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.
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:
The main asymptotic formula
The following result of Wiman 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.
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 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.
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 .
- 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.