Liouville's theorem (function theory)
The Liouville's theorem is a fundamental result in mathematical branch function theory . It is named after the French mathematician Joseph Liouville .
statement
Be a bounded , whole function , i. H. is holomorphic all over and there is a constant with for all . Then is constant .
proof
The claim follows directly from Cauchy's integral formula , cf. also the representation of the dispute between Cauchy and Liouville .
Let be bounded by, then with the integral formula and the standard estimate for curve integrals
- .
Hence the derivative is 0. Since is also connected , the claim follows.
Meaning and generalizations
Liouville's theorem provides a particularly elegant proof of the fundamental theorem of algebra .
As a consequence one immediately gets that dense in is if is holomorphic and not constant. One exacerbation of this fact is Picard's little theorem .
In the language of Riemann surfaces , Liouville's theorem means that every holomorphic function from a parabolic Riemann surface (e.g. the complex plane ) to a hyperbolic Riemann surface (e.g. the unit disk in the complex plane) must be constant .
The so-called generalized theorem of Liouville states:
Is holomorphic and there are real numbers such that for everyone
holds, then is a polynomial with .
If , that is, restricted, one obtains the "old" theorem from Liouville, since polynomials of degree less than or equal to 0 are constant.
literature
- Eberhard Freitag & Rolf Busam: Function theory 1 , Springer-Verlag, Berlin, ISBN 3-540-67641-4