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I have been studying the Fourier Transform and the convolution theorem is not clear to me.

Sometimes, I see this:

enter image description here

But other times I see this:

enter image description here

Which one is correct? Where is this constant factor from?

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  • \$\begingroup\$ These two different definitions. You should make sure which one you are working with. \$\endgroup\$
    – Eugene Sh.
    Commented Jul 26, 2017 at 17:14
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    \$\begingroup\$ And how do I know which one I am working with? I mean, what is the source of the difference between these two? \$\endgroup\$ Commented Jul 26, 2017 at 17:19
  • \$\begingroup\$ You know there are different ways to define the F.T., right? They are different because of a factor of \$1/2\pi\$ in either the forward or inverse FT, or possibly a factor of \$1/\sqrt{2\pi}\$ included in each. You need to know which definition of the F.T. and I.F.T. your source is using. \$\endgroup\$
    – The Photon
    Commented Jul 26, 2017 at 17:21
  • \$\begingroup\$ It's about the convention. The definitions are usually communicated before the usage. Take a look physicsforums.com/threads/… \$\endgroup\$
    – Eugene Sh.
    Commented Jul 26, 2017 at 17:21
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    \$\begingroup\$ I didn't know about the other definition, thank you!! \$\endgroup\$ Commented Jul 26, 2017 at 17:24

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Thanks to @ThePhoton and @EugeneSh., it turns out there are different definitions of Fourier Transforms.

https://www.physicsforums.com/threads/why-the-1-sqrt-2-pi-in-the-definition-of-the-fourier-transform.487312/

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