1994

1) Suppose that $z$ is the position vector of a particle moving on the ellipse $C:z=a\cos \omega t+ib\sin \omega t$ where $a,b,\omega$ are positive constants, $a>b$ and $t$ is the time. Determine where
(i) the velocity has the greatest magnitude. (ii) the acceleration has the least magnitude

[10M]

2) How many zeros does the polynomial $p(z)=z^4+2z^3+3z+4$ possess in (i) the first quadrant, (ii) the fourth quadrant.

[10M]

3) Test of uniform convergence in the region $|z|\le 1$ the series $\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{\cos nz}{n^3}}$

[10M]

4) Find Laurent series for (i) ${\displaystyle \frac{e^{2z}}{(z-1{)}^3}}$ about $z=1$, (ii) ${\displaystyle \frac{1}{z^2(z-3{)}^2}}$ about $z=3$.

[10M]

5) Find the residues of $f(z)=e^z{\mathrm{cosec}}^{2}z$ at all its poles in the finite plane.

[10M]

6) By means of contour integration, evaluate ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{({\log }_{e}u)}^2}{u^2+1}}du$

[10M]

1993

1) In the finite $z$ plane, show that the function $f(z)=\sec {\displaystyle \frac{1}{z}}$ has infinitely many isolated singularities in a finite intervals which includes $0.$

[10M]

2) Find the orthogonal trajectories of the family of curves in the xy-plane defined by $e-x(x\sin y-y\cos y)=\alpha$ where $\alpha$ is real constant.

[10M]

3) Prove that (by applying Cauchy Integral formula or otherwise) ${\int }_0^{2\pi }{\cos }^{2n}\theta d\theta ={\displaystyle \frac{1,3,5\dots (2n-1)}{2,4,6\cdot 2n}}2\pi$ where $n=1,2,\mathrm{3...}$

[10M]

4) If $c$ is the curve $y=x^3-3x^2+4x-1$ joining the points (1,1) and (2,3) find the value of ${\int }_c(12z^2-4iz)dz$

[10M]

5) Prove that $\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{z^n}{n(n+1)}}$ converges absolutely for $|z|\le 1$

[10M]

6) Evaluate ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{dx}{x^6+1}}$ by choosing an appropriate contour.

[10M]

1992

1) If $u=e^{-x}(xsiny-y\cos y),$ find $v$ such that $f(z)=u+iv$ is analytic. Also find $f(z)$ explicitly as a function of $z$.

[10M]

2) Let $f(z)$ be analy tic inside and on the circle $C$ defined by $|z|=R$ and let $z=e{r}^{i\theta }$ be any point in side C. Prove that $f\left(r{e}^{i\theta }\right)=1/2\pi {\int }_0^{2\pi }{\displaystyle \frac{(R^2-r^2)\left(R{e}^{i\varphi }\right))}{R^2-2Rr\cos (\theta +\varphi )+r^2}}d\varphi$

[10M]

3) Prove that all the roots of $z^7-5z^3+12=0$ lie between the circle $|z|=1$ and $|z|=2$.

[10M]

4) Find the region of convergence of the series whose $n$-th term is ${\displaystyle \frac{(-1{)}^{n-1}{z}^{2n-1}}{(2n-1)!}}$

[10M]

5) Expand $f(z)={\displaystyle \frac{1}{(z+1)(z+3)}}$ in a Laurent series valid for 6.b)(i) $|z|>3$ 6.b)(ii) $1<|z|<3$ 6.b)(iii) $|z|<1$

[10M]

6) By integrating along a suitable contour evaluate ${\int }_0^8{\displaystyle \frac{\cos mx}{x^2+1}}dx$

[10M]

1991

1) A function $f(z)$ is defined for finite values of $z$ by $f(0)=0$ and $f(z)=e^{-z^{-4}}$ everywhere else. Show that the Cauchy Riemann equation are satisfied at the origin. Show also that $f(z)$ is not analytic at the origin.

2) If $|\mathrm{a}|\ne \mathrm{R}$ show that ${\int }_{|z|=R}{\displaystyle \frac{|dz|}{|z-a||z+a|}}<{\displaystyle \frac{2\pi R}{|R^2-|a{|}^2|}}$.

3) If $J_n(t)={\displaystyle \frac{1}{2\pi }}{\int }_0^{2z}\cos (n\theta -t\sin \theta )d\theta$. Show that $e^{{\displaystyle \frac{1}{2}}(z-{\displaystyle \frac{1}{z}})}$=$J_0(t)+z{J}_1(t)+z^2{J}_2(t)$+ $\cdots$-${\displaystyle \frac{1}{z}}{J}_1(t)+{\displaystyle \frac{1}{z^2}}{J}_2(t)-{\displaystyle \frac{1}{z^3}}{J}_3(t)+\cdots$

4) Examine the nature of the singularity of $e^z$ at infinity.

5) Evaluate the residues of the function ${\displaystyle \frac{Z^3}{(Z-2)(Z-3)(Z-5)}}$ at all singularities and show that their sum is zero.

6) By integrating along a suitable contour, show that ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{e^{ax}}{1+e^x}}={\displaystyle \frac{\pi }{\sin a\pi }}$ where $0<\mathrm{a}<1$.

1990

1) Let $f$ be regular for $|Z|<R$, prove that, if $0<r<R$, $f^{\mathrm{\prime }}(0)={\displaystyle \frac{1}{\pi r}}{\int }_0^{2\pi }u(\theta )e^{-id}d\theta$, where $u(\theta )=\mathrm{Re}f\left(r{e}^{i\theta }\right)$.

2) Prove that the distance from the origin to the nearest zero of $f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ is at least ${\displaystyle \frac{r|a_0|}{M+|a_0|}}\cdot$ where $\mathrm{r}$ is any number not exceeding the radius of the convergence of the series and $M=M(r)=sup|f(z){|}_{|z|=r}$.

3) Prove that ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{x^4}{1+x^8}}dx={\displaystyle \frac{\pi }{\sqrt{2}}}\sin {\displaystyle \frac{\pi }{8}}$ using residue calculus.

4) Prove that if $\mathrm{f}=\mathrm{u}+\mathrm{i}\mathrm{v}$ is regular through out the complex plane and $au+$ bv-c $\ge 0$ for suitable constants $a$, $b$, $c$ then $\mathrm{f}$ is constant.

5) Derive a series expansion of $\log (1+e^z)$ in powers of $z$.

6) Determine the nature of singular points $\sin \left({\displaystyle \frac{1}{\cos 1/z}}\right)$ and investigate its behaviour at $z=\mathrm{\infty }$.

1989

1) Find the singularities of $\sin \left({\displaystyle \frac{1}{1-z}}\right)$ in the complex plane.