# Improper integrals over the square and modulus square of an associated Legendre function

I am trying to evaluate integrals of the type $$\int dz\, P^{\,\mu}_\nu(z)^2 \qquad \mathrm{and} \qquad \int dz\, \left|P^{\,\mu}_\nu(z)\right|^2$$ where $P^\mu_\nu$ are associated Legendre functions. The path of integration is along the imaginary axis from $-i\infty$ to $+i\infty$. In particular, I am interested in the ratio of the two integrals.

For the first integral I have an ansatz but I am not confident how valid the steps are. Using the equation for the associated Legendre function $$\frac{d}{dz}\left[(1-z^2)\frac{d}{dz}\right]P_{\nu}^\mu(z)+\left[\nu(\nu+1)-\frac{\mu^2}{1-z^2}\right]P_\nu^\mu(z)=0$$ we can show that $$\left[\nu_1(\nu_1+1)-\nu_2(\nu_2+1)\right]\int P_{\nu_1}^{\mu}(z) P_{\nu_2}^{\mu}(z) dz=(1-z^2)\left[\frac{dP_{\nu_1}^{\mu}(z)}{dz}P_{\nu_2}^{\mu}(z)-P_{\nu_1}^{\mu}(z)\frac{dP_{\nu_2}^{\mu}(z)}{dz}\right]$$ for any two solutions. This is similar to the typical derivation of Sturm-Liouville orthogonality. Treating $\nu_1$ and $\nu_2$ as continuous variables, an expansion $\nu_1=\nu_2+\delta$ will give $$\nu_1(\nu_1+1)-\nu_2(\nu_2+1)=(2\nu_2+1)\delta+\mathcal O(\delta^2)$$ and $$P_{\nu_1}^{\mu}(z)=P_{\nu_2}^{\mu}(z)+\left.\frac{dP_{\nu}^{\mu}(z)}{d\nu}\right|_{\nu=\nu_2}\delta+\mathcal O(\delta^2).$$ The limit $\delta\rightarrow0$ yields $$\int_{z_1}^{z_2} P_{\nu}^{\mu}(z) P_{\nu}^{\mu}(z) dz=\left[\frac{1-z^2}{2\nu+1}\left(\frac{ d^2P_{\nu}^{\mu}(z)}{ d\nu d z}P_\nu^\mu(z)-\frac{ d P_{\nu}^{\mu}(z)}{ d\nu}\frac{ d P_{\nu}^{\mu}(z)}{ d z}\right)\right]^{z_2}_{z_1}$$ Using this I can evaluate the integral by taking limits in the functions on the RHS. These limits will be finite for certain values of $\mu$ and $\nu$ of interest. I don't know how valid my steps are and I am not sure yet how to approach the second integral, so any comments are welcome.

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Would the same reasoning hold for $P_{\nu_1}^{\mu}(z) \bar P_{\nu_2}^{\mu}(z)$? – Andrew Aug 24 '11 at 14:46
Generically, your integrals diverge, i.e. $\int_{-L}^{L} \mathrm{d} t P_\nu^\mu(i t)^2$ grow in magnitude as $L$ grows. Could you specify which $\nu$ and $\mu$ are of interest to you? – Sasha Aug 24 '11 at 14:53
$\mu$ is a negative integer or half-integer. $\nu$ is either real and $>-1/2$ or of the form $i\rho-1/2$ where $\rho$ is real and positive. In the latter case these functions are the conical functions that have certain special properties. When the integrals diverge I am still interested what their ratio is as $L$ is sent to infinity. – matimo2 Aug 25 '11 at 9:28