Picard theorem
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In complex analysis, mathematician Charles Émile Picard's name is given to two theorems regarding the range of an analytic function.
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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.
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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.
- If f(z) is a nonconstant polynomial of degree n, the fundamental theorem of algebra guarantees that each value is taken on precisely n times (counting multiplicity). If this were not the case, applying the Great Picard theorem to g(z) = f(1/z) (which has an essential singularity at 0) would imply that in fact every value except at most one is taken on infinitely often, a contradiction.
- A recent conjecture of Bernhard Elsner (Ann. Inst. Fourier 49-1 (1999) p.330) is related to "Big Picard": Let <math>D-\{0\}</math> be the punctured unit disk in the complex plane and let <math>U_1,U_2, . . . ,U_n</math> be a finite open cover of <math>D-\{0\}</math>. Suppose that on each <math>U_j</math> there is an injective holomorphic function <math>f_j</math>, such that <math>df_j = df_k</math> on each intersection <math>U_j</math>n<math>U_k</math>. Then the differentials glue together to a meromorphic 1-form on the unit disk <math>D</math>. (In the special case where the residue is zero, the conjecture follows from Picard's theorem.)
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References
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