So I was working through the Fourier transform equations that arise. I was wondering where the radical outside the integral originated from?
$\hat{f}(k) = \sqrt{\frac{|b|}{(2 \pi)^{1-a}}} \int_{-\infty}^{\infty} dx \, f(x) \, e^{i b k x}$
and
$f(x) = \sqrt{\frac{|b|}{(2 \pi)^{1+a}}} \int_{-\infty}^{\infty} dk \, \hat{f}(k) \, e^{-i b k x}$
I was wondering where that radical originated. I always just found these equations simply stated and could not figure out where they came from and how that term came out of the integral?
Just so that you have the identity $\|f\|_2=\|\hat{f}\|_2$, called Plancherel's theorem.