IGNOU MMT 5 2026 - Evaluate the following integrals:i) .

Question

Evaluate the following integrals:
i) $I = \int_{0}^{2\pi} f(e^{i\theta}) \cos^2(\theta/2) \, d\theta$ .

ii) $I = \int_{0}^{2\pi} f(e^{i\theta}) \sin^2 \theta/2 \, d\theta$

10 Jan 2026

Answer :

Word Count : 124

Assume that (f(z)) is analytic on and inside the unit circle and has the power series expansion (f(z)=\sum_{n=0}^{\infty} a_n z^n). Then (f(e^{i\theta})=\sum_{n=0}^{\infty} a_n e^{in\theta}). Using [ \cos^2\frac{\theta}{2}=\frac{1+\cos\theta}{2}, \qquad \sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{2}, ] we write [ \int_0^{2\pi} f(e^{i\theta}),d\theta = \sum_{n=0}^{\infty} a_n \int_0^{2\pi} e^{in\theta} d\theta =2\pi a_0, ] and [ \int_0^{2\pi} f(e^{i\theta})\cos\theta,

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