Homework #1 (Math Review)

Handed out September 7, due September 14.

1. Evaluate the integral (∫_0^2*π)((d^)(θ)/(2-sin(θ))) using residues.
   Hint: pick γ=unit circle z=e(𝑖*θ), then sin(θ)=1/(2*𝑖)*(z-1/z).
   Solve here.
   

2. Evaluate (∫_-∞^∞)(cos(b*x)/(x2+a2)*(d^)(x)), a>0,b>0 by residues.
   Solve here.
   

3. Evaluate G(w)=(∫_-z^z)(1/(2*π)√(,4-ε2)/(w-ε+𝑖*δ))*(d^)(ε) for δ→0 numerically and plot the real and imaginary parts as a function of w. If you are brave, you can try to evaluate by residues as well. But that is not needed now. Plot results for -5≤w≤5.
   Solve here.
   

4. Consider the function G(z)=1/(1+e(-z)). Find the poles and residues of G(z). Use this information to compute the sum ƒ(τ)=(∑_-∞^∞)((ℇ(-𝑖*(2*n+1)*π*τ))/(𝑖*(2*n+1)*π+c))for c a real constant and 0<τ<1 (note the function is antiperiodic in τ with period 1 ). Choose c=1 and numerically calculate ƒ(τ) from the summation and from your analytic result found from solving this summation using residues.
   In particular, repeat the value for τ=0.5 and determine how large the cutoff for n is in the summation to get for digits of accuracy.
   Solve here.