Maths Homework #3

Problem 1
Contour C is a circle of radius 3 centered at the origin, f(z) = cosz
z(z−1). Compute H
C
f(z) dz
by two methods.
(a) Decompose 1
z(z−1) into partial fractions and apply Cauchy’s integral formula twice.
(b) Do not split into partial fractions. Instead, deform the contour so that integration
of f(z) is done over two small circles, one centered at the origin, and the other centered
at z = 1. Do you get the same result? Draw a picture and provide a short explanation
of how exactly the contour deformation works in this case and why it yields the correct
value.
Problem 2
Using the same method we used in class to compute R∞
−∞
e
iξx
1+x2 dx, compute the following
integral for any ξ ≥ 0:
Z∞
−∞
e
iξx
(x + 1 + i)(x − 1 + i)
dx = ...
Hint: Decomposing into partial fractions may help.
Problem 3
Contour C is a circle of radius 100 centered at 10 + 2i. Expand function f(z) = z cosz
(z−π)
3
into Laurent series about π and evaluate
I
C
z cos z
(z − π)
3
dz.
Hint: z = (z − π) + π, cos(z − π) = ...
Problem 4
(a) Expand function f(z) = sin((z−3)2
)
(z−4) into the Taylor series about 3; find first 3 non-zero
terms AND determine the radius of convergence for this series. Hint: z − 4 = (z − 3) − 1.
(b) Expand function f(z) = e
z
(z+1)
(z−2)2
into the Laurent series about 2. Find five terms (the
ones with the lower powers of (z − 2)).