1. Maximal term and central index

2. The main asymptotic formula

3. Applications

4. Later development

5. References

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

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.

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

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

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

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