Exponential integral problem

Question

I am trying to solve this

$\int_{0}^{\infty} e^{-ax^k-bx}dx$

I tried to solve this by considering this as an laplace transform of $e^{-ax^k}$ ,but I cannot find it in a laplace table or in online laplace calculator.

Please give me some help, thanks!

Answer 1

There is no closed form solution for the general case. We know this because, if such a solution were to have existed, humanity would not have been forced to invent error and Airy functions in order to be able to express its value for the special cases $k=2$ and $k=3$, respectively $($assuming, of course, that k is an integer. If not, then we can also add $k=\dfrac12$ and $k=\dfrac13$ to the list, since, in this case, a simple substitution of the form $t=\sqrt x$ or $u=\sqrt[3]x$ reduces them to the previously mentioned two$)$ For other cases, $($ generalized $)$ hypergeometric series are inevitable. I am afraid that, in your case, numerical and asymptotic approaches are the only way forward. Also, the integral you mentioned in the comments, $\displaystyle\int_0^\infty xe^{-ax^2+bx}dx$, is expressible in terms of the complementary error function $($see the first link above$)$. For positive values of a, the result is $\dfrac1{2a}-\dfrac{\sqrt\pi}4\cdot\dfrac{b~e^{C^2}\text{erfc}(c)}{a\sqrt a}$ , where $c=\dfrac b{2\sqrt a}$ .