Eigenfunctions of the Integral Equation for the Potential of the Charged Disk

The integral equation $$\mathrm{f(r,\theta )=}{\displaystyle {\int }_0^{\text{2π}}}{\displaystyle {\int }_0^1}\frac{\mathrm{u(r\prime ,\theta \prime )}}{{\mathrm{[r}}^2{\mathrm{+r\prime }}^2\text{−2rr′\hspace{0.17em}}\cos {\mathrm{\hspace{0.17em}(\theta -\theta \prime )]}}^{\frac{1}{2}}}·\frac{\mathrm{r\prime }}{{\text{(1−r′}}^2{)}^{\frac{1}{2}}}\mathrm{dr\prime d\theta \prime }$$ arises in connection with the problem of the electrostatic potential due to a charged disk. We solve the equation by computing a complete set of eigenfunctions and eigenvalues for the integral operator. The eigenfunctions have the form $F_{\mathrm{m,n}}^{\pm }{\mathrm{(r,\theta )=P}}_n^m{\text{[(1−r}}^2{)}^{\frac{1}{2}}{\mathrm{]e}}^{\mathrm{\pm im\theta }}$ , 0 ≤ m ≤ n, m + n even. Here $P_n^m\mathrm{(x)}$ is the associated Legendre function of the first kind.