(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If F(p) and G(p) are the Fourier transforms of f(x) and g(x) respectively, show that

∫f(x)g*(x)dx = ∫ F(p)G*(p)dp

where * indicates a complex conjugate. (The integrals are from -∞ to ∞)

2. Relevant equations

F(p) = ∫f(x)exp[2∏ipx]dx

G(p) = ∫g(x)exp[2∏ipx]dx

G*(p) = ∫g(x)exp[-2∏ipx]dx

f(x) = ∫F(p)exp[-2∏ipx]dp

g(x) = ∫G(p)exp[-2∏ipx]dp

g*(x) = ∫G(p)exp[2∏ipx]dp

3. The attempt at a solution

Well this question is kind of weird to me since most of the in class examples have been based on knowing the function and then using different methods of integration to find the transforms, but in this proof it's all arbitrary, obviously.

Well simply subbing definitions in I get:

∫f(x)g*(x)dx = ∫[∫F(P)exp[-2∏ipx]dp∫G(p)exp[2∏ipx]dp]dx

and

∫F(p)G*(p)dp = ∫[∫f(x)exp[2∏ipx]dx∫g(x)exp[-2∏ipx]dx]dp

Now I guess if I can show that these two lines simplify to the same thing I have my proof. However, I am not sure how to simplify this. Maybe I am forgetting some basic property of integrals?

Also I have no idea if this approach is even correct and it would be better to start someplace else.

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

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

# Fourier Transform Proof

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