GermanWikipedia - /page/entry.mu

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<title>Hankel-Transformation</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Hankel-Transformation</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">Die Hankel-Transformation ist in der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>, einem Teilgebiet der <a href="Mathematik" title="Mathematik">Mathematik</a>, eine lineare <a href="Integraltransformation" class="mw-redirect" title="Integraltransformation">Integraltransformation</a>, welche im Kern auf den <a href="Bessel-Funktion" title="Bessel-Funktion">Bessel-Funktionen</a> erster Gattung basiert. Sie ist benannt nach dem Mathematiker <a href="Hermann_Hankel" title="Hermann Hankel">Hermann Hankel</a>. Anwendungen liegen unter anderem im Bereich der <a href="Bildverarbeitung" title="Bildverarbeitung">Bildverarbeitung</a> zur Korrektur von Abbildungsfehlern.<a href="#cite_note-1">[1]</a>

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

Bei der Hankel-Transformation gibt es unterschiedliche Konventionen, sie zu definieren. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> eine <a href="Komplexwertige_Funktion" title="Komplexwertige Funktion">komplexwertige Funktion</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu >-{\tfrac {1}{2}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08240a50467b28ba0e7dd2bb474813f20b0349ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.797ex; height:3.509ex;" alt="{\displaystyle \nu >-{\tfrac {1}{2}}}" loading="lazy">.

Dann kann man die Hankel-Transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} _{\nu }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9824cb779fe934beece00cbcfad07b99eaa1c813.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.847ex; height:2.509ex;" alt="{\displaystyle \operatorname {H} _{\nu }}" loading="lazy"> der Ordnung <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"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> durch

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<annotation encoding="application/x-tex">{\displaystyle F_{\nu }(u)=\operatorname {H} _{\nu }\{f(t)\}=\int _{0}^{\infty }f(t)\cdot J_{\nu }(ut)\cdot t\,\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b2ff0a2be74135cc97486e8b881a40f5c6fefd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.777ex; height:5.843ex;" alt="{\displaystyle F_{\nu }(u)=\operatorname {H} _{\nu }\{f(t)\}=\int _{0}^{\infty }f(t)\cdot J_{\nu }(ut)\cdot t\,\mathrm {d} t}" loading="lazy"></dd></dl>

definieren, dabei sind die

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49fade6cfb3ab04000d2ef75ef7d9ceeb839feb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.938ex; height:7.343ex;" alt="{\displaystyle J_{\nu }(x):=\sum _{r=0}^{\infty }{\frac {(-1)^{r}({\frac {x}{2}})^{2r+\nu }}{\Gamma (\nu +r+1)r!}}}" loading="lazy"></dd></dl>

<a href="Besselsche_Differentialgleichung#Bessel-Funktion_erster_Gattung" class="mw-redirect" title="Besselsche Differentialgleichung">Bessel-Funktionen erster Gattung</a> und <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"> ist die <a href="Gammafunktion" title="Gammafunktion">Gammafunktion</a>. Insofern das Integral existiert, nennt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} _{\nu }\{f(t)\}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{\nu }\{f(t)\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f47b0904adfce76e3e2f175e0ebac66b9f4e4684.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.099ex; height:2.843ex;" alt="{\displaystyle \operatorname {H} _{\nu }\{f(t)\}}" loading="lazy"> die Hankel-Transformierte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. Diese Konvention der Hankel-Transformation wird überwiegend in diesem Artikel verwendet. In einzelnen Abschnitten wird jedoch die im Folgenden dargestellte Variante verwendet, worauf in den entsprechenden Abschnitten hingewiesen wird.

Eine andere Möglichkeit, die Hankel-Transformation der Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu >-{\tfrac {1}{2}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08240a50467b28ba0e7dd2bb474813f20b0349ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.797ex; height:3.509ex;" alt="{\displaystyle \nu >-{\tfrac {1}{2}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> zu definieren, ist

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<annotation encoding="application/x-tex">{\displaystyle F_{\nu }(u)=\operatorname {H} _{\nu }\{f(t)\}=\int _{0}^{\infty }f(t)\cdot {\sqrt {ut}}\cdot J_{\nu }(ut)\,\mathrm {d} t\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd8ed0e75190f6689c7c841b582b79f852bc646d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.076ex; height:5.843ex;" alt="{\displaystyle F_{\nu }(u)=\operatorname {H} _{\nu }\{f(t)\}=\int _{0}^{\infty }f(t)\cdot {\sqrt {ut}}\cdot J_{\nu }(ut)\,\mathrm {d} t\,.}" loading="lazy"></dd></dl>

Hier werden mit <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"> ebenfalls die Bessel-Funktionen erster Gattung bezeichnet und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} _{\nu }\{f(t)\}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{\nu }\{f(t)\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f47b0904adfce76e3e2f175e0ebac66b9f4e4684.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.099ex; height:2.843ex;" alt="{\displaystyle \operatorname {H} _{\nu }\{f(t)\}}" loading="lazy"> heißt auch hier Hankel-Transformierte, insofern das Integral existiert.

