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Is there a simple reason, why the integrals of the form $$ C_1(s) =\int\limits_{0}^{\pi/2} \cos(2 \nu \theta) \cos^{2s-1}(\theta) \; d \theta $$ and $$ C_2(s)= \int\limits_{0}^{\infty} \cosh(2 \nu \theta) \cosh^{-1-2s}(\theta) \; d \theta,$$ can both be expressed in terms of the Beta function, and there quotient is $C_1(s) C_2(s) = D /s$ for some constant $D$?

With simple, I mean can we write the product of the integral and transform from polar coordinates to Cartesian coordinates and use some addition theorems for cos and cosh?

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Are you sure the upper bound for integration is $\pi/2$ in the first case and $\infty$ in the second case? I do not doubt this, I'm just curious. – user20266 Jun 6 '12 at 18:02
yes that's it.. – Jun 6 '12 at 19:00

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