}}}+x{\frac {\mathrm {d} f}{\mathrm {d} x}}-(x^{2}+\nu ^{2})f=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75adc97157a36d0fdaab1a919cf1aed4fd93a972.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.25ex; height:6.176ex;" alt="{\displaystyle x^{2}{\frac {\mathrm {d} ^{2}f}{\mathrm {d} x^{2}}}+x{\frac {\mathrm {d} f}{\mathrm {d} x}}-(x^{2}+\nu ^{2})f=0}" loading="lazy"></dd></dl>

wird durch Bessel-Funktionen mit rein imaginärem Argument gelöst. Man definiert für ihre Lösung normalerweise die modifizierten Bessel-Funktionen

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<annotation encoding="application/x-tex">{\displaystyle I_{\nu }(x)=i^{-\nu }J_{\nu }(ix)=\sum _{r=0}^{\infty }{\frac {({\frac {x}{2}})^{2r+\nu }}{\Gamma (r+\nu +1)r!}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46dbe4ab91f22581f92b7a1db6dc1594ee4f9394.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.642ex; height:7.343ex;" alt="{\displaystyle I_{\nu }(x)=i^{-\nu }J_{\nu }(ix)=\sum _{r=0}^{\infty }{\frac {({\frac {x}{2}})^{2r+\nu }}{\Gamma (r+\nu +1)r!}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle I_{\nu }(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \,\varphi )\cosh {\bigl [}x\sin(\varphi ){\bigr ]}-\sin(\nu \,\varphi )\sinh {\bigl [}x\sin(\varphi ){\bigr ]}\,\mathrm {d} \varphi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09494bad915b7608160bf1e6d79cd27a33b8bf28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:66.631ex; height:5.843ex;" alt="{\displaystyle I_{\nu }(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \,\varphi )\cosh {\bigl [}x\sin(\varphi ){\bigr ]}-\sin(\nu \,\varphi )\sinh {\bigl [}x\sin(\varphi ){\bigr ]}\,\mathrm {d} \varphi }" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K_{\nu }(x)={\frac {\pi }{2}}{\frac {I_{-\nu }(x)-I_{\nu }(x)}{\sin(\nu \pi )}}={\frac {\pi }{2}}i^{\nu +1}H_{\nu }^{(1)}(ix)={\frac {\pi }{2}}(-i)^{\nu +1}H_{\nu }^{(2)}(-ix).\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51f1fd17638dbc8071c5517bb3a1217bfe5bb9ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:69.095ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}K_{\nu }(x)={\frac {\pi }{2}}{\frac {I_{-\nu }(x)-I_{\nu }(x)}{\sin(\nu \pi )}}={\frac {\pi }{2}}i^{\nu +1}H_{\nu }^{(1)}(ix)={\frac {\pi }{2}}(-i)^{\nu +1}H_{\nu }^{(2)}(-ix).\end{aligned}}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6639aa86f71ced03336ab744875d41df93a9079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.216ex; height:2.843ex;" alt="{\displaystyle K_{\nu }(x)}" loading="lazy"> ist auch als MacDonald-Funktion bekannt. Anders als die „normalen“ Besselfunktionen weisen die modifizierten Besselfunktionen kein oszillierendes, sondern ein exponentielles Verhalten auf.

Exemplarisch wird im Folgenden Bessel-Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{0}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/893d08e90ea73781dc133414d661529d0651ca80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{0}}" loading="lazy"> dargestellt:

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<annotation encoding="application/x-tex">{\displaystyle I_{0}(x)=\sum _{n=0}^{\infty }{\frac {x^{2n}}{4^{n}(n!)^{2}}}=\int _{0}^{\pi }{\frac {1}{\pi }}\cosh {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y=\int _{0}^{1}{\frac {2\cosh(xz)}{\pi {\sqrt {1-z^{2}}}}}\,\mathrm {d} z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b11471e2644a5d3778cdc65564ec1264ea1ce14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.749ex; height:7.009ex;" alt="{\displaystyle I_{0}(x)=\sum _{n=0}^{\infty }{\frac {x^{2n}}{4^{n}(n!)^{2}}}=\int _{0}^{\pi }{\frac {1}{\pi }}\cosh {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y=\int _{0}^{1}{\frac {2\cosh(xz)}{\pi {\sqrt {1-z^{2}}}}}\,\mathrm {d} z}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Airysche_Integrale">Airysche Integrale</h3></div>

