# Picard theorem

In complex analysis, mathematician Charles Émile Picard's name is given to two theorems regarding the range of an analytic function.

## Statement of the theorems

The first theorem, sometimes referred to as "Little Picard", states that if a function f(z) is entire and non-constant, the range of f(z) is either the whole complex plane or the plane minus a single point.

The second theorem, sometimes called "Big Picard" or "Great Picard" states that if f(z) has an essential singularity at a point w then on any open set containing w, f(z) takes on all possible values, with at most a single exception, infinitely often. This is a substantial strengthening of the Weierstrass-Casorati theorem, which only guarantees that the range of f is dense in the complex plane.

## Notes

• This 'single exception' is in fact needed: ez is an entire function which is never 0, and e1/z has an essential singularity at 0, but still never attains 0 as a value.
• If f(z) is a polynomial of degree n, the fundamental theorem of algebra guarantees that each value is taken on precisely n times (counting multiplicity). If this is not the case, applying the Great Picard theorem to g(z) = f(1/z) (which has an essential singularity at 0) gives that in fact every value except at most one is taken on infinitely often.
• The conjecture of Elsner (Ann. Inst. Fourier 49-1 (1999) p.330) is related to Picard's theorem: Let [itex]D-\{0\}[itex] be the punctured unit disk in the complex plane and let [itex]U_1,U_2, . . . ,U_n[itex] be a finite open cover of [itex]D-\{0\}[itex]. Suppose that on each [itex]U_j[itex] there is an injective holomorphic function [itex]f_j[itex], such that [itex]df_j = df_k[itex] on each intersection [itex]U_j[itex]n[itex]U_k[itex]. Then the differentials glue together to a meromorphic 1-form on the unit disk [itex]D[itex]. (In the special case where the residue is zero, the conjecture follows from Picard's theorem.)

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