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In
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 helpful in many branches of mathematics, including algebraic ...
, an area of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Montel's theorem refers to one of two
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s about families of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic functions is normal.


Locally uniformly bounded families are normal

The first, and simpler, version of the theorem states that a family of holomorphic functions defined on an
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s is normal if and only if it is locally uniformly bounded. This theorem has the following formally stronger corollary. Suppose that \mathcal is a family of meromorphic functions on an open set D. If z_0\in D is such that \mathcal is not normal at z_0, and U\subset D is a neighborhood of z_0, then \bigcup_f(U) is dense in the complex plane.


Functions omitting two values

The stronger version of Montel's theorem (occasionally referred to as the Fundamental Normality Test) states that a family of holomorphic functions, all of which omit the same two values a,b\in\mathbb, is normal.


Necessity

The conditions in the above theorems are sufficient, but not necessary for normality. Indeed, the family \ is normal, but does not omit any complex value.


Proofs

The first version of Montel's theorem is a direct consequence of Marty's theorem (which states that a family is normal if and only if the spherical derivatives are locally bounded) and
Cauchy's integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
. This theorem has also been called the Stieltjes–Osgood theorem, after Thomas Joannes Stieltjes and William Fogg Osgood. The Corollary stated above is deduced as follows. Suppose that all the functions in \mathcal omit the same neighborhood of the point z_1. By postcomposing with the map z\mapsto \frac we obtain a uniformly bounded family, which is normal by the first version of the theorem. The second version of Montel's theorem can be deduced from the first by using the fact that there exists a holomorphic universal covering from the unit disk to the twice punctured plane \mathbb\setminus\. (Such a covering is given by the elliptic modular function). This version of Montel's theorem can be also derived from Picard's theorem, by using Zalcman's lemma.


Relationship to theorems for entire functions

A heuristic principle known as Bloch's principle (made precise by Zalcman's lemma) states that properties that imply that an
entire function 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 ...
is constant correspond to properties that ensure that a family of holomorphic functions is normal. For example, the first version of Montel's theorem stated above is the analog of Liouville's theorem, while the second version corresponds to Picard's theorem.


See also

* Montel space * Fundamental normality test *
Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane \mathbb which is not all of \mathbb, then there exists a biholomorphic mapping f (i.e. a bijective hol ...


Notes


References

* * * {{PlanetMath attribution, title=Montel's theorem, id=5754 Compactness theorems Theorems in complex analysis