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
(1) (1.9)
(a) Use complex algebra to show that for any four integers a, b, c, and d there are integers u and v
so that
(a2 + b2 )(c2 + d2 ) = u2 + v 2
Proof. If we let z1 = a + ib and z2 = c + id then we have that
2
2
(a2 +
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
z
the domain.
Solution. We d
Math 425
1. Express
Test I Solutions
3
2
+
i
2
603
in the form a + ib. Simplify your answer as much as possible. (It may
be convenient to express
We rst write
3
2
February 25, 2011
+
i
2
3
2
+
i
2
in polar form.)
in polar form. The modulus of
3
2
i
3
+
=
Math 425
Test II Solutions
April 11, 2011
1. Evaluate the following line integrals.
(a)
C
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