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An analogue of the [[Fourier integral|Fourier integral]] for [[Bessel functions|Bessel functions]], having the form
 
An analogue of the [[Fourier integral|Fourier integral]] for [[Bessel functions|Bessel functions]], having the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\begin{equation} \label{eq1} \tag{*} f(x) = \int_0^\infty \lambda J_\nu(\lambda x) \int_0^\infty y J_\nu(\lambda y) f(y) \, dy \, d\lambda \, . \end{equation}
  
Formula (*) can be obtained from the [[Fourier–Bessel series|Fourier–Bessel series]] for the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409902.png" /> by taking the limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409903.png" />. H. Hankel (1875) established the following theorem: If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409904.png" /> is piecewise continuous, has bounded variation on any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409905.png" />, and if the integral
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Formula \eqref{eq1} can be obtained from the [[Fourier–Bessel series|Fourier–Bessel series]] for the interval $(0,l)$ by taking the limit as $l \to +\infty$. H. Hankel (1875) established the following theorem: If the function $f$ is piecewise continuous, has bounded variation on any interval $0 < x < l$, and if the integral
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409906.png" /></td> </tr></table>
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$$ \int_0^\infty \sqrt{x} |f(x)| \, dx $$
 
 
converges, then (*) is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409907.png" /> at all points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409908.png" /> is continuous, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f0409909.png" />. At a point of discontinuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099010.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099011.png" />, the right-hand side of (*) is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099012.png" />, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099013.png" /> it gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099014.png" />.
 
 
 
Analogues of the Fourier–Bessel integral (*) for other types of cylinder functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099015.png" /> are also true, but the limits in the integrals should be changed accordingly.
 
  
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converges, then \eqref{eq1} is valid for $\nu > -1/2$ at all points where $f$ is continuous, $0 < x < +\infty$. At a point of discontinuity $x_0$, $0 < x_0 < +\infty$, the right-hand side of \eqref{eq1} is equal to $[ f(x_0-) + f(x_0+)]/2$, and when $x_0 = 0$ it gives $f(0+)/2$.
  
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Analogues of the Fourier–Bessel integral \eqref{eq1} for other types of cylinder functions $Z_\nu(x)$ are also true, but the limits in the integrals should be changed accordingly.
  
 
====Comments====
 
====Comments====
In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099016.png" />, formula (*) reduces to Fourier's sine and cosine integral, respectively. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099018.png" /> formula (*) can be interpreted as a [[Fourier integral|Fourier integral]] for radial functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040990/f04099019.png" />. See also [[#References|[a1]]], p. 240.
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In case $\nu = \pm 1/2$, formula \eqref{eq1} reduces to Fourier's sine and cosine integral, respectively. In case $\nu = (n/2)-1$, where $n=1,2,\ldots$, formula \eqref{eq1} can be interpreted as a [[Fourier integral]] for radial functions on $\mathbf{R}^n$. See also [[#References|[a1]]], p. 240.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.C. Titchmarsh,  "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.C. Titchmarsh,  "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR></table>
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Latest revision as of 06:47, 24 April 2024

Hankel integral

An analogue of the Fourier integral for Bessel functions, having the form

\begin{equation} \label{eq1} \tag{*} f(x) = \int_0^\infty \lambda J_\nu(\lambda x) \int_0^\infty y J_\nu(\lambda y) f(y) \, dy \, d\lambda \, . \end{equation}

Formula \eqref{eq1} can be obtained from the Fourier–Bessel series for the interval $(0,l)$ by taking the limit as $l \to +\infty$. H. Hankel (1875) established the following theorem: If the function $f$ is piecewise continuous, has bounded variation on any interval $0 < x < l$, and if the integral

$$ \int_0^\infty \sqrt{x} |f(x)| \, dx $$

converges, then \eqref{eq1} is valid for $\nu > -1/2$ at all points where $f$ is continuous, $0 < x < +\infty$. At a point of discontinuity $x_0$, $0 < x_0 < +\infty$, the right-hand side of \eqref{eq1} is equal to $[ f(x_0-) + f(x_0+)]/2$, and when $x_0 = 0$ it gives $f(0+)/2$.

Analogues of the Fourier–Bessel integral \eqref{eq1} for other types of cylinder functions $Z_\nu(x)$ are also true, but the limits in the integrals should be changed accordingly.

Comments

In case $\nu = \pm 1/2$, formula \eqref{eq1} reduces to Fourier's sine and cosine integral, respectively. In case $\nu = (n/2)-1$, where $n=1,2,\ldots$, formula \eqref{eq1} can be interpreted as a Fourier integral for radial functions on $\mathbf{R}^n$. See also [a1], p. 240.

References

[a1] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)
How to Cite This Entry:
Fourier-Bessel integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Bessel_integral&oldid=22437
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article