View source for Fourier-Bessel integral

''Hankel integral''

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>

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

<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>

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.



====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.

====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>