[SOLVED] Amath 567, homework 5

1. Problem 1: Evaluate the integrals
1
2πi I
C
f(z)dz
where C is the unit circle centered at the origin with f(z) given below by
(a) enclosing the singular points inside C, and
(b) enclosing the singular points outside C by including the point at infinity t =
1
z
as z → ∞.
Show that you obtain the same result in both cases2. Problem 2: Find the Fourier transform of f(t) where f(t) = 1 for −a < t < a and
f(t) = 0 otherwise.
Then, do the inverse transform using techniques of contour integration, e.g. Jordan’s
lemma, principle values, etc.3. Problem 3: Consider the function
f(z) = ln(z
2 − 1)
made single-valued by restricting the angles in the following ways, with z1 = z − 1 =
r1e
iθ1 and z2 = z + 1 = r2e
iθ2
. Find where the branch cuts are for each case by
locating where the function is discontinuous. Use AB tests and show your results.