Download H3

Math 326
Homework Assignment 3
due date: Oct. 7, 2010
1. Suppose that z1 , z2 , and z3 are distinct complex numbers (i.e., no two are equal),
and that w1 , w2 , and w3 are distinct complex numbers. Show that there is a Möbius
such that f (zi ) = wi , i = 1, 2, 3. (Suggestion: Convert
transformation f (z) = az+b
cz+d
the search for a, b, c, and d into a linear algebra problem.) How do you know that the
a, b, c, d that you find satisfy ad − bc 6= 0?
2. Suppose that f : C −→ C is a function which satisfies f (z) = f ( z2 ) for all z ∈ C and
which is continuous at z = 0. Show that f is a constant function.
3. For each of the following functions f , describe the domain where they are defined
and compute their derivatives. (You can use the derivative rules – there is no need to
differentiate the functions using the limit definition.)
10
2
(b) f (z) = z + z1 .
(a) f (z) = z 3 − zez − 4.
(c) f (z) =
z2
.
ez − 2
(d) f (z) =
az + b
(with ad − bc 6= 0).
cz + d
4. For each of the following functions u(x, y) and v(x, y), check if the Cauchy-Riemann
equations hold. If they do hold, find a function f (z) so that f (x+iy) = u(x, y)+i v(x, y).
(a) u(x, y) = y, v(x, y) = −x.
(b) u(x, y) = 5x − 3y, v(x, y) = 3x + 5y.
(c) u(x, y) = cos(x) sin(y), v(x, y) = sin(x) cos(y).
(d) u(x, y) = x3 − 3xy 2 + 1, v(x, y) = 3x2 y − y 3 .
5. Let f : C −→ C be an entire function. Define a new function g(z) by g(z) = f (z).
(i.e., given a complex number z, to compute g(z) we input the complex number z into
f , and then take the complex conjugate of the answer.)
Is g also an entire function? (Prove this or give a counterexample).
1