# Liouville’s theorem

July 24, 2012 Leave a comment

We can use the Cauchy estimates from last time to prove a very beautiful theorem:

Theorem 1If is bounded (i.e., for some ), then is constant.

*Proof:*Expand as a power series at 0. We know that this representation is valid everywhere in . Write . Recall that

The Cauchy estimates tell us that

for every positive less than the radius of convergence of . Of course, the radius of convergence is infinite, so this estimate is valid for all . Thus, for , can be bounded by arbitrarily small positive numbers. That is, for all . In other words, , a constant.

It’s worth mentioning that trigonometric functions such as and are holomorphic on all of (often called **entire**) functions. Hopefully this doesn’t bother you, even in light of Liouville’s theorem. If it does, you’re remembering that is always between and . You are correct for , but Liouville’s theorem doesn’t say anything about functions on . Indeed, if you plug in complex numbers, you’ll see that can be arbitrarily large. In fact, look at for . You can use Wolfram Alpha to see that on the imaginary axis, takes on arbitrarily large values.