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.

1 Statement of the theorems
- 1.1 Little Picard
- 1.2 Big Picard
2 Notes
3 References
4 See also

[edit] Statement of the theorems

[edit] Little Picard

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.

This theorem was proved by Picard in 1879. It is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be unbounded.

Plot of the function exp(1/z), centered on the essential singularity at z=0. The hue of a point z represents the complex argument of exp(1/z), the luminance represents its absolute value. This plot shows that arbitrarily close to the singularity, all non-zero values are attained.

[edit] Big Picard

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 complex 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

The 'single exception' is in fact needed in both theorems: e^z is an entire non-constant function which is never 0, and e^1/z has an essential singularity at 0, but still never attains 0 as a value.

"Big Picard" is true in a slightly more general form that also applies to meromorphic functions: if M is a Riemann surface, w is a point on M, P¹C = C∪{∞} denotes the Riemann sphere and f : M \ {w} → P¹C is a holomorphic function with essential singularity at w, then on any open subset of M that contains w the function f attains all but at most two points of P¹C infinitely often.

As an example, the meromorphic function f(z)=1/(1-exp(1/z)) has an essential singularity at z=0 and attains the value ∞ infinitely often in any neighborhood of 0; however it does not attain the values 0 or 1.

With this generalization, "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 $U_1,U_2, \dots ,U_n$ be a finite open cover of $D - {0}$ . Suppose that on each $U j$ there is an injective holomorphic function $f j$ , such that $d f j = d f k$ on each intersection $U j$ n $U k$ . 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.)