Für die Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1/3}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{1/3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cce04af0734a382739626554c25816d7e3d62d7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.671ex; height:3.009ex;" alt="{\displaystyle K_{1/3}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2/3}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{2/3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d886e1985e3cd8e4642ad6abeb06f020f22b2e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.671ex; height:3.009ex;" alt="{\displaystyle K_{2/3}}" loading="lazy"> kann man eine Integraldarstellung angeben

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K_{1/3}(x)&={\sqrt {3}}\int _{0}^{\infty }\cos \left({\frac {3}{2}}x\left(u+{\frac {u^{3}}{3}}\right)\right)\mathrm {d} u\\K_{2/3}(x)&={\sqrt {3}}\int _{0}^{\infty }u\sin \left({\frac {3}{2}}x\left(u+{\frac {u^{3}}{3}}\right)\right)\mathrm {d} u\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/477d3563f780e392a12541988e24edcb5373f7d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:45.015ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}K_{1/3}(x)&={\sqrt {3}}\int _{0}^{\infty }\cos \left({\frac {3}{2}}x\left(u+{\frac {u^{3}}{3}}\right)\right)\mathrm {d} u\\K_{2/3}(x)&={\sqrt {3}}\int _{0}^{\infty }u\sin \left({\frac {3}{2}}x\left(u+{\frac {u^{3}}{3}}\right)\right)\mathrm {d} u\end{aligned}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Hypergeometrische_Funktion_2">Hypergeometrische Funktion</h3></div>

Auch die modifizierte Bessel-Funktion erster Gattung kann durch eine <a href="Verallgemeinerte_hypergeometrische_Funktion" title="Verallgemeinerte hypergeometrische Funktion">verallgemeinerte hypergeometrische Funktion</a> ausgedrückt werden:

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<annotation encoding="application/x-tex">{\displaystyle I_{\nu }(x)={\frac {(x/2)^{\nu }}{\Gamma (\nu +1)}}\;_{0}F_{1}(\nu +1;x^{2}/4)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bc82dcde66979e8905184d077b663417fec8b70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.733ex; height:6.509ex;" alt="{\displaystyle I_{\nu }(x)={\frac {(x/2)^{\nu }}{\Gamma (\nu +1)}}\;_{0}F_{1}(\nu +1;x^{2}/4)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beziehungen_von_Ordnungen_einer_Gattung_2">Beziehungen von Ordnungen einer Gattung</h3></div>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f151831e9842e5daee3a71075a17102d10f1cf7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.077ex; height:2.509ex;" alt="{\displaystyle K_{\nu }}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\nu }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/172f1a90292ff1160dabfbc32fd0dcb802fc9f98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.509ex;" alt="{\displaystyle I_{\nu }}" loading="lazy"> gelten die <a href="Rekursion" title="Rekursion">Rekursionsbeziehungen</a>:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\nu }{x}}K_{\nu }=-{\frac {1}{2}}\left(K_{\nu -1}-K_{\nu +1}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1c76dc386ade2a8fa6dd52a0506bcd1075b0e26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.538ex; height:5.176ex;" alt="{\displaystyle {\frac {\nu }{x}}K_{\nu }=-{\frac {1}{2}}\left(K_{\nu -1}-K_{\nu +1}\right)}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\nu }{x}}I_{\nu }={\frac {1}{2}}\left(I_{\nu -1}-I_{\nu +1}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fc27df4840e97cf3347ba15666cbaf93927687c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.881ex; height:5.176ex;" alt="{\displaystyle {\frac {\nu }{x}}I_{\nu }={\frac {1}{2}}\left(I_{\nu -1}-I_{\nu +1}\right)}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}K_{\nu }=-{\frac {1}{2}}(K_{\nu -1}+K_{\nu +1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3eb7c4b02b076bd19a73ff8772361c558abd3817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.444ex; height:5.509ex;" alt="{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}K_{\nu }=-{\frac {1}{2}}(K_{\nu -1}+K_{\nu +1})}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}I_{\nu }={\frac {1}{2}}(I_{\nu -1}+I_{\nu +1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/774c96bf6fc8b4eb30555b9a3e8f9d46c6c121ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.786ex; height:5.509ex;" alt="{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}I_{\nu }={\frac {1}{2}}(I_{\nu -1}+I_{\nu +1})}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Asymptotisches_Verhalten_2">Asymptotisches Verhalten</h3></div>

Wir nehmen wieder an, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> reell und nicht-negativ ist. Für kleine Argumente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x\ll {\sqrt {\nu +1}}}">

