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 k depends on the given points.

[12M]

1999

1) Examine the nature of the function f(z)=x2y2(x+iy)x4+y10,z≠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=2z+3z−4

transforms the circle x2+y2−4x=0 into the straight line 4u+3=0 where w=u+iv.

[20M]

4) Using Residue Theorem show that ∫∞∞xsinaxx4+4dx=π2e−asina,(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|→∞.If f(2)=5 and f(−1)=2,find f(z).

[20M]

6) What kind of singularities the following functions have?

(i) 11−ez at z=2πi

(ii) 1sinz−cosz at z=π4

(iii) cosπz(z−a)2 at z=a and z=∞.

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

[20M]

1998

1) Show that the function f(z)=x3(1+i)−y3(1−i)x2+y2,z≠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 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 fn(0), prove that (xn⌞n)2=12πi∮Cxnexz⌞nzn+1dz, where C is any closed contour surrounding the origin. Hence show that ∑∞n=0(xn⌞n)2=12π∫2π0e2xcosθ,dθ.

4) Prove that all roots of z7−5z3+12=0 lie between the circle |z|=1 and |z|=2.

5) By integrating round a suitable contour show that ∫∞0xsinmxx4+a4=π4b2e−mbsinmb, where b=a√2.

6) Using residue theorem, evaluate ∫2π0dθ3−2cosθ+sinθ.

1997

1) Prove that u=ex(xcosy−ysiny) is harmonic and find the analytic function whose real part is u.

[10M]

2) Evaluate ∮Cdzz+2 where C is unit circle Deduce that ∫2π01+2cosθ5+4cosθdθ=0

[10M]

3) If f(z)=A1z−a+A2(z−a)2+…+An(z−a)n find the residue at a for f(z)z−b where A1,A2,… A0,a and b are constants. What is the residue at infinity?

[10M]

4) Find the Laurent series for the function e1x in 0<|z|<∞, Deduce that 1π∫π0exp(cosθ)⋅cos(sinθ−nθ)dθ=1n!,(n=0,1,2,

[10M]

5) Integrating e−z2 along a suitable rectangular contour show that ∫∞0e−x2cos2bxdx=√π2e−b2

[10M]

6) Find the function f(z) analytic with in the unit circle, which takes the values a−cosθ+isinθa2−2acosθ+10≤θ≤2π on the circle.

[10M]

1996

1) Sketch the ellipse C described in the complex plane by Z=Acosλt+iBsinλt, A>B where t is a real variable and A,B,λ 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 limz→01−coszsin(z2)

[12M]

3) Show that z=0 is not a branch point for the function f(z)=sin√z√z Is it a removable singularity?

[12M]

3) Prove that every polynomial equation a0+a1z+a2z2+…+anzn=0,an≠0,n≥1 has exactly n roots.

[15M]

4) By using residue theorem, evaluate ∫∞0loge(x2+1)x2+1dx

[15M]

5) About the singularity z=−2, find the laurent expansion of (z−3)sin1z+2

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

[15M]

1995

1) Let u(x,y)=3x2y+2x2−y3−2y2. 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)=zz4+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 coshπz ∫dzCz(z2+1)

[12M]

3) Let a≥0. Evaluate the integral ∫∞0cosaxx2+1dx 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)|≤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 ∑∞n=0anzn converges at a point z0≠0. Let z1 be such that |z1|<|z0| and Z1≠0. Show that the series converges uniformly in the disc {z:|z|≤|z1|}.

[15M]