## Homework 2

Here's a hint for II.3.1 (ii) --Steve Bell:

$\frac{f(z)g(z)-f(z_0)g(z_0)}{z-z_0}=\frac{f(z)g(z)-f(z)g(z_0)+f(z)g(z_0)-f(z_0)g(z_0)}{z-z_0}=$

$f(z)\frac{g(z)-g(z_0)}{z-z_0}+g(z_0)\frac{f(z)-f(z_0)}{z-z_0}.$

Here is a hint for II.8.1 (c) --Steve Bell:

If the modulus of $f=u+iv$ is constant, then

$u^2+v^2=c.$

Take the partial derivative of this equation with respect to x to get one equation. Take it with respect to y to get another. Use the Cauchy-Riemann equations to conclude that the gradients of both u and v must be identically zero. (Note: The case c=0 is easy. If c is not zero, then it follows that it is not possible for u and v to vanish simultaneously.)

## Alumni Liaison

Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett