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Complex analysis TheoremsIn complex analysis, mathematician Charles Emile Picard's name is given to two theorems regarding the range of an analytic function.
1 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.
2 Notes
- This 'single exception' bit 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.
1 See also
Picard-Lindelöf theorem.
