Liouville's Theorem
Theorem (Liouville's Theorem)
If f is a bounded entire function then f is constant.
Proof let %alpha be any point of setc , f an entire function and let C be a circle with centre 0 and radius r >abs {%alpha} . Since f is entire, so analytic on setc we can apply Cauchy's integral formula twice to give
and
Subtracting one from the other gives
We need to estimate this integral. Since by assumption
is bounded, there exists a number
such that
for all
Also for each
we have
and so by the backwards form of the triangle inequality,
for
It follows that
for
Hence
We now show that for each positive number
we can choose
so large that
⇔
⇔
It follows that ifthen
for each
Sincecan be arbitrarily small
for all
henceis a constant function.
