# Integral of exponential function with polynomial argument

I am looking for an effcient way to evaluate

$$I = \int_{-\infty}^\infty dx\ e^{-ax+bx^2+cx^4}\\ \text{where } a,b,c \in \mathbb{R}^+$$

I have already read about a solution involving the series expansion of the exponential here Computing the integrals of the form $\exp(P(x))$, $P(x)$ a polynomial but I am looking for something computationally more efficient...

Any help is greatly appreciated!

• You are looking for a numerical method? – Fabian Jan 7 '13 at 16:48
• Well, an analytical solution in closed form would be cool, but I would also go for an efficient numerical solution. The evaluation of this integral is at the very core of the loop of my code so it better be fast ;) – user55477 Jan 7 '13 at 17:05
• any thoughts about the fact that the integral does in fact not converge? – Fabian Jan 7 '13 at 17:29
• I am somewhat flummoxed that the integral does not converge. Physically this does not make sense as it corresponds to the calculation of an expectation value. I am currently thinking about the constants a, b, and c... and about ways to rephrase my entire problem. I think I have some homework to do! Thanks for your help so far. I might get back to you here as soon as I am somewhat clearer on my problem definition. Sorry for the confusion... – user55477 Jan 7 '13 at 17:51

The answer to your question is very simple. In fact the integral does not converge as $c>0$.

• Is it possible to add a few more details regarding this response? Regards – Amzoti Jan 7 '13 at 17:32
• @Amzoti: For $c>0$ the integrand does not go to 0 for $|x|\to\infty$ and thus the integral diverges. – Fabian Jan 7 '13 at 18:02
• Thanks for the clarification! Regards – Amzoti Jan 7 '13 at 18:08

One way to begin to attack this is to understand what the parameters $a$, $b$, and $c$ represent. Obviously, $c<0$ for convergence. But what if $c$ is very small in the following sense:

$$I = \int_{-\infty}^{\infty} dx \: \exp{[-M (x^2 + a x) + c x^4]}$$

as $M \rightarrow \infty$? Then you can apply Laplace's Method and expand the integral in an asymptotic expansion in $M$. That is, a rescaling of the argument of the exponential can provide a simple approximation to the integral, in certain cases, where all you need at worst are Error functions and their derivatives.

In the most general case, you may try a simple numerical technique based on estimations of the extent of the exponential along the real axis.