An integral hidden in Gradshteyn and Ryzhik

We provide a closed-form expression for the integral N 0,4(a;m):= ∫ 0 ∞ dz (z 4+2az 2+1) m+1 where m ∈ N and a ∈ (−1,∞): N 0,4(a;m)= π 2 m+3/2(a+1) m+1/2 P m m+1/2,-m-1/2)(a) = π 2 3m+ 3 2 (a+1) m+ 1 2 x ∑ k=0 m 2k 2m−2k m−k m+k m (a + 1) k . Here P m (m+ 1 2 ,−m− 1 2 ) (a) is the Jacobi polynomial...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 1999-06, Vol.106 (2), p.361-368
Main Authors: Boros, George, Moll, Victor H.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Hypergeometric functions
Integrals
Mathematical analysis
Mathematics
Measure and integration
Rational functions
Sciences and techniques of general use
Special functions
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Summary:We provide a closed-form expression for the integral N 0,4(a;m):= ∫ 0 ∞ dz (z 4+2az 2+1) m+1 where m ∈ N and a ∈ (−1,∞): N 0,4(a;m)= π 2 m+3/2(a+1) m+1/2 P m m+1/2,-m-1/2)(a) = π 2 3m+ 3 2 (a+1) m+ 1 2 x ∑ k=0 m 2k 2m−2k m−k m+k m (a + 1) k . Here P m (m+ 1 2 ,−m− 1 2 ) (a) is the Jacobi polynomial P m ( α, β) ( a) with parameters α=m+ 1 2 and β=−(m+ 1 2 ) .
ISSN:0377-0427
1879-1778
DOI:10.1016/S0377-0427(99)00081-3