An Integral Equation for the Associated Legendre Function of the First Kind

The solution of the Fredholm homogeneous equation $$\mathrm{\psi (\xi )=\lambda }{\displaystyle {\int }_0^1}\mathrm{M(\xi ,\xi \prime )\psi (\xi \prime )d\xi \prime }$$ , where $$\mathrm{M(\xi ,\xi \prime )=}{\displaystyle {\int }_0^{\pi }}\frac{\cos \mathrm{m\phi \; d\phi }}{{\mathrm{(x}}^2\text{−2xx′}\cos {\mathrm{\phi +x\prime }}^2{)}^{\frac{1}{2}}}$$ and x = (1 − ξ²)^½ is found to be the associated Legendre function $P_n^m\mathrm{(\xi ),n+m}$ even, and the characteristic numbers of this kernel are obtained. The solution of the corresponding equation of the second kind is also found. The kernel of the homogeneous equation whose solution is $P_n^m\mathrm{(\xi )}$ , n + m odd, is obtained.