1988

1) By evaluating $\int {\displaystyle \frac{dz}{z+2}}$ over a suitable contour $\mathrm{C}$, prove that ${\int }_0^x{\displaystyle \frac{1+2\cos \theta }{5+4\cos \theta }}d\theta$

2) If $f$ is analytic in $|Z|\le R$ and $x$, $y$ lie inside the disc, evaluate the integral ${\int }_{H-R}{\displaystyle \frac{f(z)dz}{(z-x)(z-y)}}$ and deduce that a function analytic and bounded for all finite $\mathrm{z}$ is a constant.

3) If $f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ has radius of convergence $\mathrm{R}$ and prove that ${\displaystyle \frac{1}{2\pi }}{\int }_0^{2n}|f\left(r{e}^{i\theta }\right){|}^{2}d\theta =\sum _{n=0}^{\mathrm{\infty }}|a_n{|}^2{r}^{2n}$.

4) Evaluate ${\int }_c{\displaystyle \frac{Z{e}^z}{(z-a{)}^3}}$, if $a$ lies inside the closed contour $\mathrm{C}$.

5) Prove that ${\int }_0^0{e}^{-x^2}\cos (2bx)dx={\displaystyle \frac{\sqrt{\pi }}{2}}{e}^{-b^2}$; $(b>0)$ by the integrating $e^{-z^2}$ along the boundary of the rectangle $|\mathbf{x}|{\mathrm{d}}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{R},0{\mathrm{d}}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{y}{\mathrm{d}}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{b}$.

TBC

6) Prove that the coefficients $C_n$ of the expansion ${\displaystyle \frac{1}{1-z-z^2}}=\sum _{n=0}^{\mathrm{\infty }}{C}_n{z}^n$ satisfy $C_n=C_{n-1}+C_{n-2},n\ge 2$. 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 $f\left(z_0\right)={\displaystyle \frac{1}{2\pi i}}{\int }_c{\displaystyle \frac{f(z)}{z-z_0}}dz,$ where the integral is taken in the +ve sense around $\mathrm{C}$.

2) By contour integration method show that

(i) ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{dx}{x^4+a^4}}={\displaystyle \frac{\pi \sqrt{2}}{4a^3}}$, where $a>0$
(ii) ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\sin x}{x}}dx={\displaystyle \frac{\pi }{2}}$

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 ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{x\sin ax}{x^2-b^2}}dx$.

1984

1) Evaluate by contour integration method:

(i) ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{x\sin mx}{x^4+a^4}}dx$ (ii) ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{x^{a-1}\log x}{1+x^2}}dx$

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 $\mathrm{\exists }$ a point $z,$ such that $0<|z-a|<\rho$ for which $|f(z)-\eta |<ϵ$

1983

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

(i) $|z|<2$
(ii) $2<|z|<3$
(iii) $|z|>3$

2) Use the method of contour integration to evaluate ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{x^{a-1}}{1+x^2}}dx,0<a<2$.