Is there a name for the integral $\int_1^\infty e^{-a(x+\frac{b}{x})}dx$?

Question

It is known that the integral $\int_0^\infty e^{-a(x+\frac{b}{x})}dx$ where $a$ and $b$ are two constants is BesselK-like function.

Is there a name for the integral $\int_1^\infty e^{-a(x+\frac{b}{x})}dx$ ?

Answer 1

This integral corresponds to the incomplete Bessel function or leaky aquifer function $K_{-1}(a,ab)$ as defined by Harris: \begin{equation} K_\nu(x,y)=\int_1^\infty e^{-xt-\frac{y}{t}}\,\frac{dt}{t^{\nu+1}} \end{equation} Hydologist showed that ``water levels in pumped aquifer systems with finite transmissivity and leakage could be analyzed in terms of'' this integral. You can find many properties of this function, as well as numerical methods to compute it in different regions in the $(a,b)$-plane, in papers by Harris and Fripiat. Jones uses a different (but related) definition: \begin{equation} K_\nu(z,w)=\int_w^\infty e^{-z\cosh t}\cosh \nu t\,dt \end{equation} and offers a detailed analysis of its asymptotic expansion for complex values of the parameters.