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$$\int_0^{\pi} \frac{2 d \theta}{k - cos \theta}$$

So I know this theorem:

$$\int_{-\infty}^{\infty} f(x) dx = 2 \pi i \sum Res f(z)$$ which makes me believe while using this theorem, I eventually need to divide the resulting value by half.

So first things first we need to calculate the singularities and residues right?

So the singularities exist when $\cos \theta = k$. Can someone help from here?

Once we find the residues, we can calculate the integral from $-\infty \to \infty$ and then divide by two right? The variable k is throwing me off because there are some values of k that are large enough so that there will not be a singularity in the denominator right?

J. W. Tanner
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Jwan622
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    Ther is a standard procedure for evaluating such integrals and they can be found in many books. In particular Rudin's RAC gives similar examples. What you have written is comepletely out of the way. You don't even define any function of a compex variable for which residue theorem can be applied. – Kavi Rama Murthy Dec 02 '20 at 06:23
  • This is... all my book gives? – Jwan622 Dec 02 '20 at 15:46

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