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}$