Integral in regularization involving exp of a parameter in the denominator

Question

Can someone please point me a way to compute $$\int_0^\infty\frac1{s+t}\exp\left(-\alpha t+\frac{t^2\beta}{s+t}\right)dt$$ ? How about the following one?

$$\int_0^\infty ds\int_0^\infty dt\,\frac1{s+t}e^{-(s+t)}e^{-\frac{st}{s+t}}$$

Answer 1

Maybe this could help, if you define$$ I(\alpha,\beta) =\int_0^\infty\frac1{s+t}\exp\left(-\alpha t+\frac{\beta}{s+t}\right)dt $$ You could show that the function $I$ satisfy the PDE $$ \alpha \partial_\alpha I - \beta \partial_\beta I = \alpha\,s\,I-\exp(\beta s) $$ Maybe studying this PDE you could solve the integral.