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

Ähnlich wie bei der <a href="Fourier-Transformation" title="Fourier-Transformation">Fourier-Transformation</a> ist es auch bei der Hankel-Transformation unter gewissen Umständen möglich, aus der Hankel-Transformierten ihre Ausgangsfunktion zurückzugewinnen. Ein wichtiges Resultat aus der Theorie der Hankel-Transformation besagt, dass, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{1}(]0,\infty [)}">

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<annotation encoding="application/x-tex">{\displaystyle f\in L^{1}(]0,\infty [)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2589a8a2d98e68d1fe57b069d19d222beda8ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.379ex; height:3.176ex;" alt="{\displaystyle f\in L^{1}(]0,\infty [)}" loading="lazy"> eine <a href="Lebesgue-Integral" title="Lebesgue-Integral">Lebesgue-integrierbare Funktion</a> mit <a href="Beschr%C3%A4nkte_Variation" title="Beschränkte Variation">beschränkter Variation</a> ist, die Ausgangsfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> aus der Hankel-Transformierten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\nu }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3111ed8ce7dfd556b2efdb06eff374d1923f9d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.598ex; height:2.509ex;" alt="{\displaystyle F_{\nu }}" loading="lazy"> mit der inversen Integraltransformation

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<annotation encoding="application/x-tex">{\displaystyle f(t)=\operatorname {H} _{\nu }^{-1}\{F_{\nu }(u)\}=\int _{0}^{\infty }F_{\nu }(u)\cdot J_{\nu }(ut)\cdot u\,\mathrm {d} u}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0be4d0700222fc1b43e5bc1fed8cfd7fce8798e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.796ex; height:5.843ex;" alt="{\displaystyle f(t)=\operatorname {H} _{\nu }^{-1}\{F_{\nu }(u)\}=\int _{0}^{\infty }F_{\nu }(u)\cdot J_{\nu }(ut)\cdot u\,\mathrm {d} u}" loading="lazy"></dd></dl>

zurückgewonnen werden kann. Die Hankel-Transformation und ihre inverse Transformation sind also gleich. Sie kann daher als <a href="Involution_(Mathematik)" title="Involution (Mathematik)">involutive Abbildung</a> verstanden werden.

Für die alternative Definition gilt diese Aussage analog.

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

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

Die Bessel-Funktionen bilden eine <a href="Orthogonalbasis" class="mw-redirect" title="Orthogonalbasis">Orthogonalbasis</a>: Es gilt

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }J_{\nu }(ut)\cdot J_{\nu }(u't)\cdot t\,\mathrm {d} t={\frac {\delta (u-u')}{u}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b02ce6699992e24505c442c0a76489bc66f2243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.85ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{\infty }J_{\nu }(ut)\cdot J_{\nu }(u't)\cdot t\,\mathrm {d} t={\frac {\delta (u-u')}{u}}}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bee645f272e64333868e2baa275419eca4ee0718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle u'}" loading="lazy"> größer 0 und mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> als der <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a>.

<div class="mw-heading mw-heading3"><h3 id="Algebraisierung_des_besselschen_Differentialoperators">Algebraisierung des besselschen Differentialoperators</h3></div>

Sei

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78744fc04905b5cd2c35affe8f89d2c474521f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.809ex; height:6.343ex;" alt="{\displaystyle B_{\nu }(f)(r):=\left(r^{2}{\frac {\mathrm {d} ^{2}}{\mathrm {d} r^{2}}}+r{\frac {\mathrm {d} }{\mathrm {d} r}}+(r^{2}-\nu ^{2})\right)f(r)}" loading="lazy"></dd></dl>

der <a href="Besselsche_Differentialgleichung" class="mw-redirect" title="Besselsche Differentialgleichung">besselsche Differentialoperator</a>. Für die Bessel-Funktionen gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\nu }(J_{\nu })=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb0df4ad677dfd0fcc156d6b1dce244725222bdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.331ex; height:2.843ex;" alt="{\displaystyle B_{\nu }(J_{\nu })=0}" loading="lazy">. Mit Hilfe der Hankel-Transformation ist es möglich, diesen <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a> in einen Ausdruck ohne Ableitungen zu überführen. Präzise gilt

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{\nu }(L_{\nu }(f))(s)=-s^{2}\operatorname {H} _{\nu }(f)(s)\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b785c0e24ce432c8e3efd4e8c082a4a4125a9821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.636ex; height:3.176ex;" alt="{\displaystyle \operatorname {H} _{\nu }(L_{\nu }(f))(s)=-s^{2}\operatorname {H} _{\nu }(f)(s)\,,}" loading="lazy"></dd></dl>

wobei der Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\nu }(f)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1ff3ea1d12fd3f6e10f77311b95dccc9ee41663.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.774ex; height:2.843ex;" alt="{\displaystyle L_{\nu }(f)}" loading="lazy"> als

