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From the wikipedia page on Gaussian integral https://en.wikipedia.org/wiki/Gaussian_integral the following formula holds:

$$\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx =\frac12 e^f \ \sum_{\begin{smallmatrix}n,m,p=0 \\ n+p=0 \mod 2\end{smallmatrix}}^{\infty} \ \frac{b^n}{n!} \frac{c^m}{m!} \frac{d^p}{p!} \frac{\Gamma \left (\frac{3n+2m+p+1}{4} \right)}{(-a)^{\frac{3n+2m+p+1}4}}.$$

(The integral converges whenever $a<0$.) Could anyone explains how to obtain this formula? And is there any formula for the following integrals: $$\int_{-\infty}^{\infty}x^r e^{a x^4+b x^3+c x^2+d x+f}\,dx$$ where $r$ is an integer?

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    $\begingroup$ This explanation on the Wikipedia page already tells you how to get $\int_{-\infty}^\infty f(x) e^{-cx^2} dx$ when $f(x)$ is a power of $x$. For other results, expand $f(x)$ is a power series and integrate term by term. $\endgroup$ Jul 30, 2017 at 13:23
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    $\begingroup$ The integral you wrote is divergent for all real $a\neq0$. So the formula makes no sense. $\endgroup$ Jul 30, 2017 at 16:11
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    $\begingroup$ @AlexandreEremenko: the integral converges for $a<0$. But I understand where you're coming from (+1). I suspect the wikipedia page was written by an author of arxiv.org/abs/0903.2595 or clearly someone who studied that article which I find very interesting but perhaps lacking details on the integration contours in relation to the issue of convergence that you mentioned. $\endgroup$ Jul 30, 2017 at 17:09
  • $\begingroup$ I changed the tags to ones which are closer to the subject matter of the article by Morozov and Shakirov where this question comes from. $\endgroup$ Jul 30, 2017 at 18:01

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The main theme here is the attempt to generalize the Isserlis-Wick Theorem for Gaussian integrals to the case of exponentials of polynomials of higher degree. Let $$ P(x)=a_n x^n+\cdots+a_1 x+a_0 $$ be a polynomial of even degree $n$ with $\Re\ a_n>0$. Then the integral $$ J=\int_{-\infty}^{\infty}e^{-P(x)}\ dx $$ converges. One can expand $\exp(-\sum_{i=1}^{n-1}a_i x^i)$ inside the integral and factor out the constant $e^{-a_0}$ $$ J=e^{-a_0}\int_{-\infty}^{\infty}e^{-a_n x^n}\sum_{r_1,\ldots,r_{n-1}=0}^{\infty} \prod_{i=1}^{n-1}\frac{(-a_ix^i)^{r_i}}{r_i!} \ \ dx $$ and also take the sum out $$ J=e^{-a_0}\sum_{r_1,\ldots,r_{n-1}=0}^{\infty} \prod_{i=1}^{n-1}\frac{(-a_i)^{r_i}}{r_i!} \int_{-\infty}^{\infty}e^{-a_n x^n} x^{r_1+\cdots+r_{n-1}}\ dx $$ using absolute convergence.
Finally one is reduced to integrals of the form $$ \int_{-\infty}^{\infty}e^{-a_nx^n} x^r\ dx $$ which vanish if $r$ is odd and otherwise are equal to $$ 2\int_{0}^{\infty} e^{-a_nx^n}x^{r+1}\ \frac{dx}{x}=\frac{2}{n}a_n^{-\frac{r+1}{n}}\int_{0}^{\infty} e^{-t}t^{\frac{r+1}{n}}\frac{dt}{t}=\frac{2}{n}a_n^{-\frac{r+1}{n}} \Gamma(\frac{r+1}{n}) $$ after the change of variables $t=a_nx^n$.

For a no-go result about higher degree Isserlis-Wick theorems see the article "When is the Fourier transform of an elementary function elementary?" by Etingof, Kazhdan, and Polishchuk in Selecta Math 2002. Also, things work well if one expands the interaction (in QFT lingo) lower degree terms using the leading monomial as the free reference measure. If one does it the other way as suggested in Igor's comment then exchanging summation and integration is not allowed. Nevertheless, one should still be able to recover the integral from the series using Borel-Leroy summation, see the article "Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension" by Rivasseau and Wang and the follow up "Note on the intermediate field representation of $\phi^{2k}$ theory in zero dimension" by Lionni and Rivasseau.

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  • $\begingroup$ Actually I want to solve this problem: suppose $f(x)$ satisfies $\int_{-\infty}^\infty f(x)dx=1,\int_{-\infty}^\infty xf(x) dx=0 ,\int_{-\infty}^\infty x^2 f(x) dx = \theta$, find $f(x)$ such that the integral $\int f(x) \ln f(x) - f(x)dx $ reaches its maximum. It is easy to see $f(x) =e^{\alpha + \beta x + \gamma x^2}$ and we can find the values for $\alpha,\beta,\gamma$ using the Guassian integral. However if we add the constraint $\int_{-\infty}^\infty x^3 f(x) dx = q$, then $f(x) =e^{\alpha + \beta x + \gamma x^2 + \zeta x^3} $. But how to find the explicit values for the coefficients? $\endgroup$
    – sam
    Aug 1, 2017 at 6:59
  • $\begingroup$ @sam: Is this a research problem or a homework question? It might be worth formulating it precisely with all the hypotheses etc. as a separate MO question. $\endgroup$ Aug 1, 2017 at 13:16
  • $\begingroup$ I have post my question on mathoverflow.net/questions/278334/…. Thanks for your help. $\endgroup$
    – sam
    Aug 9, 2017 at 9:22

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