Picard's theorem

The sets of Picard (after Emile Picard ) are sets of function theory , a partial area of mathematics .

They are as follows:

• The Little Picard theorem states that the image of each non-constant entire function , the entire complex plane is from more than one point was taken out.
• The Big Picard theorem states that a holomorphic function with an essential singularity in every small area of this singularity often assumes every complex value infinitely with at most one exception.

Remarks

• In both sentences the possible “exception of a point” is obviously necessary. For example, does not map to, nor is any dotted neighborhood of included in the image . ${\displaystyle z\mapsto e^{z}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle z\mapsto e^{1/z}}$ ${\displaystyle 0}$
• The Small Theorem follows immediately from the Big Theorem, because an entire function is either a polynomial or it has an essential singularity in . ${\displaystyle \infty }$
• The large theorem generalizes Weierstrass-Casorati's theorem .
• A conjecture by B. Elsner is related to Picard's Great Theorem: Let be the dotted open unit disk and a finite open cover of . On each is a simple (i. E. Injective holomorphic ) function added so that at each intersection . Then the differentials merge into a meromorphic 1-form on the unit disk . (In the case that the residual vanishes, the conjecture follows from the Great Theorem.) ${\displaystyle {\dot {\mathbb {E} }}=\mathbb {E} \setminus \{0\}}$ ${\displaystyle U_{1},U_{2},\dots ,U_{n}}$ ${\displaystyle {\dot {\mathbb {E} }}}$ ${\displaystyle U_{j}}$ ${\displaystyle f_{j}}$ ${\displaystyle \mathrm {d} f_{j}=\mathrm {d} f_{k}}$ ${\displaystyle U_{j}\cap U_{k}}$ ${\displaystyle \mathbb {E} }$

proof

Using the theory of the j-function , a brief proof of Picard's little theorem can be given. Assuming be whole and omit the values , the function is ${\displaystyle f}$ ${\displaystyle a\not =b}$

${\displaystyle g(z)={\frac {f(z)-a}{b-a}}}$

whole and omits the values ​​0 and 1. The j-function now maps the upper half-plane combined with points onto a Riemann surface with an infinite number of leaves and branch points at the image points and . It follows that its inverse maps this Riemann surface (without restriction) to the closure of the standard fundamental domain. Since for all and and , or , is locally analytical for all complex values ​​except 0 and 1. It follows that the composition ${\displaystyle \mathbb {H} \cup \mathbb {Q} \cup \{\infty \}}$ ${\displaystyle 0,1}$ ${\displaystyle \infty }$ ${\displaystyle j^{-1}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle j'(\tau )\not =0}$ ${\displaystyle \rho \not =\tau \not =i}$ ${\displaystyle j'(i)=j'(\rho )=0}$ ${\displaystyle j(i)=0}$ ${\displaystyle j(\rho )=1}$ ${\displaystyle j(\infty )=\infty }$ ${\displaystyle j^{-1}}$

${\displaystyle h(z)=j^{-1}(g(z))}$

is locally analytic in every point, since it just leaves out 0 and 1. This can be expanded to a whole function, for which, however , must apply to all , da . From this it follows with Liouville's theorem that and consequently is also constant. ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle |e^{ih(z)}|<1}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle \mathrm {Im} \,h(z)>0}$ ${\displaystyle h}$ ${\displaystyle f}$

literature

• Heinrich Behnke , Friedrich Sommer: Theory of the analytical functions of a complex variable . 3. Edition. Springer-Verlag, Berlin / Heidelberg / New York 1965.

Individual evidence

1. ↑ Heinrich Behnke , Friedrich Sommer: Theory of the analytical functions of a complex variable . 3. Edition. Springer-Verlag, Berlin / Heidelberg / New York 1965, p. 490 .
2. ^ Bernhard Elsner: Hyperelliptic action integral . In: Annales de l'institut Fourier . tape 49 , no. 1 , 1999, ISSN  1777-5310 , p. 303–331 (English, numdam.org [PDF; 2.0 MB ; accessed on September 9, 2010]). P. 330.