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Fourier-Bessel integral

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Hankel integral

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

(*)

Formula (*) can be obtained from the Fourier–Bessel series for the interval by taking the limit as . H. Hankel (1875) established the following theorem: If the function is piecewise continuous, has bounded variation on any interval , and if the integral

converges, then (*) is valid for at all points where is continuous, . At a point of discontinuity , the right-hand side of (*) is equal to , and when it gives .

Analogues of the Fourier–Bessel integral (*) for other types of cylinder functions are also true, but the limits in the integrals should be changed accordingly.


Comments

In case , formula (*) reduces to Fourier's sine and cosine integral, respectively. In case , where formula (*) can be interpreted as a Fourier integral for radial functions on . 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