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I want to work out an integral as follows: enter image description here

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:

enter image description here

Then

enter image description here

So

enter image description here

Thus

enter image description here

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

enter image description here

The problem is made easier by performing below process:

enter image description here

And

enter image description here

Therefore, the integral can be rewritten as:

enter image description here

One can make use of the fact that:

enter image description here

So

enter image description here

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.

enter image description here

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.

enter image description here

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.

enter image description here

enter image description here

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

enter image description here

Then this integral is resolved. Is that correct?

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