MTH 611 Complex Analysis
Homework Assignment II
Solutions
Problem I.
(i) Expand f (z ) = (2z + 1)/(z + 1)2 as power series f (z ) = n=0 an (z 1)n . What is
the radius of convergence of the resulting series?
Solution. We compute
2
d1
(2z + 1)/(z + 1)2 =
+
MTH 611 Complex Analysis
Homework Assignment I
Due April 16, 2012
Problem I.
2i
.
(1 + 2i)(2 + i)
(ii) Find the real and imaginary parts of (1 i 3)19 .
(iii) Solve completely z 5 = 4 3i.
(i) Simplify
Solution. Part i : Compute:
(2 i)2 (1 2i)
(3 4i)(1 2i)
MTH 611 Complex Analysis
Homework Assignment IV
Solutions
Problem I.
(i) Let f (z ) be entire and suppose that f (z ) is real if and only if z is real. Show that
f (z ) can have at most one zero.
Solution. Obviously the zeros of f (z ) must be located on
MTH 611 Complex Analysis
Midterm Exam Solutions
Problem I.
(i) Is the function f (z ) = f (x + iy ) = sin x cosh y i cos x sinh y analytic?
Solution. We compute
u
v
= cos x cosh y =
= cos x cosh y,
x
y
so, as per the Cauchy-Riemann equations, the function
MTH 611 Complex Analysis
Exam Sample Problems
Problem I.
(i) Find all analytic functions f on an open, connected C such that |f |2 is harmonic
in .
1+x
y
(ii) Is f (z ) =
+i
analytic?
2 + y2
(1 + x)
(1 + x)2 + y 2
(iii) Suppose f (z ) is analytic in . Sho
MTH 611 Complex Analysis
Homework Assignment III
Solutions
Problem I.
(i) Find precise conditions under which the principal branch of the complex Log function satises the familiar identity Log (zw) = Log z + Log w.
Solution. Recall that Log z = ln |z | +