Math 412 Group Work #7 Spring 2009
Problem 1. Let
f(z) = x2+y2+i(2xy).
(a): Use the Cauchy-Riemann equations to determine where fis differentiable.
(b): Evaluate the derivative at the points z0where f0(z0) exists.
(c): Determine all singular points of the function f.
Problem 2. Suppose that f:C→Cis an entire, real-valued function. Show that f
must be a constant function.
Problem 3. Suppose that f(x+iy) = u(x, y) + iv(x, y). For what relations among
the real constants a, b, c, d with
u(x, y) = ax3+bx2y+cxy2+dy3
can v(x, y) be defined so that fis an entire function?