1988

1) By evaluating \(\int \dfrac{d z}{z+2}\) over a suitable contour \(\mathrm{C}\), prove that \(\int_{0}^{x} \dfrac{1+2 \cos \theta}{5+4 \cos \theta} d \theta\)

2) If \(f\) is analytic in \(\vert Z \vert \leq R\) and \(x\), \(y\) lie inside the disc, evaluate the integral \(\int_{H-R} \dfrac{f(z) d z}{(z-x)(z-y)}\) and deduce that a function analytic and bounded for all finite \(\mathrm{z}\) is a constant.

3) If \(f(z)=\sum_{n=0}^{\infty} a_{n} z^{n}\) has radius of convergence \(\mathrm{R}\) and prove that \(\dfrac{1}{2 \pi} \int_{0}^{2 n}\vert f\left(r e^{i \theta}\right)\vert ^{2} d \theta=\sum_{n=0}^{\infty}\vert a_{n}\vert ^{2} r^{2 n}\).

4) Evaluate \(\int_{c} \dfrac{Z e^{z}}{(z-a)^{3}}\), if \(a\) lies inside the closed contour \(\mathrm{C}\).

5) Prove that \(\int_{0}^{0} e^{-x^{2}} \cos (2 b x) d x=\dfrac{\sqrt{\pi}}{2} e^{-b^{2}}\); \((b>0)\) by the integrating \(e^{-z^{2}}\) along the boundary of the rectangle \(\vert \mathbf{x} \vert \mathrm{d}^{\prime \prime} \mathrm{R}, 0 \mathrm{d}^{\prime \prime} \mathrm{y} \mathrm{d}^{\prime \prime} \mathrm{b}\).

TBC

6) Prove that the coefficients \(C_{n}\) of the expansion \(\dfrac{1}{1-z-z^{2}}=\sum_{n=0}^{\infty} C_{n} z^{n}\) satisfy \(C_{n}=C_{n-1}+C_{n-2}, n \geq 2\). Determine \(C_n\).

1986

1) Let \(f(z)\) be single valued and analytic with in and on a closed curve \(\mathrm{C}\). If \(\mathrm{z}_{0}\) is any point interior to \(\mathrm{C}\) then show that \(f\left(z_{0}\right)=\dfrac{1}{2 \pi i} \int_{c} \dfrac{f(z)}{z-z_{0}} d z,\) where the integral is taken in the +ve sense around \(\mathrm{C}\).

2) By contour integration method show that

(i) \(\int_{0}^{\infty} \dfrac{d x}{x^{4}+a^{4}}=\dfrac{\pi \sqrt{2}}{4 a^{3}}\), where \(a>0\)
(ii) \(\int_{0}^{\infty} \dfrac{\sin x}{x} d x=\dfrac{\pi}{2}\)

1985

1) Prove that every power series represents an analytic function within its circle of convergence.

2) Prove that the derivative of a function analytic in a domain is itself an analytic function.

3) Evaluate, by the method of contour integration \(\int_{0}^{\infty} \dfrac{x \sin a x}{x^{2}-b^{2}} d x\).

1984

1) Evaluate by contour integration method:

(i) \(\int_{0}^{\infty} \dfrac{x \sin m x}{x^{4}+a^{4}} d x\) (ii) \(\int_{0}^{\infty} \dfrac{x^{a-1} \log x}{1+x^{2}} d x\)

2) Distinguish clearly between a pole and an essential singularity. If \(z=a\) is an essential singularity of a function \(f(z),\) then for an arbitrary positive integers \(\eta\), \(\in\) and \(\rho,\) prove that \(\exists\) a point \(z,\) such that \(0<\vert z-a \vert <\rho\) for which \(\vert f(z)-\eta \vert <\epsilon\)

1983

1) Obtain the Taylor and Laurent series expansions which represent the function \(\dfrac{z^{2}-1}{(z+2)(z+3)}\) in the regions:

(i) \(\vert z \vert <2\)
(ii) \(2<\vert z \vert <3\)
(iii) \(\vert z \vert >3\)

2) Use the method of contour integration to evaluate \(\int_{0}^{\infty} \dfrac{x^{a-1}}{1+x^{2}} d x, 0<a<2\).