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 \(f(z)=\dfrac{x^2y^2(x+iy)}{x^4+y^{10}},\;z\neq0,\) \(f(0)=0\)
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) \(1<\vert z\vert <2\),

(ii) \(\vert z\vert >2\). Show further that

where \(C\) is any contour enclosing the point \(z=1\) and \(z=2\).

[20M]

3) Show that the transformation \(w=\dfrac{2z+3}{z-4}\)

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 \(\int^{\infty}_{\infty} \dfrac{xsin ax}{x^4+4}\,dx=\dfrac{\pi}{2}e^{-a}sin a,(a>0).\)

[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 \(\vert z\vert \to\infty\).If \(f(2)=5\) and \(f(-1)=2\),find \(f(z)\).

[20M]

6) What kind of singularities the following functions have?

(i) \(\dfrac{1}{1-e^z}\) at \(z=2\pi i\)

(ii) \(\dfrac{1}{sin z-cos z}\) at \(z=\dfrac{\pi}{4}\)

(iii) \(\dfrac{cos\pi z}{(z-a)^2}\) at \(z=a\) and \(z=\infty\).

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

[20M]

1998

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

2) Find the Laurent expansion of \(\dfrac{z}{(z+1)(z+2)}\) 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 \((\dfrac{x^n}{\llcorner{n}})^2=\dfrac{1}{2\pi i}\oint_C \dfrac{x^ne^{xz}}{\llcorner{n}z^{n+1}}\,dz\), where \(C\) is any closed contour surrounding the origin. Hence show that \(\sum^{\infty}_{n=0} (\dfrac{x^n}{\llcorner{n}})^2=\dfrac{1}{2\pi }\int^{2\pi}_0 e^{2xcos\theta},d \theta\).

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

5) By integrating round a suitable contour show that \(\int^{\infty}_0 \dfrac{xsin mx}{x^4+a^4}=\dfrac{\pi}{4b^2}e^{-mb}sin mb\), where \(b=\dfrac{a}{\sqrt{2}}\).

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

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 \(\oint_{C} \dfrac{d z}{z+2}\) where \(\mathrm{C}\) is unit circle Deduce that \(\int_{0}^{2 \pi} \dfrac{1+2 \cos \theta}{5+4 \cos \theta} d \theta=0\)

[10M]

3) If \(f(z)=\dfrac{A_{1}}{z-a}+\dfrac{A_{2}}{(z-a)^{2}}+\ldots+\dfrac{A_{n}}{(z-a)^{n}}\) find the residue at a for \(\dfrac{f(z)}{z-b}\) where \(A_{1}, A_{2}, \ldots\) \(\mathrm{A}_{0}, \mathrm{a}\) and \(\mathrm{b}\) are constants. What is the residue at infinity?

[10M]

4) Find the Laurent series for the function \(e^{1 x}\) in \(0<\vert z\vert <\infty,\) Deduce that \(\dfrac{1}{\pi} \int_{0}^{\pi} \exp (\cos \theta) \cdot \cos (\sin \theta-n \theta) d \theta=\dfrac{1}{n !}, \quad(n=0,1,2,\)

[10M]

5) Integrating \(e^{-z^{2}}\) along a suitable rectangular contour show that \(\int_{0}^{\infty} e^{-x^{2}} \cos 2 b x d x=\dfrac{\sqrt{\pi}}{2} e^{-b^{2}}\)

[10M]

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

[10M]

1996

1) Sketch the ellipse \(C\) described in the complex plane by \(Z=A \cos \lambda t+i B \sin \lambda t\), \(A>B\) 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 \(\lim _{z \rightarrow 0} \dfrac{1-\cos z}{\sin \left(z^{2}\right)}\)

[12M]

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

[12M]

3) Prove that every polynomial equation \(\mathrm{a}_{0}+\mathrm{a}_{1} \mathrm{z}+\mathrm{a}_{2} \mathrm{z}^{2}+\ldots+\mathrm{a}_{n} \mathrm{z}^{\mathrm{n}}=0, \mathrm{a}_{\mathrm{n}} \neq 0, \mathrm{n} \geq 1\) has exactly \(\mathrm{n}\) roots.

[15M]

4) By using residue theorem, evaluate \(\int_{0}^{\infty} \dfrac{\log _{e}\left(x^{2}+1\right)}{x^{2}+1} d x\)

[15M]

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

Specify the region of convergence and the nature of singularity at \(z=-2\).

[15M]

1995

1) Let \(u(x, y)=3 x^{2} y+2 x^{2}-y^{3}-2 y^{2}\). 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 \(f(z)=\dfrac{z}{z^{4}+9}\) around \(z=0 .\) Find also the radius of convergence of the obtained series.

[12M]

3) Let \(C\) be the circle \(\vert z \vert =2\) described counter clockwise. Evaluate the integral \(\cosh \pi z\) \(\int_{C}^{d z} z\left(z^{2}+1\right)\)

[12M]

3) Let \(a \geq 0 .\) Evaluate the integral \(\int_{0}^{\infty} \dfrac{\cos a x}{x^{2}+1} d x\) 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 \(\vert f ( z )\vert \leq A \vert z \vert\) 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 \(\sum_{n=0}^{\infty} a_{n} z^{n}\) converges at a point \(z _{0} \neq 0 .\) Let \(z _{1}\) be such that \(\vert z _{1}\vert <\vert z _{0}\vert\) and \(Z _{1} \neq 0 .\) Show that the series converges uniformly in the disc \(\left\{ z :\vert z \vert \leq\vert z _{1}\vert \right\}\).

[15M]