Integrals of Exponential(Polynomial(x)) dx Form

[RDBaker]

I'm curious about the general solution to

\int_{-\infty}^{+\infty} \exp[P(x)] dx

Where P(x) is a polynomial in x with real coefficients and whose leading (highest) order is even and its leading order coefficient is negative. Intuitively these integrals ought to converge, but I'm having trouble calculating them.

I've been able to work out solutions for quadratics i.e. P(x) = -ax^2 +bx +c, but I'm thoroughly stuck w.r.t. quartic equations.

Has anyone ever seen anything like this? Gradshteyn & Rizhyk and mathematica have been of no use to me.

 

[Millennial]

http://en.wikipedia.org/wiki/Gaussian_integral#Higher-order_polynomials
It probably isn't a closed form as you would want it to be, but it is still the solution.

 

[RDBaker]

Thanks! This is great!

Next step is trying to find a reference and a text with a good exposition of this solution.

 

[BobChez]

Hi RDBaker,

I am actually interested in exactly the same problem but have had trouble finding a more comprehensive reference. Did you have any luck?

Thanks a lot!