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Which is the inverse Fourier transform of $w^a\cdot e^{-\sqrt{|w|}}$ for $a\geq 0$?

Please work with the following definitions of the Fourier transform: $$F(iw) = \int\limits_{-\infty}^{\infty} f(t)\,e^{-iwt}\,dt $$ and $$f(t) = \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} F(iw)\,e^{iwt}\,dw $$

Using $F(iw)$ instead of $F(w)$ is just notation, I am following the definitions used in the book "Signals and Systems, 2nd Edition" (Alan V. Oppenheim, Alan S. Willsky, with S. Hamid) [1].

I want to know the inverse Fourier Transform for these specific cases:

  1. $a=0$ so what is the result of $\frac{1}{2\pi} \int\limits_{-\infty}^{\infty} e^{-\sqrt{|w|}}\,e^{iwt}\,dw\,$????
  2. $a=1$ so what is the result of $\frac{1}{2\pi} \int\limits_{-\infty}^{\infty} w\,e^{-\sqrt{|w|}}\,e^{iwt}\,dw\,$????
  3. $a=2$ so what is the result of $\frac{1}{2\pi} \int\limits_{-\infty}^{\infty} w^2\,e^{-\sqrt{|w|}}\,e^{iwt}\,dw\,$????
  4. There is a general case for $a\geq0$???

Actually, I am trying to find a bound here for which any of the three first transforms will be useful, so partial answers could be been chosen as the final answer.

Beforehand, thanks you very much.

Joako
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    Your choice of $F(i\omega)$ instead of $F(\omega)$ is very strange and non standard. Why are you interested in these inverse transforms? – K.defaoite Nov 02 '21 at 14:56
  • @K.defaoite The using of $F(iw)$ instead $F(w)$ is just notation, I am using the definitions used in the book "Signals and Systems" by Oppenheim and Willsky. I am trying to found and upper bound for $\max_t |df(t)/dt|$ through the Cauchy-Schwartz Inequality for time-limited functions for which $\int_{-\infty}^\infty |w F(iw)| dw < \infty$ – Joako Nov 02 '21 at 15:03
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    No, it is not just a notation. Your equation $$F(i\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t} dt$$ would imply that the Fourier transform of a function $f$ is the function $$F:\alpha \mapsto \int_{-\infty}^{\infty} f(t)e^{-\alpha t} dt$$ when this is of course, wrong. – K.defaoite Nov 02 '21 at 15:42
  • Finding bounds can be much easier than computing exactly ... What kind of bound are you looking for ? – LL 3.14 Nov 02 '21 at 15:45
  • @K.defaoite No, this is indeed a notation, he is just defining the Fourier transform of $f$ as being $F(i\omega)$. I agree that it is strange and he should write a most common notation to ease the work of someone wanting to help him. – LL 3.14 Nov 02 '21 at 15:49
  • In the same way, going from Fourier transform to inverse Fourier transform is easy and so you could ask for the Fourier transform instead of the inverse. Similarly, changing the convention of the Fourier transform is not difficult and so you should not constrain the people wanting to help you to a specific convention. It should be an effort from your side. – LL 3.14 Nov 02 '21 at 15:59
  • @LL3.14 Is just a difference of a constant, and for me, and electrical engineer, the displayed formulas are actually the standard ones, so I asked as "please" if they could use them, if not is welcome either. Don't know why it is found so insulting, is just a question requirement. And I asking for the inverse because I don't know the function in time for which its transform (or inverse) is given by the term $e^{-\sqrt{w}}$. – Joako Nov 02 '21 at 16:08
  • If someone gives you the Fourier transform of a function, he is giving you the inverse up to a constant ... If this basis fact is unclear to you, you should first try to open a book about Fourier transform. It is not insulting, it is just advice if you want good answers. You have to show to the people that you are doing efforts on your side and you are dealing with problems at your level. In general, your question should not focus on the notations but on the step that is difficult for you, all the rest (like "please use this notation") should be removed. I suggest you edit your question – LL 3.14 Nov 02 '21 at 16:14
  • @LL3.14 Your a assuming, first, my level, second my efforts. I have already see a lot of books and looking for it on Google and I didn´t find it, and also trying it by myself without success... That is why I am asking in a mathematicians forum since as my knowledge as engineer are resulting not enough. – Joako Nov 02 '21 at 16:22
  • @LL3.14 Everybody should ask what they want if they don't know something, but making mistakes I the best way of learning something, and pointing questions and answers is a good way to move bad questions out, but if I where something trying to find something by my first time these kind of comments shows, it will discourage me to research, so please just help if you can or keep space for who could will. Been rude is a really bad habit of many intellectual people I have met, who tries to feel superior someway, but it really shows only its emotional limitations. – Joako Nov 02 '21 at 16:24
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    I am not wanting to be rude, just trying to help you to write a better question with the help of what I could observe on the site ... as you say, making mistakes is the best way of learning, and this is also true when asking questions on a forum. So the suggestion of taking more usual Fourier transform conventions or not adding "noise" to the question is only benefit to you normally. – LL 3.14 Nov 02 '21 at 17:36
  • @LL3.14 I understand, but for someone who is starting in a topic, How you think they will feel if the first thing he reads is that his question is wrong?... From someone who don't know something I will be expecting mistakes since he don't manage the topic, and from someone how now everything about so is everything perfect: Why he will be asking?... as example, I left university with 4 publications strictly related with the Fourier transform, since they were about laser optics, but since I only read then papers by "electricians", just last month I learned there others definitions, by a mistake. – Joako Nov 02 '21 at 17:42

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