# How to integrate this complex Gaussian integral?

I want to work out an integral as follows:

I am hoping to derive this integral, but don't know where to start. Dose one can use Taylor expansion? Does anyone have any advice?

I want to thank you more than I can say! Many thanks!

In my point of view, in order to integrate this integral, we first consider the integration of time t:

Then

So

Thus

Actually, it is clear that the integral has been decomposed into two parts:

The problem is made easier by performing below process:

And

Therefore, the integral can be rewritten as:

One can make use of the fact that:

So

It is worth notice that we are here considering the case b>>a, namely c<<1. How, however, can I utilize this important condition to shed light on this problem? By the way, I am so sorry that I don’t understand editing formulas at http://math.stackexchange.com/. Thus, formulas are all pictures. I am so sorry. Could anyone help me to solve this confused? Thanks! Many thanks!!

CASE 1

The independent variables k is infinite, k∈[-inf.，inf.]. Now let us investigate variation of the dependent variables exp(-c×k^2) with c<<1.

c=0.1; x=-100000:10:100000; y=exp(-(c*x).^2); plot(x,y)

Reviewing the definition of the Kronecker Delta function δij, its value equals to one for i=j and the other parts are zero. Safely, one can consider dependent variables exp(-c×k^2) to act as a Kronecker Delta function δk0.

CASE 2

The independent variables k is definite, k∈[-r.，r.]. Now let us investigate variation of the dependent variables exp(-c×k^2) with c<<1 again.

x=-12:0.01:12; c=0.1; y=exp(-(c*x).^2); y1=1-(c*x).^2+0.5*(c*x).^4-0.5*0.3333333*(c*x).^6; plot(x,y,'-r'); hold on; plot(x,y1,'--b');hold off;

So

Then this integral is resolved. Is that correct?

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