Jin Ping, Zhang,. Discussion on the relationship between convolution theorem and Fourier transform and its application [J]. Journal of Yangtze University: Science Edition, 20 16, 13( 19).

The landlord can go to HowNet or Wanfang to download it, or Baidu for help. Convolution theorem is used to solve this problem.

The basic basis is the time domain convolution theorem and frequency domain convolution theorem in the literature:

Next, insert a theorem to solve this problem, which comes from 2.3 in the reference:

So just find the function.

The Fourier transform F(iw) can be:

Next, there are three "methods" to find this integral:

Method 1: Convert the complex exponent into a real trigonometric function, which can be converted into a familiar form.

At this time, the two trigonometric functions of real part and imaginary part are applied to the multiplication and difference formula (Baidu), and then the integral is divided into four pieces, which can be integrated separately. This method is not emphasized here. If you are interested, you can try to write about this process yourself.

Method 2: "pretend to be confused", treat the complex index as a real index, that is, pretend not to know that jw is an imaginary number and think it is a positive number.

Then there is

Let p=a+jw, then

If you are seeking indefinite integral, you can use partial integral to convert it once or twice. But at this time, I also found the problem. This is not my method, but your function:

Because the integrand function is unbounded and the integral diverges at t→-∞, the integral result does not converge.

In this case, I will not introduce the third method. If the function

replace

There is still a chance to get it back. If you have any questions, please keep asking questions.