Math 425, Homework #2 Solutions
(1) Prove that for z , w C, ez = ew if and only if there exists a k Z so that z = w + i2k .
Proof. We rst consider the special case where w = 0. In this case, when we use the denition of
ex+iy = ex (cos y + i sin y ) we nd
Math 425, Homework #1 Solutions
(a) Use complex algebra to show that for any four integers a, b, c, and d there are integers u and v
(a2 + b2 )(c2 + d2 ) = u2 + v 2
Proof. If we let z1 = a + ib and z2 = c + id then we have that
Math 425, Homework #3 Solutions
(1) (4.11) Let f : D C C be an analytic function on D with D an open, convex set. Suppose that
f satises |f (z )| 1 for all z D. Show that
|f (b) f (a)| |b a|
for any a, b D.
Proof. Let C be the straight line segment connec
Math 425, Homework #4 Solutions
(1) (8.9) Dene a function f analytic in the plane minus the non-positive real axis and such that
f (x) = xx on the positive real axis. Find f (i), and f (i). Show that f () = f (z ) for all z in
Solution. We d
Test I Solutions
in the form a + ib. Simplify your answer as much as possible. (It may
be convenient to express
We rst write
February 25, 2011
in polar form.)
in polar form. The modulus of
Test II Solutions
April 11, 2011
1. Evaluate the following line integrals.
z dz , where C is the straight-line segment connecting 0 to 2 + 2i.
Answer. The curve C can be parametrized as
z (t) = t(2 + 2i)
t [0, 1].
We note that z (t) = 2 + 2