IGNOU PHE 14 2018 - Using the method of residues, show that

Question

Using the method of residues, show that

Image ignouacademy-com--p-residues-17066

01 Feb 2018

Answer :

Word Count : 515

Alright — let’s go step by step.

We are to compute

\[
I = \int_{0}^{2\pi} \frac{d\theta}{3 + 2\cos\theta}.
\]

---

### 1. Convert to a complex contour integral
Let \( z = e^{i\theta} \), \( dz = i e^{i\theta} d\theta = iz \, d\theta \), so \( d\theta = \frac{dz}{iz} \).

Also,

\[
\cos\theta = \frac{z + z^{-1}}{2}.
\]

The integral becomes:

\[
I = \oint_{|z|=1} \frac{1}{3 + 2 \cdot \frac{z+z^{-1}}{2}} \cdot \frac{dz}{iz}.
\]

Simplify denominator:
\( 3 + 2\cdot \frac{z+z^{-1}}{2} = 3 + z + z^{-1} \).

So:

\[

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