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<annotation encoding="application/x-tex">{\displaystyle L_{\nu }(f)(r)={\frac {B_{\nu }(f)(r)}{r^{2}}}-f(r)=\left({\frac {\mathrm {d} ^{2}}{\mathrm {d} r^{2}}}+{\frac {1}{r}}{\frac {\mathrm {d} }{\mathrm {d} r}}-\nu ^{2}\right)f(r)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8789f74b36bafdcd94305445249aff1a00758a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.8ex; height:6.343ex;" alt="{\displaystyle L_{\nu }(f)(r)={\frac {B_{\nu }(f)(r)}{r^{2}}}-f(r)=\left({\frac {\mathrm {d} ^{2}}{\mathrm {d} r^{2}}}+{\frac {1}{r}}{\frac {\mathrm {d} }{\mathrm {d} r}}-\nu ^{2}\right)f(r)}" loading="lazy"></dd></dl>

definiert wurde. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7229c47b5bc20ef0a1371a4f3c09459ccb6909ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =0}" loading="lazy"> stimmt dieser mit dem Radialantail des zweidimensionalen <a href="Laplace-Operator#In_zwei_Dimensionen" title="Laplace-Operator">Laplaceoperators in Polarkoordinaten</a> überein.

Dies ist eine zentrale Eigenschaft der Hankel-Transformation zum Lösen von <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichungen</a>.<a href="#cite_note-poularikas9.4-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Beziehung_zur_Fourier-Transformation">Beziehung zur Fourier-Transformation</h2></div>

Die Hankel-Transformation hat einige Analogien zur Fourier-Transformation. Insbesondere lässt sich die Hankel-Transformierte durch eine zweidimensionale <a href="Fourier-Transformation" title="Fourier-Transformation">Fourier-Transformation</a> berechnen. Sei dazu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon \mathbb {R} ^{2}\to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \phi \colon \mathbb {R} ^{2}\to \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcad46faae8726b0dfc826b87220aaec75d52c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.444ex; height:3.009ex;" alt="{\displaystyle \phi \colon \mathbb {R} ^{2}\to \mathbb {C} }" loading="lazy"> eine <a href="Radialsymmetrische_Funktion" class="mw-redirect" title="Radialsymmetrische Funktion">radialsymmetrische Funktion</a>. Das heißt, die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(r,\theta ):=\phi (r\cos(\theta ),r\sin(\theta ))}">

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<annotation encoding="application/x-tex">{\displaystyle f(r,\theta ):=\phi (r\cos(\theta ),r\sin(\theta ))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8e16922ae5d464db70f3f4e2a1443dc3dc84a7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.873ex; height:2.843ex;" alt="{\displaystyle f(r,\theta ):=\phi (r\cos(\theta ),r\sin(\theta ))}" loading="lazy"> ist unabhängig von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">, weshalb sie im Folgenden nur mit dem Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> notiert wird. Von dieser Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> wird nun mit Hilfe der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> und der Fourier-Transformation die Hankel-Transformierte beschrieben.

Um dies zu sehen, wird das Fourier-Integral

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\phi )(\xi _{1},\xi _{2})={\frac {1}{2\pi }}\int _{\mathbb {R} ^{2}}\phi (x,y)e^{-i(x\xi _{1}+y\xi _{2})}\,\mathrm {d} (x,y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc943c258ca5fb41b02134f152a2495c28a97694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.158ex; height:5.676ex;" alt="{\displaystyle {\mathcal {F}}(\phi )(\xi _{1},\xi _{2})={\frac {1}{2\pi }}\int _{\mathbb {R} ^{2}}\phi (x,y)e^{-i(x\xi _{1}+y\xi _{2})}\,\mathrm {d} (x,y)}" loading="lazy"></dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> in <a href="Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a> transformiert, was zu

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}(\phi )(s\cos(\sigma ),s\sin(\sigma ))=&{\frac {1}{2\pi }}\int _{r=0}^{\infty }r\int _{\theta =0}^{2\pi }\phi (r\cos(\theta ),r\sin(\theta ))e^{-isr\cos(\theta -\sigma )}\,\mathrm {d} \theta \,\mathrm {d} r\\=&{\frac {1}{2\pi }}\int _{r=0}^{\infty }rf(r)\int _{\alpha =0}^{2\pi }e^{-isr\cos(\alpha )}\,\mathrm {d} \alpha \,\mathrm {d} r\\=&\int _{r=0}^{\infty }rf(r)J_{0}(sr)\,\mathrm {d} r\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc8b20ef66c3d7c927ee108cc2e5c53823845da3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:77.019ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {F}}(\phi )(s\cos(\sigma ),s\sin(\sigma ))=&{\frac {1}{2\pi }}\int _{r=0}^{\infty }r\int _{\theta =0}^{2\pi }\phi (r\cos(\theta ),r\sin(\theta ))e^{-isr\cos(\theta -\sigma )}\,\mathrm {d} \theta \,\mathrm {d} r\\=&{\frac {1}{2\pi }}\int _{r=0}^{\infty }rf(r)\int _{\alpha =0}^{2\pi }e^{-isr\cos(\alpha )}\,\mathrm {d} \alpha \,\mathrm {d} r\\=&\int _{r=0}^{\infty }rf(r)J_{0}(sr)\,\mathrm {d} r\end{aligned}}}" loading="lazy"></dd></dl>

führt. Dies zeigt, dass eine Fourier-Transformation einer radialsymmetrischen Funktion immer der Hankel-Transformation einer entsprechenden Funktion entspricht. Insbesondere ist es möglich, zu einer gegebenen Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon {]0,\infty [}\to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle f\colon {]0,\infty [}\to \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1230dd8ffd11b0f8a218f69a66cebd7cee15556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.419ex; height:2.843ex;" alt="{\displaystyle f\colon {]0,\infty [}\to \mathbb {C} }" loading="lazy"> eine entsprechende radialsymmetrische Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> zu konstruieren, mit der man durch Fourier-Transformation die Hankel-Transformierte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> berechnen kann.

