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
DR ≡ {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 DR. Show that for 0 < |z| < R ,
f(z) =
1
2πi
⌠ ⌡
|ζ|=R
f(ζ)
ζ−z
dζ.
Conclude that limz → 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.
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