MODERATOR NOTE: This is the notebook used in the livestream “Ask Integration Questions to Oleg Marichev” on Wednesday, January 25 – a part of Wolfram R&D livestream series announced and scheduled here: https://wolfr.am/RDlive For questions about this livestream, please leave a comment below.
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For this integral:
\int_1^0 \pmb{H}_0\left(\sqrt{1-z^2} z\right) \, dz=\frac{2 \, _2F_3\left(1,1;\frac{5}{4},\frac{3}{2},\frac{7}{4};-\frac{1}{16}\right)}{3 \pi }
we use a Mellin Transform to solve:
InverseMellinTransform[
Integrate[
MellinTransform[StruveH[0, A*Sqrt[1 - z^2] z], A, s], {z, 0, 1},
Assumptions -> s < 1], s, A] /. A -> 1 // FullSimplify
(*(2 HypergeometricPFQ[{1, 1}, {5/4, 3/2, 7/4}, -(1/16)])/(3 \[Pi])*)
Regards M.I.
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Hi Dr. Marichev,
The attached image was taken from the table of integrals in your book “Integrals and Series, Special Functions” where Integral 7 is of interest as some cases arise in a certain physical setting of which I am currently writing a paper.
It looks like this generalized result can be derived via the use of Mellin/inverse Mellin tranforms as shown in your paper found at А. П. Прудников, Ю. А. Брычков, О. И. Маричев, “Вычисление интегралов и преобразование Меллина”, Итоги науки и техн. Сер. Мат. анал., 27, ВИНИТИ, М., 1989, 3–146; J. Soviet Math., 54:6 (1991), 1239–1341 but I wanted to verify if that’s how it was done? We would like to cite the work used to derive the result in the integral table but we weren’t sure where that source is.
My apologies as I know this is an outdated Q&A.
Best regards, Josh
