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# 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).

Shidlovskii Theorem

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## 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 Unpublished manuscript. 1999.Siegel, C. L. Transcendental Numbers.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.

E-Function

## Cite this as:

Weisstein, Eric W. "E-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/E-Function.html