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

Ebenfalls wie bei der Fourier-Transformation ist es bei der Hankel-Transformation auf analoge Weise möglich, sie auf <a href="Distribution_(Mathematik)" title="Distribution (Mathematik)">Distributionen</a> zu verallgemeinern. Im Gegensatz zur Fourier-Transformation kann die Hankel-Transformationen nicht auf dem <a href="Temperierte_Distribution" title="Temperierte Distribution">Raum der temperierten Distributionen</a> definiert werden. Daher definiert man einen neuen Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }(]0,\infty [)}">

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<annotation encoding="application/x-tex">{\displaystyle H_{\nu }(]0,\infty [)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce606ae036428c1976dbf97e8d2648d83abdae29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.658ex; height:2.843ex;" alt="{\displaystyle H_{\nu }(]0,\infty [)}" loading="lazy"> und erklärt die Hankel-Transformation für Distributionen auf seinem <a href="Dualraum" title="Dualraum">Dualraum</a>.

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2eb08629e9bd6bdc9f2550b56603c9c602a81f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.751ex; height:2.176ex;" alt="{\displaystyle \nu \in \mathbb {R} }" loading="lazy">, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }(]0,\infty [)}">

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<annotation encoding="application/x-tex">{\displaystyle H_{\nu }(]0,\infty [)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce606ae036428c1976dbf97e8d2648d83abdae29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.658ex; height:2.843ex;" alt="{\displaystyle H_{\nu }(]0,\infty [)}" loading="lazy"> definiert durch

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<annotation encoding="application/x-tex">{\displaystyle H_{\nu }(]0,\infty [):=\left\{\phi \in C^{\infty }(]0,\infty [)\left|\forall k,m\in \mathbb {N} _{0}:\sup _{x\in {]0,\infty [}}\left|x^{m}\left({\tfrac {1}{x}}{\tfrac {\mathrm {d} }{\mathrm {d} x}}\right)^{k}(x^{-\nu -{\frac {1}{2}}}\phi (x))\right|<\infty \right.\right\}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/294bbf02c1de2544cbf7e18a1349327ae790498c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:86.355ex; height:7.509ex;" alt="{\displaystyle H_{\nu }(]0,\infty [):=\left\{\phi \in C^{\infty }(]0,\infty [)\left|\forall k,m\in \mathbb {N} _{0}:\sup _{x\in {]0,\infty [}}\left|x^{m}\left({\tfrac {1}{x}}{\tfrac {\mathrm {d} }{\mathrm {d} x}}\right)^{k}(x^{-\nu -{\frac {1}{2}}}\phi (x))\right|<\infty \right.\right\}\,.}" loading="lazy"></dd></dl>

Auf diesem Vektorraum wird zusätzlich eine <a href="Topologischer_Raum" title="Topologischer Raum">Topologie</a> in Form eines Konvergenzbegriffs definiert. Eine Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\phi _{j})\subset H_{\nu }(]0,\infty [)}">

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<annotation encoding="application/x-tex">{\displaystyle (\phi _{j})\subset H_{\nu }(]0,\infty [)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79791d985cd814f5c3759adf1598880550a920e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.861ex; height:3.009ex;" alt="{\displaystyle (\phi _{j})\subset H_{\nu }(]0,\infty [)}" loading="lazy"> konvergiert genau dann gegen Null, wenn

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<annotation encoding="application/x-tex">{\displaystyle \lim _{j\to \infty }\sup _{x\in {]0,\infty [}}\left|x^{m}\left({\tfrac {1}{x}}{\tfrac {\mathrm {d} }{\mathrm {d} x}}\right)^{k}(x^{-\nu -{\frac {1}{2}}}\phi _{j}(x))\right|=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8db1b30f47bfc033b7e9a40ce889b5c081d32e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.019ex; width:41.258ex; height:6.509ex;" alt="{\displaystyle \lim _{j\to \infty }\sup _{x\in {]0,\infty [}}\left|x^{m}\left({\tfrac {1}{x}}{\tfrac {\mathrm {d} }{\mathrm {d} x}}\right)^{k}(x^{-\nu -{\frac {1}{2}}}\phi _{j}(x))\right|=0}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51a8ba1ee0d4d1f1c842396f0deda1019a261052.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.859ex; height:2.509ex;" alt="{\displaystyle k,m\in \mathbb {N} _{0}}" loading="lazy"> gilt. Durch Bilden des <a href="Dualraum" title="Dualraum">topologischen Dualraums</a> erhält man den Distributionenraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }'(]0,\infty [)}">

