Liouville's theorem (complex analysis)

In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all is constant. Equivalently, non-constant holomorphic functions on have unbounded images. The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits two or more complex numbers must be constant. This important theorem has several proofs. A standard analytical proof uses the fact that holomorphic functions are analytic. Another proof uses the mean value property of harmonic functions. The proof can be adapted to the case where the harmonic function is merely bounded above or below. See Harmonic function#Liouville's theorem. There is a short proof of the fundamental theorem of algebra based upon Liouville's theorem. A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if and are entire, and everywhere, then for some complex number . Consider that for the theorem is trivial so we assume . Consider the function . It is enough to prove that can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of is clear except at points in . But since is bounded and all the zeroes of are isolated, any singularities must be removable. Thus can be extended to an entire bounded function which by Liouville's theorem implies it is constant. Suppose that is entire and , for . We can apply Cauchy's integral formula; we have that where is the value of the remaining integral. This shows that is bounded and entire, so it must be constant, by Liouville's theorem. Integrating then shows that is affine and then, by referring back to the original inequality, we have that the constant term is zero. The theorem can also be used to deduce that the domain of a non-constant elliptic function cannot be .