Wiman-Valiron theory is a mathematical theory invented by Anders Wiman as a tool to study the behavior of arbitrary entire functions.
After the work of Wiman, the theory was developed by other mathematicians, and extended to more general classes of analytic functions.
The main result of the theory is an asymptotic formula for the function and its derivatives near the point where the maximum modulus of this function is attained.
By definition, an entire function can be represented by a power series which is convergent for all complex
The terms of this series tend to 0 as
there is a term of maximal modulus.
Its modulus is called the maximal term of the series:
is the exponent for which the maximum is attained; if there are several maximal terms, we define
and is called the central index.
was first proved by Borel, and a more precise estimate due to Wiman reads [1]
there exist arbitrarily large values of
In fact, it was shown by Valiron that the above relation holds for "most" values of
for which it does not hold has finite logarithmic measure:
Improvements of these inequality were subject of much research in the 20th century.
[2] The following result of Wiman [3] is fundamental for various applications: let
is attained; by the Maximum Principle we have
like a monomial: there are arbitrarily large values of
is an arbitrary positive number, and the o(1) refers to
This is useful for studies of entire solutions of differential equations.
Another important application is due to Valiron [4] who noticed that the image of the Wiman-Valiron disk contains a "large" annulus (
This implies the important theorem of Valiron that there are arbitrarily large discs in the plane in which the inverse branches of an entire function can be defined.
A quantitative version of this statement is known as the Bloch theorem.
This theorem of Valiron has further applications in holomorphic dynamics: it is used in the proof of the fact that the escaping set of an entire function is not empty.
In 1938, Macintyre [5] found that one can get rid of the central index and of power series itself in this theory.
Macintyre replaced the central index by the quantity
and proved the main relation in the form
This statement does not mention the power series, but the assumption that
The final generalization was achieved by Bergweiler, Rippon and Stallard [6] who showed that this relation persists for every unbounded analytic function
defined in an arbitrary unbounded region
The key statement which makes this generalization possible is that the Wiman-Valiron disk is actually contained in