I am having problem with the inverse transformation of a fourier transformed function which should give the function itself.(adsbygoogle = window.adsbygoogle || []).push({});

Let

[itex]f=f(x)[/itex] and let f be fourier transformable (whatever that implies)

Let

[itex]\tilde{f}(k)=∫^{\infty}_{-\infty}dx e^{-ikx}f(x)[/itex] (1)

then we should have:

[itex]f(x)=∫^{\infty}_{-\infty}dk e^{ikx}f(k)[/itex] (2)

This implies:

[itex]f(x)=∫^{\infty}_{-\infty}dk e^{ikx}∫^{\infty}_{-\infty}dx' e^{-ikx'}f(x')[/itex] (3)

Note that x'≠x

My solution is as follows:

[itex]f(x)=∫^{\infty}_{-\infty}dk e^{ikx}∫^{\infty}_{-\infty}dx' e^{-ikx'}f(x')[/itex] (4)

[itex]f(x)=∫^{\infty}_{-\infty}dk ∫^{\infty}_{-\infty}dx' e^{-ik(x'-x)}f(x')[/itex] (5)

[itex]f(x)=∫^{\infty}_{-\infty}dx' ∫^{\infty}_{-\infty}dk e^{-ik(x'-x)}f(x')[/itex] (6)

[itex]f(x)=∫^{\infty}_{-\infty}dx' δ(x'-x)f(x')[/itex] (7)

[itex]f(x)=f(x)[/itex] (8)

Is this correct? Step (6) to (7) bothers me. And what about the change in integration variables? I guess that is correct as well?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Inverse Fourier Transformation of a Fourier Transformation

**Physics Forums | Science Articles, Homework Help, Discussion**