derivative of the fourier transform
my book defines the fourier transform as F(ξ) = integral (-infinite to infinite) f(x)e^iξx dx.
my book takes the derivative of the fourier transform and does:
DF(ξ) = integral (D_ξ (f(x) e^iξx)) dx = integral (ixf(x)e^iξx) dx = ixF(ξ), where D_ξ is the partial derivative with respect to ξ
my book signifies the fourier transform by f with a small hat on top. i put F instead since i don't know how to type f with a small hat on top. so the result of the differentiation is the following: ixf(ξ) and the whole thing has a small hat on top of it.
what i don't understand is why isn't the derivative iξf(ξ) all with a hat on top? from the definition of the fourier transform i gathered that you take whatever is in front of the e^iξx and you replace the x with ξ and you put a hat on it. in the derivative, the function in front of e^iξx is ixf(x), but couldn't it be written as g(x) = ixf(x)? then by the definition the integral would be g(ξ) with a hat on top which would be iξf(ξ) with a hat on top wouldn't it? my apologies if this is messy to read. thanks.