This is a chemically inspired problem, but basically all it consists of is computing integrals.
I have an integral like this. $$f=\int\int{\int{K_0(x_a,x_b,\frac{it}{\hbar}-\frac{\beta}{2})K_1(x_b,x_a,\frac{-it}{\hbar}-\frac{\beta}{2})}}dx_adx_bdt$$
My kernels $K$'s are:
$$K_0=\sqrt{\frac{m}{2\pi t_0}}\exp{(-S_0)}$$
$$K_1=\sqrt{\frac{m}{2\pi t_1}}\exp{(-S_1)}$$
$S's$ correspond to action (in the sense of physics) which is:
$$S_n(x_a,x_b,t_n)=\frac{m(x_a-x_b)^2}{2 t_n}-\frac{(x_a+x_b)k_nt_n}{2}-\frac{k_n^2t_n^3}{24m}$$
Where $n$ is either 0 or 1. The problem is the integral above doesn't seem to be converging with the imaginary argument for $t$'s.I am trying to integrate with respect to: $x_a$, $x_b$ and $t$ from -Inf to +Inf. My final answer for integral should look something like this:
$$\exp{\frac{k_0^2k_1^2\hbar^2\beta^3}{24m(k_0-k_1)^2}}$$
One can set $\hbar$ to 1. Is it a gaussian integral with respect to $x_a$ and $x_b$? One has to be careful because there are also imaginary parts in the exponent. How does one reach at the-above final answer for the integral? Really confused! Any help is appreciated since the integral doesn't look gaussian to me.