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

单词

Dirichlet Integrals

释义

Dirichlet Integrals

There are several types of integrals which go under the name of a ``Dirichlet integral.'' The integral

(1)

appears in Dirichlet's Principle.

The integral

(2)

where the kernel is the Dirichlet Kernel, gives the

th partial sum of the Fourier Series.

Another integral is denoted

(3)

for

, ...,

There are two types of Dirichlet integrals which are denoted using the letters , , , and . The type 1 Dirichletintegrals are denoted , , and , and the type 2 Dirichlet integrals are denoted , , and .

The type 1 integrals are given by


			(4)

where

is the Gamma Function. In the case

(5)

where the integration is over the Triangle

bounded by the

-axis,

-axis, and line

and

isthe Beta Function.

The type 2 integrals are given for -D vectors and , and ,

(6)

(7)

(8)

where

			(9)
			(10)

and

are the cell probabilities. For equal probabilities,

. The Dirichlet

integral can be expanded asa Multinomial Series as


	(11)

For small

and

can be expressed analytically either partially or fully for general arguments and

			(12)
			(13)

where

(14)

is a Hypergeometric Function.

			(15)
			(16)

where

(17)

References

Sobel, M.; Uppuluri, R. R.; and Frankowski, K. Selected Tables in Mathematical Statistics, Vol. 4: Dirichlet Distribution--Type 1. Providence, RI: Amer. Math. Soc., 1977.

Sobel, M.; Uppuluri, R. R.; and Frankowski, K. Selected Tables in Mathematical Statistics, Vol. 9: Dirichlet Integrals of Type 2 and Their Applications. Providence, RI: Amer. Math. Soc., 1985.

Weisstein, E. W. ``Dirichlet Integrals.'' Mathematica notebook DirichletIntegrals.m.

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。