I am currently facing an integral I have no clue how to solve it. I believe it is rather exoctic, but I hope you might have some good advice:
$$\int_0^{\infty} x e^{-ax^2-bx^{-1}} \, \mathrm{d}x$$ $$ a,b\in\mathbb{C},\,\, \Re(b)=0,\,\,\Im(b)\in\left[-\infty,+\infty\right], \,\,a\neq0$$ As my knowledge is about of a science undergraduate student I tried sofar subsitutions, expansion to the complex numbers (integration of an analytic function), partial integration. Also I have tried to solve it by reshaping it in various ways and looking for a solution in the book "Table of Integrals, Series, And Products".
I need to understand the integration, so the answer alone won't help me very much. That is why I would like to know how your strategy on solving it would be, and what advice you can give me. The actual solving I need and want to do by myself.
Thank you for your help
If anything is unclear please let me know (english is not my best language)