I have the following integral for which I want a closed-form solution:

$$ I_n(kr,\cos\theta) = \int_{C} \frac{e^{ikrt}P_n(t)}{t-\cos\theta} dt, $$

where $P_n$ is a Legendre polynomial and $C$ is a line in the complex plane starting at $t=-1$ and terminating at $t\rightarrow i\infty$. For reference, $|kr| > n$.

This is an integral which results from playing around with spherical Hankel functions. I went back to Watson's book on Bessel functions to understand where the contour comes from and to try to reverse-engineer a closed-form expression in terms of Bessel functions using his methods, namely, studying the conditions under which the expression

$$ \nabla_\nu \left[f(z)\int_{a}^{b}e^{izt}T(t)dt\right]=0, $$

is satisfied, where Bessel's differential operator is defined by $\nabla_\nu=z^2\frac{d^2}{dz^2} + z\frac{d}{dz} + z^2-\nu^2$. I was unable to reduce the problem to a simpler one using these methods, i.e. substituting $I_n$ above for the integral in the brackets and trying to solve for $f$.

I have also tried several integrations by parts but none of those got me anywhere either. Mathematica also seems to have no idea how to express this integral in closed form. It has no problem giving me numerical answers, which is good because I know this integral should be convergent under the conditions stated above.

Does anyone have ideas for tackling this integral or wish to take a crack at it themselves?

EDIT (6/24/18): I realized I should probably point out that the integral representation of the spherical Hankel function of order $n$ is given by

$$ h_n^{(2)}(kr)=\int_C e^{ikrt}P_n(t)dt, $$

where $C$ is the line contour defined above. Essentially, I am looking for this integral with a simple pole at $t=\cos\theta$ that is highly likely off the integration path.



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