I want to evaluate $$\int_{\cos\theta}^1 \left( P_{-1/2+i\tau}(x) \right)^2 dx,$$ where $P$ is the Legendre function of the first kind, $i$ is the imaginary unit, and $\tau$ is a real number.
Are their any analytical or simple formulae to evaluate this ?
There is a following formula $$(\beta-\alpha)(\alpha+\beta+1)\int_{\cos\theta}^1 P_\alpha(x) P_\beta(x)dx = \sin^2\theta \left[ P_\alpha(\cos\theta)P'_\beta(\cos\theta)-P_\beta(\cos\theta)P'_\alpha(\cos\theta) \right],$$ where ' denotes the differentiation with respect to the argument and $\alpha \neq \beta$. Since $P_{-1/2+i\tau}(x) = P_{-1/2-i\tau}(x)$ (property of Mehler function) is known, the above formula may seem applicable. However, it is not usable because $\alpha + \beta + 1 = 0$ unfortunately.
I appreciate any hints and comments.
Thank you very much in advance.
Ryuichi
For numerical evaluation, the given formula could be evaluated with the same computational load for the evaluation of $P_{-1/2+i\tau}(x)$. From definition,
$$P_{-1/2+i\tau}(x) = \sum_{n=0}^{\infty} \prod_{k=0}^{n-1} \left\{(1/2+k)^2 + \tau^2 \right\}\frac{(\frac{1-x}{2})^n}{(n!)^2}.$$
Here we define a function $f(n,\tau)$ $$f(n,\tau) = \frac{1}{(n!)^2} \prod_{k=0}^{n-1} \left\{(1/2+k)^2 + \tau^2 \right\}.$$
Therefore, $$\left\{ P_{-1/2+i\tau}(x) \right\}^2 = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}f(n,\tau)f(m,\tau) \left( \frac{1-x}{2} \right)^{n+m}.$$
By integrating for x from $\cos\theta$ to $1$, $$\int_{\cos\theta}^{1}\left\{ P_{-1/2+i\tau}(x) \right\}^2 dx = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}f(n,\tau)f(m,\tau) \int_{\cos\theta}^{1}\left(\frac{1-x}{2} \right)^{n+m}dx \\ = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{f(n,\tau)f(m,\tau)}{2^{n+m}(n+m+1)} (1-\cos\theta)^{n+m+1}. $$
Although the accuracy and the speed of convergence are questioned, the given formula could be evaluated.
If there are better solutions, please post a comment. Thank you in advance.
Ryuichi