I wonder whether one could reduce the following integral to some combination of special (e.g. Bessel) functions: $$\int_{m}^{\infty}d\epsilon \exp\left(-s\epsilon\right)\left( \epsilon^{2}-m^{2}\right)^{1/2}\left( \epsilon^{2}-\lambda m^{2}\right)^{1/2}$$

The integral without the last term in the integrand reduces to a well know integral representation of $K_{1}$, but I can not figure out if there is a clever variable substitution which brings the integral above to something better known.

  • $\begingroup$ I cannot even do $\int_{4}^{\infty }\!{{\rm e}^{-\epsilon}}\sqrt {{\epsilon}^{2}-16} \sqrt {{\epsilon}^{2}-4}\,{\rm d}\epsilon$ $\endgroup$ – Gerald Edgar Jul 18 '18 at 13:57

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