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I would like to calculate the closed-form expression for the following integral: \begin{align} \int_0^\infty x^{a} K_0(b\sqrt{x}) K_\nu(c\sqrt{x}) dx, \end{align} where $a,b,c$ and $\nu$ are all positive constants. Note that $K_{\mu}(\cdot)$ is the modified Bessel function of second kind (also called the Macdonald function).

Shin Tensai
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  • I have worked out this integration with Mathematica. But the result seems strange that it contains the ratio of two infinity parameters (i.e., $\frac{\infty}{\infty}$).

    Are there any references to solve this?

     – Shin Tensai Oct 22 '19 at 07:15
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    Applying this method gives $$4^a b^{-2 a - 2} G_{2, 2}^{2, 2} {\left( \left( \frac c b \right)^2 \middle | {-a, -a \atop -\frac \nu 2, \frac \nu 2} \right)},$$ which can be expanded into a sum of two ${_2 F_1}$ functions by Slater's theorem. The integral converges for $\nu < 2 a + 2$. – Maxim Oct 22 '19 at 16:18
  • @Maxim Thank you very much! – Shin Tensai Nov 06 '19 at 08:53

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