Approximation for the following integral needed

Question

I have following integral $$\int_0^{\infty}e^{-ax-bx^m}dx$$ where $a>0, b>0, m>1$. I can get an approximation for the above integral when $b$ is small. However, I want to get an expression for the case when $b$ is large. Any help in this regard will be highly appreciated. Thanks in advance.

Answer 1

Expanding the factor $e^{-ax}$ in series and integrating term-by-term yields

$$ \int_0^\infty e^{-bx^m} e^{-ax}\,dx = \frac{b^{-1/m}}{m} \sum_{k=0}^{\infty} \frac{(-a)^k \operatorname{\Gamma}\!\left(\frac{k+1}{m}\right)}{k!} b^{-k/m}. $$

This series also serves as an asymptotic series as $b \to +\infty$.