Given the continuous Fourier Transform $X(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}dt$
A sufficient condition for it to exist is $E_{x}=\int_{-\infty}^{\infty}|x(t)|^{2}dt < +\infty$.
We can write $X(f) = R(f) + jI(f)$ where R(f) is the real part and I(f) is the imaginary part.
Assuming x(t) is a real function, we can then write:
$R(f)=\int_{-\infty}^{\infty}x(t)cos(2\pi ft)dt$
$I(f)=-\int_{-\infty}^{\infty}x(t)sin(2\pi ft)dt$
The book states that from these two expressions above it is clearly visible that:
$R(f)=R(-f)$
$I(f)=-I(-f)$
I don't see it. Shouldn't it depends on whether the x(t) is an even or odd function?
E.g. in case of R(f), $cos(2\pi ft)$ is even, so:
- odd times even = odd hence R(f)=-R(-f)
- even times even = even hence R(f)=R(-f)
What's wrong?