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.