Computing $ \int_{-\infty}^{\infty} e^{-ax^4+bx^2} dx$ I was wondering if it was possible to compute $ \int_{-\infty}^{\infty} e^{-ax^4+bx^2} dx$  explicitly.
Completing the square doesn't seem to work, since there is no factor of $x$ in order to use the Gaussian integral.
Any hint would be greatly appreciated.
Thanks!

Your integral is related by a scaling transformation to
$$\int_{-\infty}^{\infty}
\mathrm{e}^{-z(x^4/8-x^2+1)}\mathrm{d}x
= \pi(I_{-1/4}(z)+I_{1/4}(z))\text{,}$$
where $\Re z > 0$ and $I$ denotes a modified Bessel function. To prove this equality,

*

*Write $\mathrm{e}^{zx^2}$ as a Maclaurin series,

*Integrate termwise using Euler's gamma function,

*Use the duplication formula for the gamma function,

*Write the series as a linear combination of two confluent hypergeometric functions, and

*Simplify to the modified Bessel functions.

