Picard theorem
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- For the theorem on existence and uniqueness of solutions of differential equations, see Picard's existence theorem.
In complex analysis, the Picard theorem, named after Charles Émile Picard, is either of two distinct yet related theorems, both of which pertain to the range of an analytic function.
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[edit] Statement of the theorems
The first theorem, also 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, also 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.
[edit] 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.
- "Little Picard" follows from "Big Picard" because an entire function is either a polynomial or it has an essential singularity at infinity.
- A recent conjecture of Bernhard Elsner (Ann. Inst. Fourier 49-1 (1999) p.330) is related to "Big Picard": Let D − {0} be the punctured unit disk in the complex plane and let
be a finite open cover of D − {0}. Suppose that on each Uj there is an injective holomorphic function fj, such that dfj = dfk on each intersection UjnUk. Then the differentials glue together to a meromorphic 1-form on the unit disk D. (In the special case where the residue is zero, the conjecture follows from Picard's theorem.)
[edit] References
- Conway, John B. (1978). Functions of One Complex Variable I, 2nd edition, Springer. ISBN 0-387-90328-3.
