Contour Integral, Where am I going wrong?

BigFlorida · Jan 17, 2016

So, my book (Mathematical Methods in the Physical Sciences 3rd ed by Boas) proposed a problem that I have really been struggling with:

Handwritten derivation of a contour integral of sin^2(theta) over 5 + 3cos(theta) from 0 to 2pi, substituting z = e^(i theta), with a unit circle diagram marking z = e^(i theta) and the angle theta.

Handwritten derivation applying the residue theorem to a contour integral of (z - z^-1)^2 / ((z+3)(z+1/3)), with residue R(-1/3) computed by direct substitution to give 8/9 and a final result of 32pi/9.

I know it is probably just an algebra mistake, but I have gone over it multiple times and cannot seem to find my mistake. Any ideas? The answer is supposed to be 2Pi/9.
Thank you in advance for any help.

Ray Vickson · Jan 17, 2016

BigFlorida said:

So, my book (Mathematical Methods in the Physical Sciences 3rd ed by Boas) proposed a problem that I have really been struggling with:
View attachment 94420 View attachment 94421

I know it is probably just an algebra mistake, but I have gone over it multiple times and cannot seem to find my mistake. Any ideas? The answer is supposed to be 2Pi/9.
Thank you in advance for any help.

The image is too small and blurry to make out properly, so I will not even try. Why don't you do what others here do, and type it out?

Contour Integral, Where am I going wrong?

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