In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
Little Picard Theorem: If a function f : C → C is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point.
Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by λ, and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If f omits two values, then the composition of f with the inverse of the modular function maps the plane into the unit disc which implies that f is constant by Liouville's theorem.
This theorem is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be unbounded. Many different proofs of Picard's theorem were later found and Schottky's theorem is a quantitative version of it. In the case where the values of f are missing a single point, this point is called a lacunary value of the function.
Great Picard's Theorem: If an analytic function f has an essential singularity at a point w, then on any punctured neighborhood of w, f(z) takes on all possible complex values, with at most a single exception, infinitely often.
This is a substantial strengthening of the Casorati–Weierstrass theorem, which only guarantees that the range of f is dense in the complex plane. A result of the Great Picard Theorem is that any entire, non-polynomial function attains all possible complex values infinitely often, with at most one exception.
The "single exception" is needed in both theorems, as demonstrated here:
Great Picard's theorem is true in a slightly more general form that also applies to meromorphic functions:
Great Picard's Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P^{1}(C) = C ∪ {∞} denotes the Riemann sphere and f : M\{w} → P^{1}(C) is a holomorphic function with essential singularity at w, then on any open subset of M containing w, the function f(z) attains all but at most two points of P^{1}(C) infinitely often.
Example: The function f(z) = 1/(1 − e^{1/z}) is meromorphic on C* = C - {0}, the complex plane with the origin deleted. It 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 Theorem follows from Great Picard Theorem because an entire function is either a polynomial or it has an essential singularity at infinity. As with the little theorem, the (at most two) points that are not attained are lacunary values of the function.
The following conjecture is related to "Great Picard's Theorem":^{ [1] }
Conjecture: Let {U_{1}, ..., U_{n}} be a collection of open connected subsets of C that cover the punctured unit disk D \ {0}. Suppose that on each U_{j} there is an injective holomorphic 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 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 the conjecture follows from the "Great Picard's Theorem".
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f(z) has a root at w, then f(z)/(z−w), taking the limit value at w, is an entire function. On the other hand, neither the natural logarithm nor the square root is an entire function, nor can they be continued analytically to an entire function.
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem.
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of Dexcept for a set of isolated points, which are poles of the function. The term comes from the Ancient Greek meros (μέρος), meaning "part".
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.
In complex analysis, a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.
In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
In mathematics, complex geometry is the study of complex manifolds, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant. Equivalently, non-constant holomorphic functions on have unbounded images.
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z_{0} is an isolated singularity of a function f if there exists an open disk D centered at z_{0} such that f is holomorphic on D \ {z_{0}}, that is, on the set obtained from D by taking z_{0} out.
In mathematics, the value distribution theory of holomorphic functions is a division of mathematical analysis. It tries to get quantitative measures of the number of times a function f(z) assumes a value a, as z grows in size, refining the Picard theorem on behaviour close to an essential singularity. The theory exists for analytic functions of one complex variable z, or of several complex variables.
In the mathematical field of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl has called it "one of the few great mathematical events of century." The theory describes the asymptotic distribution of solutions of the equation f(z) = a, as a varies. A fundamental tool is the Nevanlinna characteristic T(r, f) which measures the rate of growth of a meromorphic function.
In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality.
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 French mathematician Paul Montel, and give conditions under which a family of holomorphic functions is normal.
In mathematics, a doubly periodic function is a function defined on the complex plane and having two "periods", which are complex numbers u and v that are linearly independent as vectors over the field of real numbers. That u and v are periods of a function ƒ means that
In complex analysis, a subfield of mathematics, a lacunary value or gap of a complex-valued function defined on a subset of the complex plane is a complex number which is not in the image of the function.
Bloch's Principle is a philosophical principle in mathematics stated by André Bloch.