Fourier-Bessel series

The expansion of a function $f$ in a series

$$\tag{* } f ( x) = \ \sum _ {m = 1 } ^ \infty c _ {m} J _ \nu \left ( x _ {m} ^ {( \nu ) } \cdot { \frac{x}{a} } \right ) ,\ \ 0 < x < a,$$

where $f$ is a function given on the interval $( 0, a)$, $J _ \nu$ is the Bessel function of order $\nu > - 1/2$( cf. Bessel functions), and the $x _ {m} ^ {( \nu ) }$ are the positive zeros of $J _ \nu$ taken in increasing order; the coefficients $c _ {m}$ have the following values:

$$c _ {m} = \ { \frac{2}{a ^ {2} J _ {\nu + 1 } ^ {2} ( x _ {m} ^ {( \nu ) } ) } } \int\limits _ { 0 } ^ { a } rf ( r) J _ \nu \left ( x _ {m} ^ {( \nu ) } \cdot { \frac{r}{a} } \right ) dr.$$

If $f$ is a piecewise-continuous function given on an interval $( 0, a)$ and if the integral

$$\int\limits _ { 0 } ^ { a } \sqrt r | f ( r) | dr < \infty ,$$

then the Fourier–Bessel series converges and its sum is equal to $[ f ( x + ) + f ( x - )]/2$ at each interior point $x$ of $( 0, a)$ at which $f$ locally has bounded variation.