The theorems of Picardy , of the mathematician Picardy Emile, are two:

The small theorem of Picardy says that a nonconstant whole function takes any complex number like value, except perhaps one.

The great theorem of Picardy says that a function having an essential singularity takes, on any vicinity of this singularity, any complex number an infinity of times like value, except perhaps one.

Remarks

• “except perhaps one” in these statements is necessary, as the examples of the whole function $e^z$ and the essential singularity show it $e^\; \{1/z\}$.

• the small theorem results immediately from large, because any whole function is either polynomial or it has an essential singularity ad infinitum.

• the great theorem of Picardy generalizes the Théorème of Weierstrass-Casorati.

• a recent conjecture of Bernhard Elsner is related to the great theorem of Picardy: That is to say $D-\; \backslash \; \{0\; \backslash \}$ the disc unit épointé and $U\_1,\; U\_2,\dots ,\; U\_n$ a covering of $D-\; \backslash \; \{0\; \backslash \}$ by the open ones. On each open $U\_j$ an injective holomorphic function such as $df\_j\; is$ f\_j$=\; df\_k$ on all the intersections $U\_j$n$U\_k$. Then these differentials are restuck in a 1-form méromorphe on the disc $D$. (If the residue is null, the conjecture rises from the great theorem of Picardy.)

References