Representation of parametric integral through special functions

Question

I am trying to find out for which values of $\alpha>0, \beta>0$ below integral is infinite when $B\to \infty$: $$ \lim_{B\to \infty}\int_{1}^B z^{-2}e^{-\frac{\beta}{z}}e^{-\alpha z}dz $$ For $\beta=0$ it can be represented through the Gamma-function, hence the limit is finite for all $\alpha>0$. However, for $\beta>0$, it should not converge for all $\alpha>0$, I think. However, I am not able to relate above integral to any special function known to me.

Answer 1

First I'll write the integral as $$\int_1^\infty t^{-2}\exp\left(-at-\frac{b}{t}\right)\mathrm dt=a\int_a^\infty \exp\left(-s-\frac{ab}{s}\right)\frac{\mathrm ds}{s^2}$$ Using an integral representation on DLMF, $$K_{\nu}(z)=\frac{1}{2}(z/2)^\nu\int_0^\infty\exp\left(-t-\frac{z^2}{4t}\right)\frac{\mathrm dt}{t^{\nu+1}}$$ We have, $$a\int_a^\infty \exp\left(-s-\frac{ab}{s}\right)\frac{\mathrm ds}{s^2}=2\sqrt{\frac{a}{b}}~K_1(2\sqrt{ab})-a\int_0^a\exp\left(-s-\frac{ab}{s}\right)\frac{\mathrm ds}{s^2}$$ This form should be very useful for numerical implementation, since $K_1$ is very fast to compute and the other integral is over a finite region and so should be tractable for numerical methods.