July 24, 2012 Leave a comment
We can use the Cauchy estimates from last time to prove a very beautiful theorem:
Theorem 1 If 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.