“Gaussian Integral”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gaussian Integral

释义

Gaussian Integral

The Gaussian integral, also called the Probability Integral, is the integral of the 1-D Gaussian over. It can be computed using the trick of combining two 1-D Gaussians


			(1)

and switching to Polar Coordinates,


			(2)

However, a simple proof can also be given which does not require transformation to Polar Coordinates (Nicholas andYates 1950).

The integral from 0 to a finite upper limit can be given by theContinued Fraction

(3)

The general class of integrals of the form

(4)

can be solved analytically by setting

			(5)
			(6)
			(7)

Then


			(8)

For

, this is just the usual Gaussian integral, so

(9)

For

, the integrand is integrable by quadrature,

(10)

To compute

for

, use the identity



			(11)

For

Even,



			(12)

(13)

is Odd, then



			(14)

(15)

The solution is therefore

(16)

The first few values are therefore

			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)

A related, often useful integral is

(24)

which is simply given by

(25)

References

Nicholas, C. B. and Yates, R. C. ``The Probability Integral.'' Amer. Math. Monthly 57, 412-413, 1950.

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。