2000

1) Show that any four given points of the complex plane can be carried by a bilinear transformation to positions $1,-1,k$ and $-k$ where the value of $\mathrm{k}$ depends on the given points.

[12M]

1999

1) Examine the nature of the function $f(z)={\displaystyle \frac{x^2{y}^2(x+iy)}{x^4+y^{10}}},z\ne 0,$ $f(0)=0$
in a region including the origin and hence show that Cauchy-Riemann equations are satisfied at the origin but $f(z)$ is not analytic there.

[20M]

2) For the function

find the Laurent series foe the domain

(i) $1<|z|<2$,

(ii) $|z|>2$. Show further that

where $C$ is any contour enclosing the point $z=1$ and $z=2$.

[20M]

3) Show that the transformation $w={\displaystyle \frac{2z+3}{z-4}}$

transforms the circle $x^2+y^2-4x=0$ into the straight line $4u+3=0$ where $w=u+iv$.

[20M]

4) Using Residue Theorem show that ${\int }_{\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{xsinax}{x^4+4}}dx={\displaystyle \frac{\pi }{2}}{e}^{-a}sina,(a>0).$

[20M]

5) The function $f(z)$ has a double pole at $z=0$ with residue 2,a simple pole at $z=1$ with residue 2, is analytic at all other finite points of the plane and is bounded as $|z|\to \mathrm{\infty }$.If $f(2)=5$ and $f(-1)=2$,find $f(z)$.

[20M]

6) What kind of singularities the following functions have?

(i) ${\displaystyle \frac{1}{1-e^z}}$ at $z=2\pi i$

(ii) ${\displaystyle \frac{1}{sinz-cosz}}$ at $z={\displaystyle \frac{\pi }{4}}$

(iii) ${\displaystyle \frac{cos\pi z}{(z-a{)}^2}}$ at $z=a$ and $z=\mathrm{\infty }$.

In case (iii) above what happens when $a$ is an integer(including $a=0$)?

[20M]

1998

1) Show that the function $f(z)={\displaystyle \frac{x^3(1+i)-y^3(1-i)}{x^2+y^2}},z\ne 0$ $f(0)=0$ is continuous and $C-R$ conditions are satisfied at z=0, but $f^{\prime }(z)$ does not exist at z=0.

2) Find the Laurent expansion of ${\displaystyle \frac{z}{(z+1)(z+2)}}$ about the singularity $z=-2$. Specify the region of convergence and the nature of singularity ar $z=-2$.

3) By using the integral representation of $f^n(0)$, prove that $({\displaystyle \frac{x^n}{⌞n}}{)}^2={\displaystyle \frac{1}{2\pi i}}{\oint }_C{\displaystyle \frac{x^n{e}^{xz}}{⌞n{z}^{n+1}}}dz$, where $C$ is any closed contour surrounding the origin. Hence show that $\sum _{n=0}^{\mathrm{\infty }}({\displaystyle \frac{x^n}{⌞n}}{)}^2={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{e}^{2xcos\theta },d\theta$.

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

5) By integrating round a suitable contour show that ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{xsinmx}{x^4+a^4}}={\displaystyle \frac{\pi }{4b^2}}{e}^{-mb}sinmb$, where $b={\displaystyle \frac{a}{\sqrt{2}}}$.

6) Using residue theorem, evaluate ${\int }_0^{2\pi }{\displaystyle \frac{d\theta }{3-2cos\theta +sin\theta }}$.

1997

1) Prove that $\mathrm{u}={\mathrm{e}}^{\mathrm{x}}(\mathrm{x}\cos \mathrm{y}-\mathrm{y}\sin \mathrm{y})$ is harmonic and find the analytic function whose real part is $u$.

