A note on dual integral equations involving associated Legendre function
The author obtains a closed form solution of the following dual integral equations: \[ \int^ \infty_ 0\sigma^{-1}A(\sigma)P^ \mu_{- 1/2+i\sigma}(\cosh\alpha)\tanh c\sigma d\sigma=f(\alpha),\quad 0<\alphaa, \tag{2} \] with \(\text{Re} \mu<1/2\). He first finds a closed form solution of the pair of dual integral equations consisting of (1) and (2) with \(g(\alpha)=0\) and then the solution of the pair consisting of (1) with \(f(\alpha)=0\) and (2). The desired solution of equations (1) and (2) is then obtained by combining solutions of these two pairs. The method used here is formal and no attempt is made to find precise conditions under which the solution obtained is valid.
scientific article; zbMATH DE number 89253