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Math Help - derivative of the fourier transform

  1. #1
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    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.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by oblixps View Post
    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.

    Look at the derivation of the FT of a derivative, you should have something like:

    \displaystyle \frac{d}{d\xi}\left( \mathfrak{F}(f(x))(\xi) \right)=-i \widehat{xf(x)}(\xi)=-i \mathfrak{F}(xf(x))(\xi)

    (You should not have the minus sign as you appear to have a definition of the FT at variance with what I regard as the usual definition)

    CB
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  3. #3
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    thanks for clearing this up. I was just a little confused the first time I encountered this in my textbook.
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