Liouville's Theorem (complex analysis)
In complex analysis, Liouville's Theorem states that a bounded holomorphic function on the entire complex plane must be constant. It is named after Joseph Liouville. Picard's Little Theorem is a stronger result.
Let be a holomorphic function. Suppose there exists some real number such that for all . Then is a constant function.
We use Cauchy's Integral Formula.
Pick some ; let denote the simple counterclockwise circle of radius centered at . Then Since is holomorphic on the entire complex plane, can be arbitrarily large. It follows that , for every point . Now for any two complex numbers and , so is constant, as desired.
It follows from Liouville's theorem if is a non-constant entire function, then the image of is dense in ; that is, for every , there exists some that is arbitrarily close to .
Suppose on the other hand that there is some not in the image of , and that there is a positive real such that has no point within of . Then the function is holomorphic on the entire complex plane, and it is bounded by . It is therefore constant. Therefore is constant.
Picard's Little Theorem offers the stronger result that if avoids two points in the plane, then it is constant. It is possible for an entire function to avoid a single point, as avoids 0.