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<annotation encoding="application/x-tex">{\displaystyle 0<x\ll {\sqrt {\nu +1}}}</annotation>

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<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}I_{\nu }(x)&\approx {\frac {1}{\Gamma (\nu +1)}}\left({\frac {x}{2}}\right)^{\nu }\\\\K_{\nu }(x)&\approx {\begin{cases}-\left(\ln \left({\frac {x}{2}}\right)+\gamma \right)&{\text{wenn }}\nu =0\\\\{\frac {\Gamma (\nu )}{2}}\left({\frac {2}{x}}\right)^{\nu }&{\text{wenn }}\nu >0\,.\end{cases}}\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{\nu }(x)&\approx {\frac {1}{\Gamma (\nu +1)}}\left({\frac {x}{2}}\right)^{\nu }\\\\K_{\nu }(x)&\approx {\begin{cases}-\left(\ln \left({\frac {x}{2}}\right)+\gamma \right)&{\text{wenn }}\nu =0\\\\{\frac {\Gamma (\nu )}{2}}\left({\frac {2}{x}}\right)^{\nu }&{\text{wenn }}\nu >0\,.\end{cases}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6e72caffa47b93e6ee560d71df3b96b82a2596e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:41.882ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}I_{\nu }(x)&\approx {\frac {1}{\Gamma (\nu +1)}}\left({\frac {x}{2}}\right)^{\nu }\\\\K_{\nu }(x)&\approx {\begin{cases}-\left(\ln \left({\frac {x}{2}}\right)+\gamma \right)&{\text{wenn }}\nu =0\\\\{\frac {\Gamma (\nu )}{2}}\left({\frac {2}{x}}\right)^{\nu }&{\text{wenn }}\nu >0\,.\end{cases}}\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle x\gg |\nu ^{2}-1/4|}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{\nu }(x)&\approx {\frac {1}{\sqrt {2\pi x}}}e^{x}\left(1+{\mathcal {O}}\left({\frac {1}{x}}\right)\right)\\\\K_{\nu }(x)&\approx {\sqrt {\frac {\pi }{2x}}}e^{-x}\left(1+{\mathcal {O}}\left({\frac {1}{x}}\right)\right)\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2ebb90fab3ed5d217f31556e8b88a19380861ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:36.308ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}I_{\nu }(x)&\approx {\frac {1}{\sqrt {2\pi x}}}e^{x}\left(1+{\mathcal {O}}\left({\frac {1}{x}}\right)\right)\\\\K_{\nu }(x)&\approx {\sqrt {\frac {\pi }{2x}}}e^{-x}\left(1+{\mathcal {O}}\left({\frac {1}{x}}\right)\right)\,.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Sphärische_Besselfunktionen:_jμ,_yμ,_hμ(1,2)">Sphärische Besselfunktionen: jμ, yμ, hμ(1,2)</h2></div>

Die <a href="Helmholtz-Gleichung" title="Helmholtz-Gleichung">Helmholtz-Gleichung</a> in <a href="Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a> führt nach Separation der Variablen auf die Radialgleichung

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<annotation encoding="application/x-tex">{\displaystyle x^{2}{\frac {\mathrm {d} ^{2}f_{\mu }(x)}{\mathrm {d} x^{2}}}+2x{\frac {\mathrm {d} f_{\mu }(x)}{\mathrm {d} x}}+[x^{2}-\mu (\mu +1)]f_{\mu }(x)=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08c9ebd8d68202e770f63575e17505fdce1673b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:51.768ex; height:6.343ex;" alt="{\displaystyle x^{2}{\frac {\mathrm {d} ^{2}f_{\mu }(x)}{\mathrm {d} x^{2}}}+2x{\frac {\mathrm {d} f_{\mu }(x)}{\mathrm {d} x}}+[x^{2}-\mu (\mu +1)]f_{\mu }(x)=0}" loading="lazy">.</dd></dl>

Nach der Substitution

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<annotation encoding="application/x-tex">{\displaystyle f_{\mu }(x)={\frac {1}{\sqrt {x}}}u_{\mu }(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2285ecc3c554d595914110e53bd1556f2815897d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.394ex; height:6.176ex;" alt="{\displaystyle f_{\mu }(x)={\frac {1}{\sqrt {x}}}u_{\mu }(x)}" loading="lazy"></dd></dl>

erhält man die Besselsche Differentialgleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\nu =\mu +1/2)}">

