Picard theorem

The sets of Picard (after Émile Picard ) are sets of function theory, a sub- area of mathematics.

They are as follows:

• The Little Theorem Picard says that the image of each non-constant entire function is the entire complex plane, from which a maximum of one point was taken out.

• The big set of Picard says that a holomorphic function with an essential singularity in every little area of this singularity each complex value with at most one exception infinitely often assumes.

Comments

• In both sets the possible " exception of a point " obviously necessary. For example, does not reflect on, just is not in the image of a punctured neighborhood of each included.

• The little theorem follows immediately from the Great set, because an entire function is either a polynomial or she has an essential singularity in.

• The big set generalizes the theorem of Weierstrass Casorati.

• A conjecture of B. Elsner is related to the large set of Picard: Be the dotted open unit disk and a finite open cover of. On each a simple (ie injective holomorphic ) function is given, so that on each intersection. Then the differentials merge into a meromorphic 1-form on the unit disk. ( In the case that the residue disappears, followed by the assumption of the Great set. )