As suggested from this topic, we are interested in evaluating the following complex integral
The trick here is to use the Fourier transform. Thanks to ZY for teaching me this interesting technique.
In , the Fourier transform of function , denoted by , is defined to be
If we apply the Fourier transform twice to a function, we get a spatially reversed version of the function. Precisely,
where denotes the inverse Fourier transform.
Now we consider the following function
The Fourier transform of can be estimated as follows
Now by scaling
Thus, this is about to say that
Similarly, the following integral
can be easily computed via the choice of characteristic function with the fact that