How to rewrite this improper integral in term of any special function $\int\limits_{x = 0}^{ + \infty } {\frac{{\exp \left( { – ax – \frac{b}{x}} \right)}}{{1 + cx}}dx}$ or at least written it as an infinite sum ?
This is a follow up post from the previous one
How to evaluate $\int\limits_0^\infty {\frac{{{e^{ – x – \frac{1}{x}}}}}{{1 + x}}dx}$ in form of special function?
That is $\int\limits_{x = 0}^{ + \infty } {\frac{{\exp \left( { – x – \frac{1}{x}} \right)}}{{1 + x}}} = {K_0}\left( 2 \right)$
Where ${K_M}\left( x \right)$ denote the modified Bessel function of the second kind
I hope that I could generalized the old result !
Notice that $a,b,c$ are all positive number.
Best Answer
Assuming $0<c<1$ $$I=\int\limits_{0}^{ \infty } {\frac{{\exp \left( { - ax - \frac{b}{x}} \right)}}{{1 + cx}}dx}=\sum_{n=0}^\infty (-1)^n c^n \int\limits_{0}^{ \infty } x^n\exp \left( { - ax - \frac{b}{x}} \right)\,dx$$ $$I=2\sum_{n=0}^\infty (-1)^n c^n a^{-\frac{n+1}{2} } b^{\frac{n+1}{2}} K_{-(n+1)}\left(2 \sqrt{a} \sqrt{b}\right) \qquad \Re(b)>0\land \Re(a)>0$$ that you could make nicer.