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

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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 mofulus 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 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

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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

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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

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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. The final improvement 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  .

References

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  1. ^ Wiman, A. (1914). "Uber den Zusammenhang dem Maximalbetrage einer analytischen Funktion und dem Grossten Gleide der Zugehorihen taylor'schen Reihe". Acta Mathematica. 37: 305–326 (German). doi:10.1007/BF02401837. S2CID 121155803.
  2. ^ Hayman, W. (1974). "The local growth of power series: a survey of the Wiman-Valiron method". Canadian Math. Bull. 17 (3): 317–358. doi:10.4153/CMB-1974-064-0. S2CID 53382194.
  3. ^ Wiman, A. (1916). "Uber den Zuzammenhang zwischen dem Maximalbetrage einer analytischen Funktionen und dem grossten Betrage bei gegebenem Argumente der Funktion". Acta Mathematica. 41: 1-28 (German). doi:10.1007/BF02422938. S2CID 122491610.
  4. ^ Valiron, G. (1954). Fonctions analytiques. Paris: Presses Universitaires de France.
  5. ^ Macintyre, A. (1938). "Wiman's method and the "flat regions" of integral functions". Quarterly J. Math.: 81–88. doi:10.1093/qmath/os-9.1.81.
  6. ^ Bergweiler, W.; Rippon, Ph.; Stallard, G. (2008). "Dynamics of meromorphic functions with direct or logarithmic singularities". Proc. London Math. Soc. 97 (2): 368–400. arXiv:0704.2712. doi:10.1112/plms/pdn007. S2CID 16873707.