E-Function
المؤلف:
Siegel, C. L
المصدر:
Transcendental Numbers. New York: Chelsea, 1965.
الجزء والصفحة:
...
18-11-2018
1104
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 ![<span style=]()
{q_n}_(n>=1)" src="http://mathworld.wolfram.com/images/equations/E-Function/Inline11.gif" style="height:14px; width:40px" /> 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 Unpublished manuscript. 1999.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.
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