In complex analysis, Liouville's theorem states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a positive number M {\displaystyle M} such that | f ( z ) | ≤ M {\displaystyle |f(z)|\leq M} for all z ∈ C {\displaystyle z\in \mathbb {C} } is constant. Equivalently, non-constant holomorphic functions on C {\displaystyle \mathbb {C} } have unbounded images. The theorem is named after Joseph Liouville, although the theorem was first proven by Cauchy in 1844. 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.
Statement Liouville's theorem: Every holomorphic function f : C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } for which there exists a positive number M {\displaystyle M} such that | f ( z ) | ≤ M {\displaystyle |f(z)|\leq M} for all z ∈ C {\displaystyle z\in \mathbb {C} } is constant. More succinctly, Liouville's theorem states that every bounded entire function must be constant.
Proof 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 f {\displaystyle f} is merely bounded above or below. See Harmonic function#Liouville's theorem. Another approach to prove the theorem is
Corollaries
Fundamental theorem of algebra There is a short proof of the fundamental theorem of algebra using Liouville's theorem.
No entire function dominates another entire function A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if f {\displaystyle f} and g {\displaystyle g} are entire, and | f | ≤ | g | {\displaystyle |f|\leq |g|} everywhere, then f = α g {\displaystyle f=\alpha g} for some complex number α {\displaystyle \alpha } . Consider that for g = 0 {\displaystyle g=0} the theorem is trivial so we assume g ≠ 0 {\displaystyle g\neq 0} . Consider the function h = f / g {\displaystyle h=f/g} . It is enough to prove that h {\displaystyle h} can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of h {\displaystyle h} is clear except at points in g − 1 ( 0 ) {\displaystyle g^{-1}(0)} . But since h {\displaystyle h} is bounded and all the zeroes of g {\displaystyle g} are isolated, any singularities must be removable. Thus h {\displaystyle h} can be extended to an entire bounded function which by Liouville's theorem implies it is constant.
If f is less than or equal to a scalar times its input, then it is linear Suppose that f {\displaystyle f} is entire and | f ( z ) | ≤ M | z | {\displaystyle |f(z)|\leq M|z|} , for M > 0 {\displaystyle M>0} . We can apply Cauchy's integral formula; we have that
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