PHZ 3113, Section 3924, Fall 2013, Homework 7
Due at the start of class on Friday, October 18. Half credit will be available for
homework submitted after the deadline but no later than the start of class on Monday,
October 21.
Answer all questions. Please write neatly and include your name on the front page of your
answers. You must also clearly identify all your collaborators on this assignment. To gain
maximum credit you should explain your reasoning and show all working.
1. Part of a circuit consists of two branches in parallel. In one branch, a resistor, R, is in series
with a capacitor, C. The other branch consists simply of an inductor, L.
a) Find the total impedance, Z, of this part of the circuit.
b) Find the resonant frequency, ω, of this part of the circuit.
c) Evaluate Z at the resonant frequency.
2. Find a Laurent series expansion for:
1
,
z(z − 1)(z − 2)
valid in the annulus between |z| = 1 and |z| = 2. What is the residue of f (z) at z = 0?
f (z) =
3. This question concerns simple poles and residues.
a) Identify the poles of:
zeiz
,
f (z) = 2
z − c2
and find the residue at each.
b) Hence evaluate:
I
f (z)dz,
C
around the contour |z| = 2c.
c) Show how you can evaluate:
by considering the contour integral:
∫ ∞
sin(x)
dx,
x
0
I
eiz
dz,
z
around the upper half plane.
d) What contour would you use if i were replaced by −i in c)?
4. Use the generalized Cauchy integral formula to evaluate:
I
I
e3z dz
sin 2zdz
,
,
3
C z − ln 2
C (6z − π)
where C is the square with vertices ±1 ± i.