### A Theorem of Abel and Its Application to the Development of a Function in Terms of Bessel's Functions

Transactions of the American Mathematical Society, Vol. 8, No. 1 (Jan. 1907), 92-106

Presented to the Society December 29, 1905 and February 24, 1906. Received for publication June 8, 1906.

Introduction

The object of this paper is twofold: first to deduce directly from the properties of double integrals, the formula first given by Abel,
\[\frac{2\mu}{\pi}\int_0^1 dt \int_0^1 \frac{f'(\lambda u t)\lambda d\lambda}{\sqrt{1-t^2} \cdot \sqrt{1-\lambda^2}} = f(u)-f(0);\]
and second to apply this formula to obtain sufficient conditions under which a function is developable in a series of Bessel's functions of multiple values of the argument.