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<annotation encoding="application/x-tex">{\displaystyle (\nu =\mu +1/2)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08950d6ffec6d8eb3c900ccbb4f1a131d2ee9a72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.869ex; height:2.843ex;" alt="{\displaystyle (\nu =\mu +1/2)}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}{\frac {\mathrm {d} ^{2}u_{\mu }(x)}{\mathrm {d} x^{2}}}+x{\frac {\mathrm {d} u_{\mu }(x)}{\mathrm {d} x}}+\left[x^{2}-\left(\mu +{\frac {1}{2}}\right)^{2}\right]u_{\mu }(x)=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c59898f5b0bf75c37b6068e107d621a1dd1e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.081ex; height:7.509ex;" alt="{\displaystyle x^{2}{\frac {\mathrm {d} ^{2}u_{\mu }(x)}{\mathrm {d} x^{2}}}+x{\frac {\mathrm {d} u_{\mu }(x)}{\mathrm {d} x}}+\left[x^{2}-\left(\mu +{\frac {1}{2}}\right)^{2}\right]u_{\mu }(x)=0}" loading="lazy">.</dd></dl>

Für die Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mu }(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83627ad492ce579b73b09ce567547e1c2d9efac7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.502ex; height:3.009ex;" alt="{\displaystyle f_{\mu }(x)}" loading="lazy"> der Radialgleichung werden üblicherweise die sphärischen Bessel-Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\mu }(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0da36600857514bc114f8a7376349f9c02a078a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:5.347ex; height:3.009ex;" alt="{\displaystyle j_{\mu }(x)}" loading="lazy">, die sphärischen Neumann-Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{\mu }(x)=n_{\mu }(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dac74cb73dc23bffa3e95b7471e9c4f342e7ff70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.357ex; height:3.009ex;" alt="{\displaystyle y_{\mu }(x)=n_{\mu }(x)}" loading="lazy"> und die sphärischen Hankel-Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\mu }^{(1,2)}(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc140339001c24778d27523a68faa26d955aa4e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.091ex; height:3.676ex;" alt="{\displaystyle h_{\mu }^{(1,2)}(x)}" loading="lazy"> definiert:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&j_{\mu }(x)\quad ={\sqrt {\frac {\pi }{2x}}}J_{\mu +1/2}(x)\\&y_{\mu }(x)\quad ={\sqrt {\frac {\pi }{2x}}}Y_{\mu +1/2}(x)\\&h_{\mu }^{(1,2)}(x)={\sqrt {\frac {\pi }{2x}}}H_{\mu +1/2}^{(1,2)}=j_{\mu }(x)\pm iy_{\mu }(x)\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dad14639153f7cbdb7dc8d0b5065b41ad6b2624c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:42.055ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}&j_{\mu }(x)\quad ={\sqrt {\frac {\pi }{2x}}}J_{\mu +1/2}(x)\\&y_{\mu }(x)\quad ={\sqrt {\frac {\pi }{2x}}}Y_{\mu +1/2}(x)\\&h_{\mu }^{(1,2)}(x)={\sqrt {\frac {\pi }{2x}}}H_{\mu +1/2}^{(1,2)}=j_{\mu }(x)\pm iy_{\mu }(x)\end{aligned}}}" loading="lazy">.</dd></dl>

Es gelten die alternativen Darstellungen für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42411e85d874a733209223302bbd8d5e3ad04cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&j_{m}(x)\quad =(-x)^{m}\left({\frac {1}{x}}{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{m}\ {\frac {\sin x}{x}}\\&y_{m}(x)\quad =-(-x)^{m}\left({\frac {1}{x}}{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{m}\ {\frac {\cos x}{x}}\\&h_{m}^{(1,2)}(x)=\mp i(-x)^{m}\left({\frac {1}{x}}{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{m}{\frac {e^{\pm ix}}{x}}\\\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44ba75c073a447dc81245b11d8775d3291c4f242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:37.521ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}&j_{m}(x)\quad =(-x)^{m}\left({\frac {1}{x}}{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{m}\ {\frac {\sin x}{x}}\\&y_{m}(x)\quad =-(-x)^{m}\left({\frac {1}{x}}{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{m}\ {\frac {\cos x}{x}}\\&h_{m}^{(1,2)}(x)=\mp i(-x)^{m}\left({\frac {1}{x}}{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{m}{\frac {e^{\pm ix}}{x}}\\\end{aligned}}}" loading="lazy"></dd></dl>

Die sphärischen Bessel- und Hankelfunktionen werden beispielsweise für die Behandlung des kugelsymmetrischen Potentialtopfs in der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> benötigt.

