# Fourier-Bessel integral

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Hankel integral

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

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

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.

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%E2%80%93Bessel_integral&oldid=22438