Help solving this question (Fourier Transform)

[mario99]

I am not against your method of analyzing. It can be helpful in many circumstances. It is just not going to help either the OP nor I. Why? Because the OP needs the steps to find the solution and I can give him these steps. My method of solving is complicated somehow (unfortunately it was beyond the OP knowledge) and you have never used it (I think). So, even if your analyzing method was good enough to realize that my solution was wrong, it would not be able to tell me where I went wrong.

I've mentioned more than once that I don't know the answer. E.g., from my post #18: "if only because I don't know the answer myself."

Yes, you did that, but you wrote many posts before post #18.

Verifying solutions is often much easier than finding them - look up NP problems in computer science, for example.

Fully agreed.

I did my best, but please let me know which part you did not find convincing and why -- we can then discuss it in more detail.

All parts. Because I am stupid!
Go ahead please. (But don't say I am stupid if I did not understand you again.)

Sorry, but it looks to me like you have.

I have understood it as that "Don't Post" wrong answers! (It was only the first attempt of warming up!) So, let it be wrong!

Thank you a lot professor blamocur for verifying that my answer was wrong because I did the contour integration perfectly (I think I did that). And I would be so happy if someone can tell me where I went wrong, or at least give us the final solution of that nasty convolution integral.

Who will be the super hero?

 

[YehiaMedhat]

Hey guys I've already forgotten about it
Calm down, no need for that

 

[mario99]

Hey guys I've already forgotten about it
Calm down, no need for that

It is a normal discussion between two adult gentlemen. Don't misunderstand it.

You also Yahia, should answer post #15! (It is a bad habit to ignore questions.)

 

[mario99]

@blamocur

I have done the contour integration perfectly. But why did I get wrong result?

Here is the reason: Because I violated the main rule of contour integration. The integrand must be smooth everywhere, except at the poles. But [imath]|t-x|[/imath] is not smooth at [imath]t[/imath], so applying contour integration is an invalid method of solving that convolution integral.

Splitting the absolute value into left and right will not change the fact that the original function is not smooth at \(\displaystyle t\). Therefore, if you solve the original integral by contour integration by either splitting it into two parts or not, everything will be normal, except that you will get incorrect result.

I am still wondering how the OP got that nasty integral?!