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<annotation encoding="application/x-tex">{\displaystyle H_{\nu }'(]0,\infty [)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55b23fb72f06f302d1015524e0bd8e0f5389ea55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.658ex; height:2.843ex;" alt="{\displaystyle H_{\nu }'(]0,\infty [)}" loading="lazy">, auf dem man die Hankel-Transformation definieren kann. Beispielsweise sind alle <a href="Distribution_mit_kompaktem_Tr%C3%A4ger" class="mw-redirect" title="Distribution mit kompaktem Träger">Distributionen mit kompaktem Träger</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ]0,\infty [}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103a5a305a61f6370460c3c4c6882164cd4e0ff4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.814ex; height:2.843ex;" alt="{\displaystyle ]0,\infty [}" loading="lazy">, wie die Delta-Distribution eine ist, in dem Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }'(]0,\infty [)}">

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\nu }'(]0,\infty [)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55b23fb72f06f302d1015524e0bd8e0f5389ea55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.658ex; height:2.843ex;" alt="{\displaystyle H_{\nu }'(]0,\infty [)}" loading="lazy"> enthalten.

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

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in H_{\nu }'(]0,\infty [)}">

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\in H_{\nu }'(]0,\infty [)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf79a590e79fd755f58b91b0d2961344351cdb38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.135ex; height:2.843ex;" alt="{\displaystyle T\in H_{\nu }'(]0,\infty [)}" loading="lazy"> ist die Hankel-Transformation für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in H_{\nu }(]0,\infty [)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi \in H_{\nu }(]0,\infty [)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7075f7781fa97ff2163562928c4001d076685a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.884ex; height:2.843ex;" alt="{\displaystyle \phi \in H_{\nu }(]0,\infty [)}" loading="lazy"> definiert durch

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{\nu }(T)(\phi ):=T(\operatorname {H} _{\nu }(\phi ))\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc9233988c389074fbdd0dfd004e30a6bae06228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.753ex; height:2.843ex;" alt="{\displaystyle \operatorname {H} _{\nu }(T)(\phi ):=T(\operatorname {H} _{\nu }(\phi ))\,.}" loading="lazy"></dd></dl>

Der Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} _{\nu }(\phi )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6abfbfd97127e72f80918d000983f3110c88b1a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.041ex; height:2.843ex;" alt="{\displaystyle \operatorname {H} _{\nu }(\phi )}" loading="lazy"> ist wieder eine Hankel-Transformation einer Funktion und daher definiert. Aufgrund der Konstruktion des Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }(]0,\infty [)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\nu }(]0,\infty [)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce606ae036428c1976dbf97e8d2648d83abdae29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.658ex; height:2.843ex;" alt="{\displaystyle H_{\nu }(]0,\infty [)}" loading="lazy"> wird hier allerdings die Konvention <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \operatorname {H} _{\nu }(\phi )=\int _{0}^{\infty }f(t){\sqrt {ut}}J_{\nu }(ut)\,\mathrm {d} t}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \operatorname {H} _{\nu }(\phi )=\int _{0}^{\infty }f(t){\sqrt {ut}}J_{\nu }(ut)\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04052454cbd18afa726afd8606c458e3440d7876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.841ex; height:3.343ex;" alt="{\displaystyle \textstyle \operatorname {H} _{\nu }(\phi )=\int _{0}^{\infty }f(t){\sqrt {ut}}J_{\nu }(ut)\,\mathrm {d} t}" loading="lazy"> für die Transformation verwendet.

Wie bei der Fourier-Transformation für Distributionen führt man auch die Hankel-Transformation nicht auf der Distribution selbst aus, sondern sie wird auf der Testfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> berechnet.

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

<th>Signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)\,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a471e94c3479917173461b7045481220c91b6182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.315ex; height:2.843ex;" alt="{\displaystyle f(t)\,}" loading="lazy">

</th>

<th>Hankel-Transformierte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}(u):=\operatorname {H} _{0}(f)(u)\,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{0}(u):=\operatorname {H} _{0}(f)(u)\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f20ceb821fdb3d4243e7c079415ac0c8b3acd53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.844ex; height:2.843ex;" alt="{\displaystyle F_{0}(u):=\operatorname {H} _{0}(f)(u)\,}" loading="lazy">

</th></tr>

<tr>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 1\,}" loading="lazy">

</td>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12ed8e3df96895058697bedba7eb57ccfe24835f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.067ex; height:2.843ex;" alt="{\displaystyle \delta (u)/u\,}" loading="lazy">, gültig für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a898503ce5010770d91b6ad0ade3cfbc67ae94e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle u\neq 0}" loading="lazy">

</td></tr>

<tr>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6f2d875a72046fad4c0329ab5b02931ceb24b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.552ex; height:2.843ex;" alt="{\displaystyle 1/t\,}" loading="lazy">

