How do I go about evaluating the following integral:
∫π0cosnθ1−2rcosθ+r2dθ
for n∈N, and r∈(0,1)?
My tack has been to rewrite, using the exponential form of cos, as:
∫π0eiθ+e−iθ2(1−reiθ)(1−re−iθ)dθ
and letting z=eiθ and dθ=1izdz, we get
12i∫|z|=1zn+1zn(1−rz)(z−r)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
∫π0cosnθ1−2rcosθ+r2dθ=πrn1−r2.
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)1−2pcos(x)+p2,|p|<1
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