Picard theorem
Encyclopedia
In complex analysis
, the term Picard theorem (named after Charles Émile Picard
) refers to either of two distinct yet related theorem
s, both of which pertain to the range
of an analytic function
.
The first theorem, also referred to as "Little Picard", states that if a complex function
f(z) 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.
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.
consisting of a plane minus two points, and
the upper half-space, on which the modular group
acts by Moebius transformations. From the theory of modular curve
s, the j-invariant
is a holomorphic map from the upper half-space
that is invariant under the action of 
. Its derivative 
vanishes only on the orbit of i and 
, so that its restriction to 
induces a covering map 
.
This covering space is isomorphic to an open set of the disc, as the half plane and the disc are conformally equivalent (see Moebius transformation#Subgroups of the Moebius group).
Suppose now that we have an entire function f which misses two points. This function can be thought of as a map
, and can therefore be factored by the cover j holomorphically to the disc. This gives a bounded map holomorphic map from the plane to itself, which, by Liouville theorem, is therefore constant hence the original function f must have been constant.
at a point w, then on any open set
containing w, the function 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.
Complex analysis
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 number theory and applied mathematics; as well as in physics,...
, the term Picard theorem (named after Charles Émile Picard
Charles Émile Picard
Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...
) refers to either of two distinct yet related theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s, both of which pertain to the range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
of an analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
.
Little Picard
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f(z) is entire
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
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.
This theorem was proved by Picard in 1879. It is a significant strengthening of Liouville's theorem
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| ≤ M for all z in C is constant.The theorem is considerably improved by...
which states that the image of an entire non-constant function must be unbounded.
Overview of the proof
We call X the Riemann surfaceRiemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
consisting of a plane minus two points, and
the upper half-space, on which the modular groupModular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
acts by Moebius transformations. From the theory of modular curveModular curve
In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...
s, the j-invariant
J-invariant
In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.We haveThe modular discriminant \Delta is defined as \Delta=g_2^3-27g_3^2...
is a holomorphic map from the upper half-space
that is invariant under the action of . Its derivative vanishes only on the orbit of i and , so that its restriction to induces a covering map .This covering space is isomorphic to an open set of the disc, as the half plane and the disc are conformally equivalent (see Moebius transformation#Subgroups of the Moebius group).
Suppose now that we have an entire function f which misses two points. This function can be thought of as a map
, and can therefore be factored by the cover j holomorphically to the disc. This gives a bounded map holomorphic map from the plane to itself, which, by Liouville theorem, is therefore constant hence the original function f must have been constant.Big Picard
The second theorem, also called "Big Picard" or "Great Picard", states that if an analytic function f(z) has an essential singularityEssential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...
at a point w, then on any open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
containing w, the function 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
Weierstrass–Casorati theorem
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of meromorphic functions near essential singularities...
, which only guarantees that the range of f is dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in the complex plane.