M417

1.

Suppose that f(z) is an entire (analytic in the entire complex plane) function. Suppose that there is are constants A and B such that |f(z)| ≤ A + B |z|^N. Prove that f^(N+1)(z) = 0. Conclude that f(z) is a polynomial of degree ≤ N.

2.

Let f(z) be analytic in the domain D_R ≡ {z | 0 < |z| < R }. Show that for 0 < ϵ < |z| < R ,

f(z) = 1 2πi ⌠ ⌡ |ζ|=R  f(ζ) ζ−z  dζ− 1 2πi ⌠ ⌡ |ζ|=ϵ  f(ζ) ζ−z  dζ.

Both integrals are taken in the positive direction.

3.

Let f be as in problem 2. Suppose that in addition f is bounded in the domain D_R. Show that for 0 < |z| < R ,

f(z) = 1 2πi ⌠ ⌡ |ζ|=R  f(ζ) ζ−z  dζ.

Conclude that lim_{z → 0} f(z) exists.

N.B.

The function f(z) can be extended to be analytic in the domain { |z| ≤ R}. This shows that an isolated singularity at which an analytic function remains bounded (in a deleted neighborhood) is removable.