Fourier-Bessel integral

Hankel integral

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

$$ \tag{*} f(x) = \int_0^\infty \lambda J_\nu(\lambda x) \int_0^\infty y J_\nu(\lambda y) f(y) \, dy \, d\lambda \, . $$

Formula (*) 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 (*) 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 (*) 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 (*) 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 (*) reduces to Fourier's sine and cosine integral, respectively. In case $\nu = (n/2)-1$, where $n=1,2,\ldots$, formula (*) 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)