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 . ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle c\in \mathbb {R} }$ ${\displaystyle |f(z)|\leq c}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle f}$

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 ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$ ${\displaystyle c\in \mathbb {R} }$

${\displaystyle \left|f'(z)\right|=\left|{\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U_{r}(z)}{\frac {f(\zeta )}{\left(\zeta -z\right)^{2}}}\mathrm {d} \zeta \right|\leq {\frac {1}{2\pi }}\cdot 2\pi r\cdot {\frac {c}{r^{2}}}\rightarrow 0\left(r\rightarrow \infty \right)}$.

Hence the derivative is 0. Since is also connected , the claim follows. ${\displaystyle \mathbb {C} }$

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 . ${\displaystyle f(\mathbb {C} )}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$

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 . ${\displaystyle \mathbb {C} }$

The so-called generalized theorem of Liouville states:

Is holomorphic and there are real numbers such that for everyone ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$ ${\displaystyle b,c,d}$ ${\displaystyle z\in \mathbb {C} }$

${\displaystyle |f(z)|\leq b\cdot |z|^{d}+c}$

holds, then is a polynomial with . ${\displaystyle f}$ ${\displaystyle \deg(f)\leq d}$

If , that is, restricted, one obtains the "old" theorem from Liouville, since polynomials of degree less than or equal to 0 are constant. ${\displaystyle d=0}$ ${\displaystyle f}$

literature

• Eberhard Freitag & Rolf Busam: Function theory 1 , Springer-Verlag, Berlin, ISBN 3-540-67641-4