# Recursive algorithm for integration of complex functions

Integrals of the form $\int_{0}^{\pi} d\theta \sin\theta f(r(\theta)) j_{\ell_{1}} (a r(\theta))y_{\ell_{2}}(br(\theta))P_{\ell_{3}} ^{m} (\cos(\theta))P_{\ell_{4}} ^{m}(\cos(\theta))$, where $f$ is a $C^{\infty}$ complex function, $\ell_{i}$ $(i=1,\ldots,4)$ integers, $a$,$b$ complex constant values, $j_{\ell_{1}}$ spherical bessel function (of first kind), $y_{\ell_{2}}$ spherical neumann function (spherical bessel function of second type, $P_{\ell} ^{m}(x)$ the Legendre function and $r$ real function with $r(0) = r(\pi)$, appear in various boundary value problems. They are usually solved using numerical integration. However there are cases that the function to be integrated oscillates rapidly, while the results is zero. Is there any analytic formula or recursive algorithm that does not include numerical handling?

EDIT: The function $r$ is an arbitrary continuous function, but is not differentiatable in a finite number of discrete points (finite critical points). I need a method (if there is any) that will be generally valid and can produce results bounded to any arbitrarily small interval.

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Have you tried stationary phase methods? –  Alex R. Feb 3 '11 at 1:19
Residue theorem is always a good candidate (your function is $\C^\infty,$ so has a Fourier series, so...) –  Igor Rivin Feb 3 '11 at 3:15