Good day! I'm studying right now some transformation and I encountered the following equation:
$$(2\pi)^{-n/2} \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \exp\left(-\frac{1}{2} \|a\|^2\right)\cdot\cos\left[\varepsilon f(t)\right]\,da_0\cdots da_n = \exp-\frac{1}{2}\varepsilon^2 \sum_0^n t^{2k}$$
where $n$ is the degree of a polynomial and $a$ is the coefficients of the polynomial... my question is that..is there a way by any means that a certain series of integration can be converted to summation?... if there is, what method should I consider?
I found this on net (fourier cosine transformation):
$$\sqrt{\frac{2}{\pi}}\int\limits_{0}^{\infty}e^{-at^{2}}cos(xt)dt = \frac{1}{\sqrt{2a}}e^{-\frac{x^{2}}{4a}}$$ and $$cos x = \sum\limits_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{2n!}$$
would it help?