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften_2">Eigenschaften</h3></div>

<ul><li>Für die sphärischen Bessel-Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\mu }}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f321b23df64c46e52f3902f58aaf47fe8359fe15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:2.208ex; height:2.843ex;" alt="{\displaystyle j_{\mu }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{\mu }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2957545f10d89e02ebf6454fdb8a78e5b3b1b0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.363ex; height:2.343ex;" alt="{\displaystyle y_{\mu }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\mu }^{(1)}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fa5f98588c3d588a334a2f69c30b148b6fb8086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.673ex; height:3.676ex;" alt="{\displaystyle h_{\mu }^{(1)}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\mu }^{(2)}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8c1e8b9e7f622eff3c567ba3e96751bdd40107a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.673ex; height:3.676ex;" alt="{\displaystyle h_{\mu }^{(2)}}" loading="lazy"> gelten die <a href="Rekursion" title="Rekursion">Rekursionsbeziehungen</a>:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {2\mu +1}{x}}\omega _{\mu }(x)\;\,=\omega _{\mu -1}(x)+\omega _{\mu +1}(x)\\&(2\mu +1)\omega '_{\mu }(x)\,=\mu \omega _{\mu -1}(x)-(\mu +1)\omega _{\mu +1}(x)\\&{\frac {\mathrm {d} }{\mathrm {d} x}}(x\omega _{\mu }(x))\quad =x\omega _{\mu -1}(x)-\mu \omega _{\mu }(x)\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f770051f2fa1d34bdd05fb80c3442e8975423dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:45.695ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}&{\frac {2\mu +1}{x}}\omega _{\mu }(x)\;\,=\omega _{\mu -1}(x)+\omega _{\mu +1}(x)\\&(2\mu +1)\omega '_{\mu }(x)\,=\mu \omega _{\mu -1}(x)-(\mu +1)\omega _{\mu +1}(x)\\&{\frac {\mathrm {d} }{\mathrm {d} x}}(x\omega _{\mu }(x))\quad =x\omega _{\mu -1}(x)-\mu \omega _{\mu }(x)\end{aligned}}}" loading="lazy">.</dd></dl>

<ul><li>Für die <a href="Wronski-Determinante" title="Wronski-Determinante">Wronski-Determinante</a> gilt</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle W(j_{\mu },y_{\mu })={\frac {1}{i}}W(j_{\mu },h_{\mu }^{(1)})=-W(y_{\mu },h_{\mu }^{(1)})={\frac {1}{x^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84df7ffdbb1beca8441dc46b8c92f305a851b37d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:48.591ex; height:5.509ex;" alt="{\displaystyle W(j_{\mu },y_{\mu })={\frac {1}{i}}W(j_{\mu },h_{\mu }^{(1)})=-W(y_{\mu },h_{\mu }^{(1)})={\frac {1}{x^{2}}}}" loading="lazy">.</dd></dl>

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Hankel-Transformation" title="Hankel-Transformation">Hankel-Transformation</a></div>

Die Hankel-Transformation ist eine <a href="Integraltransformation" class="mw-redirect" title="Integraltransformation">Integraltransformation</a>, die eng mit der <a href="Fourier-Transformation" title="Fourier-Transformation">Fourier-Transformation</a> verwandt ist. Der <a href="Integralkern" class="mw-redirect" title="Integralkern">Integralkern</a> der Hankel-Transformation ist die Bessel-Funktion erster Gattung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22b79b47e08e4e16510d309639ef56a24c28696c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.509ex;" alt="{\displaystyle J_{n}}" loading="lazy">, das heißt, der Integraloperator lautet:

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<annotation encoding="application/x-tex">{\displaystyle H_{n}[f](s)=\int _{0}^{\infty }J_{n}(ts)tf(t)\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cd6286755495c71b0277b7a3e8c1286fc3d8f28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.594ex; height:5.843ex;" alt="{\displaystyle H_{n}[f](s)=\int _{0}^{\infty }J_{n}(ts)tf(t)\mathrm {d} t}" loading="lazy">.</dd></dl>

Eine besondere Eigenschaft der Hankel-Transformation ist, dass mit ihr der Bessel-Operator in einen algebraischen Ausdruck (eine Multiplikation) überführt werden kann.

