1988

1) By evaluating ∫dzz+2 over a suitable contour C, prove that ∫x01+2cosθ5+4cosθdθ

2) If f is analytic in |Z|≤R and x, y lie inside the disc, evaluate the integral ∫H−Rf(z)dz(z−x)(z−y) and deduce that a function analytic and bounded for all finite z is a constant.

3) If f(z)=∑∞n=0anzn has radius of convergence R and prove that 12π∫2n0|f(reiθ)|2dθ=∑∞n=0|an|2r2n.

4) Evaluate ∫cZez(z−a)3, if a lies inside the closed contour C.

5) Prove that ∫00e−x2cos(2bx)dx=√π2e−b2; (b>0) by the integrating e−z2 along the boundary of the rectangle |x|d′′R,0d′′yd′′b.

TBC

6) Prove that the coefficients Cn of the expansion 11−z−z2=∑∞n=0Cnzn satisfy Cn=Cn−1+Cn−2,n≥2. Determine Cn.

1986

1) Let f(z) be single valued and analytic with in and on a closed curve C. If z0 is any point interior to C then show that f(z0)=12πi∫cf(z)z−z0dz, where the integral is taken in the +ve sense around C.

2) By contour integration method show that

(i) ∫∞0dxx4+a4=π√24a3, where a>0
(ii) ∫∞0sinxxdx=π2

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 ∫∞0xsinaxx2−b2dx.

1984

1) Evaluate by contour integration method:

(i) ∫∞0xsinmxx4+a4dx (ii) ∫∞0xa−1logx1+x2dx

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 η, ∈ and ρ, prove that ∃ a point z, such that 0<|z−a|<ρ for which |f(z)−η|<ϵ

1983

1) Obtain the Taylor and Laurent series expansions which represent the function z2−1(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 ∫∞0xa−11+x2dx,0<a<2.