E-Function

For any  (where  denotes the set of algebraic numbers), let  denote the maximum of moduli of all conjugates of . Then a function

is said to be an E-function if the following conditions hold (Nesterenko 1999).

1. All coefficients  belong to the same number field  of finite degree over Q.

2. If  is any positive number, then  as .

3. For any , there exists a sequence of natural numbers  such that  for , ...,  and that .

Every E-function is an entire function, and the set of E-functions is a ring under the operations of addition and multiplication. Furthermore, if  is an E-function, then  and  are E-functions, and for any algebraic number , the function  is also an E-function (Nesterenko 1999).

REFERENCES:

Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.

Siegel, C. L. Transcendental Numbers. New York: Chelsea, 1965.