</td>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d2a4311b6a03eb7c26b050a02550ca07be5ed8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.042ex; height:2.843ex;" alt="{\displaystyle 1/u\,}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/946383a7c6d1876177c662a95b369ced2ad99cd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle t\,}" loading="lazy">

</td>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cdabd3fc0f350e53f7141d3edeccff348f7f186.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.904ex; height:3.176ex;" alt="{\displaystyle -1/u^{3}\,}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t^{3}\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3847885057ef3d475c291b46f7ad03a3ca65d06c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.281ex; height:2.676ex;" alt="{\displaystyle t^{3}\,}" loading="lazy">

</td>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9/u^{5}\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/291f9ecb448f7e326819b66a576bca75e09c9749.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.096ex; height:3.176ex;" alt="{\displaystyle 9/u^{5}\,}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2392b00c5d9cbd5011348cc747d1c2315e8fcf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.902ex; height:2.343ex;" alt="{\displaystyle t^{m}\,}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2^{m+1}\Gamma (m/2+1)}{u^{m+2}\Gamma (-m/2)}}\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/115c6deccab8facd296e0a65091d74eca8711235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.791ex; height:6.676ex;" alt="{\displaystyle {\frac {2^{m+1}\Gamma (m/2+1)}{u^{m+2}\Gamma (-m/2)}}\,}" loading="lazy">, gültig für ungerades <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {t^{2}+z^{2}}}}\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/929a04c890b402ac0c92c627df3244d0089bc779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.426ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {t^{2}+z^{2}}}}\,}" loading="lazy">

</td>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e^{-u|z|}}{u}}={\sqrt {\frac {2|z|}{\pi u}}}K_{-1/2}(u|z|)\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d37c9814d0ee29e4ad0ce6998815b892f17711a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.715ex; height:7.676ex;" alt="{\displaystyle {\frac {e^{-u|z|}}{u}}={\sqrt {\frac {2|z|}{\pi u}}}K_{-1/2}(u|z|)\,}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{t^{2}+z^{2}}}\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2e590bb8d8cc8574bbba434ab5ebc2c072cb3ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.102ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{t^{2}+z^{2}}}\,}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc4769f614419ebc973b2ac7a8b734f632ed0f2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.641ex; height:2.843ex;" alt="{\displaystyle K_{0}(uz)\,}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/169fae60c23a2027ece2aa7fd4b5047492887e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\mathrm {i} at}/t\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5623c057a1c5b51e7112dcabc31b9e3cd7248143.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.626ex; height:3.176ex;" alt="{\displaystyle e^{\mathrm {i} at}/t\,}" loading="lazy">

</td>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} /{\sqrt {a^{2}-u^{2}}}\quad (a>0,u<a)\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34ca7b02aab78df92c79d74b6e19907c0d7bc7f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.343ex; height:3.509ex;" alt="{\displaystyle \mathrm {i} /{\sqrt {a^{2}-u^{2}}}\quad (a>0,u<a)\,}" loading="lazy">

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {u^{2}-a^{2}}}\quad (a>0,u>a)\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/addc36e633f3b16f297d64004c281179b8009ed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.859ex; height:3.509ex;" alt="{\displaystyle 1/{\sqrt {u^{2}-a^{2}}}\quad (a>0,u>a)\,}" loading="lazy">

</td></tr>

<tr>

<msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{-a^{2}t^{2}/2}\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2e8bb19fca758bbb6f35d74f903cd0b6921a6d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.752ex; height:3.009ex;" alt="{\displaystyle e^{-a^{2}t^{2}/2}\,}" loading="lazy">

</td>

<mfrac>

<msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e^{-u^{2}/(2a^{2})}}{a^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbdb4b95864fbeff11a8105973ff9675d6cf112c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.827ex; height:6.343ex;" alt="{\displaystyle {\frac {e^{-u^{2}/(2a^{2})}}{a^{2}}}}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5eb3b643b7aba3e7e448b38798ad28d2dcdec5fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.017ex; height:3.176ex;" alt="{\displaystyle -t^{2}f(t)\,}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}F_{\nu }}{du^{2}}}+{\frac {1}{u}}{\frac {dF_{\nu }}{du}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31641afc5af8075ef5f65712d4466d9e940477b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.363ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}F_{\nu }}{du^{2}}}+{\frac {1}{u}}{\frac {dF_{\nu }}{du}}}" loading="lazy">

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

In diesem Abschnitt wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17c650a012565485bca044f8d6ee5e0fdfa84362.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.089ex; height:2.843ex;" alt="{\displaystyle K_{n}(z)}" loading="lazy"> die <a href="Bessel-Funktion" title="Bessel-Funktion">Bessel-Funktionen</a> zweiter Gattung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_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">-ter Ordnung, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> die <a href="Gammafunktion" title="Gammafunktion">Gammafunktion</a>, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> wieder die Delta-Distribution bezeichnet. In der Tabelle auf der rechten Seite werden noch zusätzlich einige Paare von Hankel-Transformationen gelistet.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Die_Hyperbel_1/t">Die Hyperbel 1/t</h3></div>

