Fourier-Bessel integral

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)