Homework #1
Analytic Function Theory 3450:625
Dr. Norfolk
1. Suppose that one value of z 1/3 = 3 5i. In the form a + ib, nd the exact values of
(a) z
(b) The other 2 values of z 1/3 .
2. Let P (x) = an xn + an1 xn1 + + a0 be a polynomial of actual degree
Homework #4
Analytic Function Theory 3450:625
Dr. Norfolk
1. Let a, b C . Dene a sequence < zn > via
I
zn1 + zn2
z0 = a, z1 = b and zn =
for n 2.
2
2
1n 1
2
+ a + b for n 0.
Show that zn = (a b)
3
2
3
3
2. Find the points of discontinuity for the functio
Homework #2
Analytic Function Theory 3450:625
Dr. Norfolk
1. Describe the set of z for which Log [(1 + i)z] = Log (1 + i) + Log (z), both algebraically and
by a sketch.
2. Consider the mapping w = z 2 . Given the region 0 Re w 1, 0 Im w 1, describe the
re
Homework #7 Analytic Function Theory 3450:625 Dr. Norfolk
Determine, with proof, the region of convergence of the following series:
(n!)n z 2
1.
n
n=1
2.
sin
n=1
+ iy
3
n
, where y is real.
einz
3.
n
n=1 1 + 2
2tn z n , where tn is the largest power of 2
Homework #6
Analytic Function Theory 3450:625
Dr. Norfolk
1. Use the denitions of the complex trigonometric functions to prove that
sin(z + w) = sin(z ) cos(w) + cos(z ) sin(w) .
2. Prove that
|z |=3
eiz
6e3
dz
z3 + 2
25
z 2 dz , where is the triangle wi
Homework #5
Analytic Function Theory 3450:625
Dr. Norfolk
1. Given f : D D, where D is some domain in C . There exists 0 < 1 such that
I
|f (z1 ) f (z2 )| |z1 z2 | for z1 , z2 D.
(a) Prove that f (z ) is continuous on D.
(b) Choose z0 D, and dene < zn > v
Homework #3
Analytic Function Theory 3450:625
Dr. Norfolk
1. Show that every circle through the origin can be described by the parametrization
z = z0 + z0 ei , 0 2 , z0 = 0.
2. The standard polar representation of a cardioid is r = a(1 + cos ).
Show that