PRACTICE PROBLEMS, COMPLEX VARIABLES
SPRING 2015
(1) Let f be a non-constant entire function. Show that M (r) = max 0 ≤ θ < 2π f (reiθ )
is a strictly increasing function.
(2) By comparing an z n with an an + · · · + a1 z + a0 on circles |z| = R of large radius,
prove the Fundamental
of Algebra using Rouché’s theorem.
P∞ Theorem
n
(3) Suppose f (z) = n=0 an z , with an > 0 for all n. If the radius of convergence of
the series is one, show that f has a singularity at z = 1. Hint: Suppose f were
analytic at z = 1. Expand f about z = 1/2 to get a series that converges for some
x > 1. What does this imply about the original series?
1
that is
(4) Find the Laurent series about z = 0 for
(z − 1)(z − 3)
(a) valid for |z| < 1.
(b) valid Z
for 1 < |z| < 3
∞
dx
√ .
(5) Calculate
(1
+
x) x
0
1