[10M]

2) Evaluate ${\oint }_C{\displaystyle \frac{dz}{z+2}}$ where $\mathrm{C}$ is unit circle Deduce that ${\int }_0^{2\pi }{\displaystyle \frac{1+2\cos \theta }{5+4\cos \theta }}d\theta =0$

[10M]

3) If $f(z)={\displaystyle \frac{A_1}{z-a}}+{\displaystyle \frac{A_2}{(z-a{)}^2}}+\dots +{\displaystyle \frac{A_n}{(z-a{)}^n}}$ find the residue at a for ${\displaystyle \frac{f(z)}{z-b}}$ where $A_1,A_2,\dots$ ${\mathrm{A}}_0,\mathrm{a}$ and $\mathrm{b}$ are constants. What is the residue at infinity?

[10M]

4) Find the Laurent series for the function $e^{1x}$ in $0<|z|<\mathrm{\infty },$ Deduce that ${\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }\exp (\cos \theta )\cdot \cos (\sin \theta -n\theta )d\theta ={\displaystyle \frac{1}{n!}},(n=0,1,2,$

[10M]

5) Integrating $e^{-z^2}$ along a suitable rectangular contour show that ${\int }_0^{\mathrm{\infty }}{e}^{-x^2}\cos 2bxdx={\displaystyle \frac{\sqrt{\pi }}{2}}{e}^{-b^2}$

[10M]

6) Find the function $f(z)$ analytic with in the unit circle, which takes the values ${\displaystyle \frac{a-\cos \theta +i\sin \theta }{a^2-2a\cos \theta +1}}0\le \theta \le 2\pi$ on the circle.

[10M]

1996

1) Sketch the ellipse $C$ described in the complex plane by $Z=A\cos \lambda t+iB\sin \lambda t$, $A>B$ where $t$ is a real variable and $A,B,\lambda$ are positive constants. If $C$ is the trajectory of a particle with $z(t)$ as the position vector of the particle at time $t$, identify with justification (i) the two positions where the acceleration is maximum, and (ii) the two positions where the velocity is minimum.

[12M]

2) Evaluate $\underset{z\to 0}{lim}{\displaystyle \frac{1-\cos z}{\sin \left(z^2\right)}}$

[12M]

3) Show that $z=0$ is not a branch point for the function $f(z)={\displaystyle \frac{\sin \sqrt{z}}{\sqrt{z}}}$ Is it a removable singularity?

[12M]

3) Prove that every polynomial equation ${\mathrm{a}}_0+{\mathrm{a}}_1\mathrm{z}+{\mathrm{a}}_2{\mathrm{z}}^2+\dots +{\mathrm{a}}_n{\mathrm{z}}^{\mathrm{n}}=0,{\mathrm{a}}_{\mathrm{n}}\ne 0,\mathrm{n}\ge 1$ has exactly $\mathrm{n}$ roots.

[15M]

4) By using residue theorem, evaluate ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\log }_e(x^2+1)}{x^2+1}}dx$

[15M]

5) About the singularity $z=-2,$ find the laurent expansion of $(z-3)\sin {\displaystyle \frac{1}{z+2}}$

Specify the region of convergence and the nature of singularity at $z=-2$.

[15M]

1995

1) Let $u(x,y)=3x^{2}y+2x^2-y^3-2y^2$. Prove that $u$ is a harmonic function. Find a harmonic function $v$ such that $u+iv$ is an analytic function of $z$.

[12M]

2) Find the Taylor series expansion of the function $f(z)={\displaystyle \frac{z}{z^4+9}}$ around $z=0.$ Find also the radius of convergence of the obtained series.

[12M]

3) Let $C$ be the circle $|z|=2$ described counter clockwise. Evaluate the integral $\cosh \pi z$ ${\int }_C^{dz}z(z^2+1)$

[12M]

3) Let $a\ge 0.$ Evaluate the integral ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\cos ax}{x^2+1}}dx$ with the aid of residues.

[15M]

4) Let $f$ be analytic in the entire complex plane. Suppose that there exists a constant $A>0$ such that $|f(z)|\le A|z|$ for all $z$. Prove that there exists a complex number a such that $f(z)=az$ for all $z$.

[15M]

5) Suppose a power series $\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ converges at a point $z_0\ne 0.$ Let $z_1$ be such that $|z_1|<|z_0|$ and $Z_1\ne 0.$ Show that the series converges uniformly in the disc $\{z:|z|\le |z_1|\}$.

[15M]