Für die Hankel-Transformierte nullter Ordnung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{t}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4d2851f22ca31e9ddea8662f3c79833cb29093e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{t}}}" loading="lazy"> gilt

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {H} _{0}\left({\frac {1}{t}}\right)(s)=&\int _{0}^{\infty }t\cdot {\frac {1}{t}}\cdot J_{0}(st)\mathrm {d} t\\=&\int _{0}^{\infty }J_{0}(st)\,\mathrm {d} t\\=&{\frac {1}{s}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fe8b17d8a6c9b7dc6fdd4f3724a10a590ab2522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:33.848ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {H} _{0}\left({\frac {1}{t}}\right)(s)=&\int _{0}^{\infty }t\cdot {\frac {1}{t}}\cdot J_{0}(st)\mathrm {d} t\\=&\int _{0}^{\infty }J_{0}(st)\,\mathrm {d} t\\=&{\frac {1}{s}}\end{aligned}}}" loading="lazy">.</dd></dl>

Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{t}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4d2851f22ca31e9ddea8662f3c79833cb29093e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{t}}}" loading="lazy"> ist also ein <a href="Fixpunkt_(Mathematik)" title="Fixpunkt (Mathematik)">Fixpunkt</a> der Hankel-Transformation.

<div class="mw-heading mw-heading3"><h3 id="Die_Gaußsche_Glockenkurve">Die Gaußsche Glockenkurve</h3></div>

In diesem Abschnitt wird die Berechnung der Hankel-Transformation von der <a href="Gau%C3%9Fsche_Glockenkurve" class="mw-redirect" title="Gaußsche Glockenkurve">gaußschen Glockenkurve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {x^{2}}{2}}}}">

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/630cac2d37290162fe668cd5126cec22e58906a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.994ex; height:4.009ex;" alt="{\displaystyle e^{-{\frac {x^{2}}{2}}}}" loading="lazy"> mit Hilfe der Fourier-Transformation skizziert. Da die Funktion <a href="Analytische_Funktion" title="Analytische Funktion">analytisch</a> ist, kann sie auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \cong \mathbb {R} ^{2}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \cong \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0746a9a1a63d1a9766d88ccb945d92daae386517.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.509ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} \cong \mathbb {R} ^{2}}" loading="lazy"> fortgesetzt werden und ist dort sogar radialsymmetrisch. Daher kann die Hankel-Transformierte mit der Fourier-Transformation über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"> berechnet werden. Für die Fourier-Transformation ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {x^{2}}{2}}}}">

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/630cac2d37290162fe668cd5126cec22e58906a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.994ex; height:4.009ex;" alt="{\displaystyle e^{-{\frac {x^{2}}{2}}}}" loading="lazy"> ein Fixpunkt, woraus folgt, dass die Hankel-Transformierte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {x^{2}}{2}}}}">

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/630cac2d37290162fe668cd5126cec22e58906a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.994ex; height:4.009ex;" alt="{\displaystyle e^{-{\frac {x^{2}}{2}}}}" loading="lazy"> ebenfalls wieder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {x^{2}}{2}}}}">

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/630cac2d37290162fe668cd5126cec22e58906a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.994ex; height:4.009ex;" alt="{\displaystyle e^{-{\frac {x^{2}}{2}}}}" loading="lazy"> ist. Also ist die gaußsche Glockenkurve ebenfalls ein Fixpunkt der Hankel-Transformation.<a href="#cite_note-poularikas9.4-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Die_Delta-Distribution">Die Delta-Distribution</h3></div>

In diesem Beispiel wird die Hankel-Transformation nullter Ordnung der Delta-Distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> berechnet. Es gilt

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {H} _{0}({\tfrac {1}{u}}\delta _{0})(\phi )=&\operatorname {H} _{0}(\delta _{0})({\tfrac {1}{u}}\phi )=\delta _{0}(\operatorname {H} _{0}({\tfrac {1}{u}}\phi ))=\operatorname {H} _{0}({\tfrac {1}{u}}\phi )(0)\\=&\int _{0}^{\infty }{\tfrac {u}{u}}J_{0}(0)\phi (u)\,\mathrm {d} u\\=&\int _{0}^{\infty }\phi (u)\,\mathrm {d} u\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7232f924484ebf418c914557c2f683c7c5d6756a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:55.135ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {H} _{0}({\tfrac {1}{u}}\delta _{0})(\phi )=&\operatorname {H} _{0}(\delta _{0})({\tfrac {1}{u}}\phi )=\delta _{0}(\operatorname {H} _{0}({\tfrac {1}{u}}\phi ))=\operatorname {H} _{0}({\tfrac {1}{u}}\phi )(0)\\=&\int _{0}^{\infty }{\tfrac {u}{u}}J_{0}(0)\phi (u)\,\mathrm {d} u\\=&\int _{0}^{\infty }\phi (u)\,\mathrm {d} u\end{aligned}}}" loading="lazy">.</dd></dl>

Der Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{0}^{\infty }\phi (u)\mathrm {d} u}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{0}^{\infty }\phi (u)\mathrm {d} u}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e41556e6365033405c168d1f868eca5406f82f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.923ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{0}^{\infty }\phi (u)\mathrm {d} u}" loading="lazy"> ist als Distribution, die von der konstanten Einsfunktion erzeugt wird, zu verstehen.

