“Stirling's Approximation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Stirling's Approximation

释义

Stirling's Approximation

Stirling's approximation gives an approximate value for the Factorial function or the GammaFunction for . The approximation can most simply be derived for an Integer by approximating the sum over the terms of the Factorial with an Integral, so that


			(1)

The equation can also be derived using the integral definition of the Factorial,

(2)

Note that the derivative of the Logarithm of the integrand can be written

(3)

The integrand is sharply peaked with the contribution important only near

. Therefore, let

where

, and write

(4)

Now,


			(5)



			(6)

Taking the Exponential of each side then gives

(7)

Plugging into the integral expression for

then gives


			(8)

Evaluating the integral gives

			(9)
			(10)

Taking the Logarithm of both sides then gives

(11)

This is Stirling's Series with only the first term retained and, for large

, it reduces toStirling's approximation

(12)

Gosper notes that a better approximation to

(i.e., one which approximates the terms in Stirling's Seriesinstead of truncating them) is given by

(13)

This also gives a much closer approximation to the Factorial of 0,

, yielding

instead of 0 obtained with the conventional Stirling approximation.

See also Stirling's Series

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。