# Associated Legendre Polynomial Identity

1. Jul 15, 2014

### HasuChObe

Does anyone know how to prove this identity? I don't quote understand why the associated Legendre function is allowed to have arguments where $|x|>1$.
$$h_n(kr)P_n^m(\cos\theta)=\frac{(-i)^{n+1}}{\pi}\int_{-\infty}^\infty e^{ikzt}K_m(k\rho\gamma(t))P_n^m(t)\,dt$$

where
$$\gamma(t)=\begin{cases} \sqrt{t^2-1} & |t|\ge 1 \\ -i\sqrt{1-t^2} & |t|<1 \end{cases} \\ \rho=\sqrt{x^2+y^2} \\ r=\sqrt{\rho^2+z^2} \\ \cos\theta=\frac{z}{r}$$

and $k$ is the wavenumber, $h_n(x)$ is the spherical hankel function of the first kind, and $K_m(x)$ is the modified cylindrical bessel function of the second kind.

Last edited: Jul 15, 2014
2. Jul 15, 2014

### Dr Transport

could not find it in my copy of Watson, can you give a reference where you found this integral to start??

3. Jul 16, 2014

### HasuChObe

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