Picard theorem

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 after Charles Émile Picard) refers to either of two distinct yet related theorems, both of which pertain to the range of an analytic 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 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.

Notes

* The 'single exception' is in fact needed in both theorems: "ez" is an entire non-constant function which is never 0, and "e1/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.

* 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 punctured unit disk in the complex plane and let "U"1, "U"2, …,"Un" be a finite open cover of "D"{0}. Suppose that on each "Uj" there is an injective holomorphic function "fj", such that "dfj" = "dfk" on each intersection "Uj" ∩ "Uk". 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

*


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Picard–Lindelöf theorem — In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard s existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to certain initial value problems.The… …   Wikipedia

  • Picard group — In mathematics, the Picard group of a ringed space X, denoted by , is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the… …   Wikipedia

  • Theorem of the cube — In mathematics, the theorem of the cube is a foundational result in the algebraic geometry of a complete variety. It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry. The specific… …   Wikipedia

  • Picard, Émile — (1856 1941)    mathematician Among the numerous studies that Émile Picard, who was born and educated in Paris, began were those involving uniform and multiform analytical functions and the functions of several complex variables. His theorem… …   France. A reference guide from Renaissance to the Present

  • Charles Émile Picard — Born 24 July 1856(1856 07 24) Paris …   Wikipedia

  • Peano existence theorem — In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy Peano theorem, named after Giuseppe Peano and Augustin Louis Cauchy, is a fundamental theorem which guarantees the… …   Wikipedia

  • Fermat's Last Theorem in fiction — The famous problem in number theory known as Fermat s Last Theorem has repeatedly received attention in fiction and popular culture.* In The Royale , an episode of , Captain Picard states that the theorem had gone unsolved for 800 years. At the… …   Wikipedia

  • Frobenius theorem (differential topology) — In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first order homogeneous linear partial differential equations. In modern geometric terms …   Wikipedia

  • Liouville's theorem (complex analysis) — In complex analysis, Liouville s theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that | f ( z )| ≤ M for all… …   Wikipedia

  • Montel's theorem — In complex analysis, an area of mathematics, Montel s theorem refers to one of two theorems about families of holomorphic functions. These are named after Paul Montel, and give conditions under which a family of holomorphic functions is normal.… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”