Suppose $a>0$ , I am interested in a solution of the following definite integral:
$$\int_{1}^{\infty}\frac{\exp({-az})}{z \sqrt{z^2-1}}{\,dz}$$
Thank you.
2026-04-02 20:08:07.1775160487
Definite integration of a exponential function mixed with rational functions
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We notice that $I'(a)=-\displaystyle\int_1^\infty\frac{e^{-az}}{\sqrt{z^2-1}}dx$, which, by letting $x=\cosh t$, can be rewritten as $-\text{Bessel K}_0(a)$. Then, integrating with regards to a, we get $I(a)=\dfrac\pi2\bigg\{1-a~\bigg[K_0(a)L_{-1}(a)+$ $+K_1(a)L_0(a)\bigg]\bigg\}$, where L is the Struve function.