Sunday, 24 November 2013

integration - Evaluating intp0ifraccosntheta12rcostheta+r2dtheta




How do I go about evaluating the following integral:



π0cosnθ12rcosθ+r2dθ



for nN, and r(0,1)?



My tack has been to rewrite, using the exponential form of cos, as:



π0eiθ+eiθ2(1reiθ)(1reiθ)dθ




and letting z=eiθ and dθ=1izdz, we get



12i|z|=1zn+1zn(1rz)(zr)dθ



This would have a singularity when z=eiθ=r or when zr=reiθ=1. Since r(0,1) neither of these can occur. So we just need to integrate the above, but I can't see any way about this?



Is my approach valid or along the right lines, and how can I proceed?


Answer



The contour integration way:




Residue theorem reveals within a second that



π0cosnθ12rcosθ+r2dθ=πrn1r2.



Note: To make everything simpler (compared with what you tried in your post) it is enough to use in the numerator einθ.



The real method way:



Exploit carefully the well-known series result (which can be proved by real methods)




n=1pnsin(nx)=psin(x)12pcos(x)+p2,|p|<1


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