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.

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 https://www.mathnet.ru/links/8611c07b73d37d0d5e47e71bce3f69ec/intm83.pdf 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My apologies as I know this is an outdated Q&A.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

Attachments:

MarichevIntegralTable.png