For any alpha in A (where A denotes the set of algebraic numbers), let |alpha|^_ denote the maximum of moduli of all conjugates of alpha. Then a function


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

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

2. If epsilon>0 is any positive number, then |c_n|^_=O(n^(epsilonn)) as n->infty.

3. For any epsilon>0, there exists a sequence of natural numbers {q_n}_(n>=1) such that q_nc_k in Z_K for k=0, ..., n and that q_n=O(n^(epsilonn)).

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 f(z) is an E-function, then f^'(z) and int_0^zf(t)dt are E-functions, and for any algebraic number alpha, the function f(alphaz) is also an E-function (Nesterenko 1999).

See also

Shidlovskii Theorem

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Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.Siegel, C. L. Transcendental Numbers. New York: Chelsea, 1965.

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Cite this as:

Weisstein, Eric W. "E-Function." From MathWorld--A Wolfram Web Resource.

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