PHGN311 Practice Exam #1 2013
Show your work on all problems to receive full credit. All problems have equal value.
1. Find the region of convergence for the following complex series and sketch the
region in the complex plane:
a.
∞
∑
n!z n
n=1
b.
n 2 (z −1+ i) n
n=1
2n
∞
∑
2. Consider a box with height 200cm, width 200cm and length 100cm. Without using a
calculator, estimate how much the diagonal length, h, changes when the dimensions
€ change to 201cm, 202cm
€ and 99cm.
h
3. The van der Waals equation of state is a relationship between the pressure (P),
temperature (T) and volume (V) of a gas which improves upon the ideal gas law. It is
given by:
P=
RT
a
− 2
V −b V
# ∂P &
# ∂P &
where R, a, and b are constants. Obtain expressions for: % ( and % ( and show
$ ∂T 'V
$ ∂V 'T
€
2
2
∂ P
∂ P
=
∂V∂T ∂T∂V
€
i
€
π
4. Sketch the region of the complex z plane where Re[e 2 z] > 2 .
5. Using complex variables, find the general solution to the homogeneous differential
d4z
€ in real form since its likely in physical
equation:
+ 4z = 0 . Write the solution
dx 4
problems we want a real solution.
6. Consider the Fourier Trig Series of the periodic triangle wave function in the figure.
€
Which of the following are true (could be
more than one):
1. There is no constant term
2. There are no Sine terms
3. There are no Cosine terms
4. There will be a Gibb’s phenomena feature
at the origin (x=0).