Liouville's theorem (complex analysis)

The set of Liouville is a fundamental result in mathematical branch function theory. It is named after the French mathematician Joseph Liouville.

Statement

Be a bounded entire function, i.e., is holomorphic at all and there is one constant with all. Then is constant.

Evidence

The assertion follows directly from the integral formula of Cauchy, see also the presentation of the dispute between Cauchy and Liouville.

Be limited by, the following applies to the integral formula and the standard estimate for line integrals

Therefore, the derivative is equal to 0 and since is continuous, the assertion follows.

Importance and generalizations

The set of Liouville provides a particularly elegant proof of the fundamental theorem of algebra.

As a corollary we obtain immediately that close in, when holomorphic and not constant. An aggravation of this fact is the small set of Picard.

In the language of Riemann surfaces of Liouville's theorem implies that any holomorphic function of a parabolic Riemann surface (such as the complex plane ) on a hyperbolic Riemann surface (such as the unit disk in the complex plane ) must be constant.

The so-called generalized Liouville's theorem states:

Is holomorphic and there are real numbers such that for all

Applies, then with a polynomial.

Is so limited, we get the "old" set of Liouville, for polynomials of degree less than or equal 0 are constant.