# Wiman-Valiron theory

**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.

## Maximal term and central index

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
, so for each
there is a term of maximal modulus.
This term depends on
.
Its modulus is called the *maximal term* of the series:

Here
is the exponent for which the maximum is attained; if there are several maximal terms, we define
as the
largest exponent of them. This number
depends on
, it is denoted by
and is called the *central index*.

Let

be the maximum modulus of the function . Cauchy's inequality implies that for all . The converse estimate was first proved by Borel, and a more precise estimate due to Wiman reads [1]

in the sense that for every there exist arbitrarily large values of for which this inequality holds. In fact, it was shown by Valiron that the above relation holds for "most" values of : the exceptional set 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 main asymptotic formula

The following result of Wiman [3] is fundamental for various applications: let be the point for which the maximum in the definition of is attained; by the Maximum Principle we have . It turns out that behaves near the point like a monomial: there are arbitrarily large values of such that the formula

holds in the disk

Here
is an arbitrary positive number, and the o(1) refers to
,
where
is the exceptional set described above. This disk is usually called the *Wiman-Valiron disk*.

## Applications

The formula for for near can be differentiated so we have an asymptotic relation

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 ( where both and are arbitrarily large). 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.

## Later development

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 is entire was used by Macintyre.

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 in the complex plane, under the only assumption that is bounded for . The key statement which makes this generalization possible is that the Wiman-Valiron disk is actually contained in for all non-exceptional .

## References

- Wiman, A. (1914). "Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Gliede der zugehörigen taylor'schen Reihe".
*Acta Mathematica*.**37**: 305–326 (German). - Hayman, W. (1974). "The local growth of power series: a survey of the Wiman-Valiron method".
*Canadian Mathematical Bulletin*.**17**(3): 317–358. - Wiman, A. (1916). "Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Betrage bei gegebenem Argumente der Funktion".
*Acta Mathematica*.**41**: 1–28 (German). - Valiron, G. (1949).
*Lectures on the general theory of integral functions*. NY: Chelsea, reprint of the 1923 ed. - Macintyre, A. (1938). "Wiman's method and the "flat regions" of integral functions".
*Quarterly J. Math.*: 81–88. - Bergweiler, W.; Rippon, Ph.; Stallard, G. (2008). "Dynamics of meromorphic functions with direct or logarithmic singularities".
*Proc. London Math. Soc*.**97**: 368–400.