Trigonometric Integral Help: Solving for a<1 Using Complex Contour Method

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Homework Help Overview

The discussion revolves around evaluating the integral \(\int_0^{2\pi} \frac{\cos x}{a - \cos x} \, dx\) for the case where \(a < 1\), using complex contour integration methods. The original poster has transformed the integral into a contour integral over the unit circle but is uncertain about handling the poles that lie on the unit circle when \(a < 1\.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to evaluate the integral using complex analysis but expresses concern about the poles being on the unit circle for \(a < 1\). Some participants suggest alternative methods, such as the substitution \(u = \tan(x/2)\), while others note the complications that arise with this approach. There is also a mention of the integral potentially diverging for \(a < 1\.

Discussion Status

The discussion is ongoing, with various approaches being explored. Some participants are questioning the necessity of using complex methods, while others are trying to clarify the implications of the poles on the unit circle. The original poster is seeking further insights into the principal value for \(a < 1\.

Contextual Notes

There is an assumption that \(a\) is a parameter greater than 0, and the discussion includes considerations of divergence for \(a < 1\. The complexity of the integral and the behavior of the poles are central to the conversation.

von_biber
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need some assistance with the following integral:

\int_0^{2\pi} cosx/(a-cosx), a-parameter (say a>0)

i've converted it into a complex contour integral over z=e^(ix):

~ \int_{|z|=1} dz (z^2+1)/[z(z^2-2az+1)]

which is easily evaluated for a>1. my question regards a<1 - i am not sure how to solve it in this case, because the the 2 poles

z_1=a+Sqrt[a^2-1], z_2=a-Sqrt[a^2-1]

are exactly on the unit circle and off the real axis. thanks for any suggestions!
 
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Hmm..is it really necessay to venture out in complexity here?
Rational expressions in the trig. functions are easily handled by the transformation u=tan(x/2).
 
it get's a bit complicated with this tan(x/2) substitution, with complex substitution it's very easily evaluated for a>1 and i'd like to find out what happens for a<1.
 
Well, for a<1, the integral will readily diverge, so no wonder if you get into some problems.
 
anyway, what is the principal value for a<1?
thanks for your help, btw. i really appreciate it.
 

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