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

Bessel-Funktionen wurden von Bessel 1824 ausführlich behandelt,<a href="#cite_note-2">[2]</a> tauchten aber auch schon vorher bei speziellen physikalischen Problemen auf, zum Beispiel bei <a href="Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> (Schwingung schwerer Ketten 1738), <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (Membranschwingung 1764), in der <a href="Himmelsmechanik" title="Himmelsmechanik">Himmelsmechanik</a> bei <a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> (1770) und bei <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>, in der Wärmeleitung bei <a href="Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> (Wärmeausbreitung im Zylinder 1822) und <a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a> (1823).<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a>

<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 Stegun</a>: Handbook of Mathematical Functions. Dover, New York 1972, <a rel="nofollow" class="external text" href="https://www.math.sfu.ca/~cbm/aands/page_355.htm">S. 355</a>.</li>

<li>J. H. Graf, E. Gubler: Einleitung in die Theorie der Bessel’schen Funktionen. <a rel="nofollow" class="external text" href="http://resolver.library.cornell.edu/math/1927697">Erster Band</a> <a rel="nofollow" class="external text" href="https://resolver.library.cornell.edu/math/1927697a">Zweiter Band</a>. K. J. Wyss, Bern 1900</li>

<li><a href="Carl_Gottfried_Neumann" title="Carl Gottfried Neumann">Carl Gottfried Neumann</a>: Theorie der Besselschen Funktionen, ein Analogon zur Theorie der Kugelfunktionen. B. G. Teubner, Leipzig 1867.</li>

<li><a href="Paul_Schafheitlin" title="Paul Schafheitlin">Paul Schafheitlin</a> <a rel="nofollow" class="external text" href="http://www.archive.org/details/dietheoriederbes027703mbp">Die Theorie der Besselschen Funktionen</a>. B. G. Teubner, Leipzig 1908.</li>

<li><a href="G._N._Watson" class="mw-redirect" title="G. N. Watson">G. N. Watson</a> A Treatise on the Theory of Bessel functions, Cambridge University Press 1922, 1944, <a rel="nofollow" class="external text" href="https://archive.org/details/treatiseontheory00watsuoft">Archive</a></li></ul>

Besselfunktionen werden in vielen Lehrbüchern der Theoretischen Physik behandelt z. B.:

<ul><li><a href="John_David_Jackson_(Physiker)" title="John David Jackson (Physiker)">John David Jackson</a>: Classical Electrodynamics. John Wiley, New York NY 1962 (3. edition. ebenda 1999, ISBN 0-471-30932-X; deutsch: 4. überarbeitete Auflage. de Gruyter, Berlin u. a. 2006, ISBN 3-11-018970-4).</li>

<li>Wolfgang Nolting: Grundkurs Theoretische Physik 5/2. Quantenmechanik – Methoden und Anwendungen, 6. Auflage, Springer-Lehrbuch, 2006, ISBN 978-3-540-26035-6</li>

<li><a href="Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> Vorlesungen über Theoretische Physik, Band 6: Partielle Differentialgleichungen der Physik, Harri Deutsch 1992, ISBN 3-87144-379-4.</li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <cite style="font-style:italic">Bessel-Operator</cite>. In: Guido Walz (Hrsg.): <cite style="font-style:italic">Lexikon der Mathematik</cite>. 1. Auflage. Spektrum Akademischer Verlag, Mannheim/Heidelberg 2000, ISBN 978-3-8274-0439-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bessel-Funktion&rft.atitle=Bessel-Operator&rft.btitle=Lexikon+der+Mathematik&rft.date=2000&rft.edition=1&rft.genre=book&rft.isbn=9783827404398&rft.place=Mannheim%2FHeidelberg&rft.pub=Spektrum+Akademischer+Verlag" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Friedrich Wilhelm Bessel: <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_Un4EAAAAYAAJ/page/n121">Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne entsteht.</a> In: Abhandlungen der Berliner Akademie der Wissenschaften 1824, Math. Classe, S. 1–52, Berlin 1826.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Jacques Dutka: On the early history of Bessel functions. In: <a href="Archive_for_History_of_Exact_Sciences" title="Archive for History of Exact Sciences">Archive for History of Exact Sciences</a>. Band 49, 1995, S. 105–134.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> G. N. Watson: Theory of Bessel Functions. Cambridge University Press, 1944, Kapitel 1 (zur Geschichte).

</li>

</ol>

<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/BesselDifferentialEquation.html">Bessel Differential Equation.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-12-14" href="https://de.wikipedia.org/wiki/?title=Bessel-Funktion&oldid=262404657">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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