My integral table has this definite integral:

$$\int_{-\infty}^\infty e^{-\big(a x^2 + b x + c\big)} dx = \sqrt{\frac{\pi}{a}}e^{\frac{b^2-4ac}{4a}}$$

I'd like to solve this similar definite integral:

$$\int_{-\infty}^\infty e^{-i\big(a x^2 + b x + c(x)\big)} dx~~=~~~~?$$

Where $x$ is real valued, $a$ and $b$ are constants, and $c(x)$ is an arbitrary function of $x$, and $i=\sqrt{-1}$. The big difference between my equation and the one I found on the table is that, $c(x)\ne c$.

What is the solution to this? Or, if it isn't easy to solve, what steps should I take? There are a few ways I can limit $c(x)$; for example, it could be Gaussian white noise. However, I'd prefer to solve the general case. Or is there an approximation?

  • $\begingroup$ Use Fresnel integral as a reference. $\endgroup$ – Gustave Jul 30 '18 at 23:45
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    $\begingroup$ I may be wrong, but I think you have to be more specific about the nature of $c(x)$ to answer this question $\endgroup$ – angryavian Jul 30 '18 at 23:46
  • $\begingroup$ @MarkViola op has a function of x in $c(x)$ that is unknown $\endgroup$ – Isham Jul 30 '18 at 23:53
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    $\begingroup$ if $c(x)$ can be any function of x then it's just like trying to solve the general case $$\int_{-\infty}^\infty e^{c(x)}dx$$ $\endgroup$ – Isham Jul 30 '18 at 23:57
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    $\begingroup$ Right. There is no way to help without specification of $c(x)$. Just imagine that $c(x) = \sin (1/x)$ or $1$ or $\tan (x^2)$ or ... $\endgroup$ – David G. Stork Jul 31 '18 at 1:23

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