Im Bereich der Physik notiert man die Delta-Distribution oftmals unpräzise als reellwertige Funktion und nicht als <a href="Funktional" title="Funktional">Funktional</a>. In diesem Fall kürzt sich die Berechnung der Hankel-Transformation auf

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{0}({\tfrac {1}{u}}\delta _{0})(t)=\int _{0}^{\infty }\delta (u)\cdot {\frac {1}{u}}\cdot u\cdot J_{0}(tu)\,\mathrm {d} u=J_{0}(0)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c858ff3bdb2ed01e6c478944d0f406c5ab15b87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.672ex; height:5.843ex;" alt="{\displaystyle \operatorname {H} _{0}({\tfrac {1}{u}}\delta _{0})(t)=\int _{0}^{\infty }\delta (u)\cdot {\frac {1}{u}}\cdot u\cdot J_{0}(tu)\,\mathrm {d} u=J_{0}(0)=1}" loading="lazy">.</dd></dl>

Möchte man umgekehrt die Hankel-Transformierte der konstanten Einsfunktion berechnen, stößt man beim Einsetzen in die Integraldarstellung auf ein divergentes Integral. Aufgrund von <a href="Dichte_Teilmenge" title="Dichte Teilmenge">Dichtheitsargumenten</a> ist es trotzdem möglich, die Delta-Distribution als Hankel-Transformierte der konstanten Einsfunktion aufzufassen.

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

<ul><li>Larry C. Andrews, Bhimsen K. Shivamoggi: <cite style="font-style:italic">Integral Transforms for Engineers</cite>. SPIE Press, University of Central Florida, 1999, ISBN 978-0-8194-3232-2, Kapitel 7.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Hankel-Transformation&rft.atitle=Kapitel+7&rft.au=Larry+C.+Andrews%2C+Bhimsen+K.+Shivamoggi&rft.btitle=Integral+Transforms+for+Engineers&rft.date=1999&rft.genre=bookitem&rft.isbn=9780819432322&rft.pub=SPIE+Press%2C+University+of+Central+Florida" style="display:none"> </li></ul>

<ul><li>Alexander D. Poularikas: <cite style="font-style:italic">The Transforms and Applications Handbook</cite>. 2. Auflage. CRC Press, 2000, ISBN 978-0-8493-8595-7, Kapitel 9.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Hankel-Transformation&rft.atitle=Kapitel+9&rft.au=Alexander+D.+Poularikas&rft.btitle=The+Transforms+and+Applications+Handbook&rft.date=2000&rft.edition=2.&rft.genre=bookitem&rft.isbn=9780849385957&rft.pub=CRC+Press" style="display:none"> </li></ul>

<ul><li>L. S. Dube, J. N. Pandey: <a rel="nofollow" class="external text" href="https://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.tmj/1203529246">On the Hankel transform of distributions</a> Tohoku Math. J. (2) Volume 27, Number 3 (1975), 337–354.</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> <a href="Bernd_J%C3%A4hne" title="Bernd Jähne">Bernd Jähne</a>: <cite style="font-style:italic">Digitale Bildverarbeitung</cite>. 6. Auflage. Springer, ISBN 978-3-540-24999-3, S. 219 bis 223.<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:Hankel-Transformation&rft.au=Bernd+J%C3%A4hne&rft.btitle=Digitale+Bildverarbeitung&rft.edition=6.&rft.genre=book&rft.isbn=9783540249993&rft.pages=219+bis+223&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-poularikas9.4-2">↑ <a href="#cite_ref-poularikas9.4_2-0">a</a> <a href="#cite_ref-poularikas9.4_2-1">b</a> Alexander D. Poularikas: <cite style="font-style:italic">The Transforms and Applications Handbook</cite>. 2. Auflage. CRC Press, 2000, ISBN 978-0-8493-8595-7, Kapitel 9.4.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Hankel-Transformation&rft.atitle=Kapitel+9.4&rft.au=Alexander+D.+Poularikas&rft.btitle=The+Transforms+and+Applications+Handbook&rft.date=2000&rft.edition=2.&rft.genre=bookitem&rft.isbn=9780849385957&rft.pub=CRC+Press" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Alexander D. Poularikas: <cite style="font-style:italic">The Transforms and Applications Handbook</cite>. 2. Auflage. CRC Press, 2000, ISBN 978-0-8493-8595-7, Kapitel 9.11.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Hankel-Transformation&rft.atitle=Kapitel+9.11&rft.au=Alexander+D.+Poularikas&rft.btitle=The+Transforms+and+Applications+Handbook&rft.date=2000&rft.edition=2.&rft.genre=bookitem&rft.isbn=9780849385957&rft.pub=CRC+Press" style="display:none">

</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/HankelTransform.html">Hankel Transform.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li></ul></div><div><div class="zim-footer">

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