Picard theorem

From Academic Kids

In complex analysis, mathematician Charles ╔mile 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 <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.)

See also

Picard-Lindel÷f theorem

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