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单词 IntegrationOfDifferentialBinomial
释义

integration of differential binomial


Theorem.  Let a, b, c, α, β be given real numbers and  αβ0.  The antiderivative

I=xa(α+βxb)c𝑑x

is expressible by of the elementary functionsMathworldPlanetmath only in the three cases:  (1)a+1b+c,  (2)a+1b,  (3)c

In accordance with P. L. Chebyshev (1821-1894), who has proven this theorem, the expression  xa(α+βxb)cdx  is called a differential binomial.

It may be worth noting that the differential binomial may be expressed in terms of the incomplete beta functionDlmfDlmfDlmfMathworld and the hypergeometric functionDlmfDlmfDlmfMathworldPlanetmath. Define y=βxb/α. Then we have

I=1bαa+1b+cβ-a+1bBy(1+ab,c-1)
=11+aαa+1b+cβ-a+1by1+abF(a+1b,2-c;1+a+bb;y)

Chebyshev’s theorem then follows from the theorem on elementary cases of the hypergeometric function.

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更新时间:2025/5/4 21:04:11