- Picard theorem
:"For the theorem on existence and uniqueness of solutions of differential equations, see Picard's existence theorem."In

complex analysis , the term**Picard theorem**(named afterCharles Émile Picard ) refers to either of two distinct yet relatedtheorem s, both of which pertain to the range of ananalytic function .**tatement of the theorems****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.

**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 anyopen 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.**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 function s: if "M" is aRiemann surface , "w" is a point on "M",**P**^{1}**C**=**C**∪{∞} denotes theRiemann sphere and "f" : "M" {"w"} →**P**^{1}**C**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**P**^{1}**C**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.

* The following

conjecture [*Citation|last = Elsner|first = Bernhard|year = 1999|journal = Annales de l'institut Fourier|volume = 49|number = 1|pages = 303–331|title = Hyperelliptic action integral|url = http://archive.numdam.org/ARCHIVE/AIF/AIF_1999__49_1/AIF_1999__49_1_303_0/AIF_1999__49_1_303_0.pdf*] is related to "Big Picard": Let "D"{0} be the puncturedunit disk in the complex plane and let "U"_{1}, "U"_{2}, …,"U_{n}" be a finite open cover of "D"{0}. Suppose that on each "U_{j}" there is an injectiveholomorphic function "f_{j}", such that "df_{j}" = "df_{k}" on each intersection "U_{j}" ∩ "U_{k}". Then the differentials glue together to a meromorphic 1-form on the unit disk $D$. (It is clear that the differentials glue together to a holomorphic 1-form "g dz" on "D"{0}. In the special case where the residue of "g" at 0 is zero, then the conjecture follows from "Big Picard".)**Notes****References***

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