Liouville's theorem in complex analysis states that every bounded (i.e., there exists a real number M such that |f(z)| ≤ M for all z in C) entire function (a holomorphic function f(z) defined on the whole complex plane C) must be constant. Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... The text or formatting below is generated by a template which has been proposed for deletion. ... In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

Liouville's theorem can be used to give an elegant short proof for the fundamental theorem of algebra. The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. ...

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 complex analysis, mathematician Charles Emile Picards name is given to two theorems regarding the range of an analytic function. ...

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. In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...

Proof

Given f, we have

by Taylor series about 0, which implies As the degree of the taylor series rises, it approaches the correct function. ...

where C_r is the circle about 0 of radius r. By moving the absolute value inside of the integral, we find

Now we can use the assumption that |f(z)| ≤ M for all z (since f is given to be bounded), and the fact that |z|=r on the circle C_r. We get

Then,

Let r now tend to infinity so the circle C_r gets ever larger. If k is greater than 0, M/r^k tends to zero and so a_k must be zero.

However, if k=0, r⁰ = 1 (r ≠ 0 as r tends to infinity), so a₀ is the only term left in the Taylor series, and we have our result.

See also

• Joseph Liouville.

Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ...

External links

• Liouville's theorem (http://planetmath.org/?op=getobj&from=objects&id=1145) on PlanetMath.