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For Liouville's theorem in Hamiltonian mechanics, see Liouville's theorem (Hamiltonian).

In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f(z)| ≤ M for all z in C is constant.

Liouville's theorem can be used to give an elegant short proof for the fundamental theorem of algebra.

The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits at least two complex numbers must be constant. In the language of Riemann surfaces, the theorem can be generalized as follows: if M is a parabolic Riemann surface (such as the complex plane C) and N is a hyperbolic one (such as an open disk), then every holomorphic function f : M → N must be constant.

Contents 1 Proof 1.1 A corollary 2 Remarks 3 See also 4 External links

The theorem follows from the fact that holomorphic functions are analytic. Since the given f is entire, it can be represented by its Taylor series about 0

where (Cauchy's integral formula)

and C_r is the circle about 0 of radius r > 0. We can estimate directly

where in the second inequality we have invoked the assumption that |f(z)| ≤ M for all z. But the choice of r in the path integrals used, is arbitrary. Therefore, letting r tend to infinity gives a_k = 0 for all k ≥ 1. Thus f(z) = a₀ and this proves the theorem.

A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if f and g are entire, and |f| ≤ |g| everywhere, then f = α g for some complex number α. To show this, consider the function

It is enough to have that h is entire, in which case the result follows by Liouville's theorem. The holomorphy of h is clear except at points in the discrete fiber g^-1(0). Now if g(a) = 0, then f(a) = 0. Analyticity then implies h can be continuously, therefore holomorphically, extended over a. Thus h can be extended over g^-1(0) to an entire function.

Let {∞} ∪ C be the one point compactification of the complex plane C. In place of holomorphic functions defined on regions in C, one can consider regions in {∞} ∪ C. Viewed this way, the only possible singularity for entire functions, defined on C ⊂ {∞} ∪ C, is the point ∞. If an entire function f is bounded in a neighborhood of ∞, then ∞ is a removable singularity of f, i.e. f cannot blow up or behave erratically at ∞. In light of the power series expansion, it is not surprising that Liouville's theorem holds.

Similarly, if an entire function has a pole at ∞, i.e. blows up like zⁿ in some neighborhood of ∞, then f is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if |f(z)| ≤ M|zⁿ| for |z| sufficiently large, then f is a polynomial of degree at most n. A proof can be sketched as follows. Again take the Taylor series representation of f

By assumption, we must have

for |z| sufficiently large. So

for |z| sufficiently large. Since f is entire, appealing to the identity theorem proves the claim.

• Joseph Liouville

• Liouville's theorem on PlanetMath
• Eric W. Weisstein, Liouville’s Boundedness Theorem at MathWorld.
• Module for Liouville’s Theorem by John H. Mathews