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 giveandSubtracting one from the other gives

We need to estimate this integral. Since by assumptionis bounded, there exists a number such thatfor allAlso for eachwe haveand so by the backwards form of the triangle inequality,forIt follows thatfor

Hence

We now show that for each positive numberwe can chooseso large that

⇔⇔

It follows that ifthenfor eachSincecan be arbitrarily smallfor allhenceis a constant function.