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Consider a function $f(t)$ with Fourier Transform $F(s)$. So $$F(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f(t) \ dt$$

What is the Fourier Transform of $f'(t)$? Call it $G(s)$.So $$G(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f'(t) \ dt$$

Would we consider $\frac{d}{ds} F(s)$ and try and write $G(s)$ in terms of $F(s)$?

NebulousReveal
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3 Answers3

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A simpler way, using the anti-transform:

$$f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) \, e^{i \omega t} d\omega$$

$$f'(t) = \frac{d}{dt}\!\left( \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) \, e^{i \omega t} d\omega \right)= \frac{1}{2\pi} \int_{-\infty}^{\infty} i \omega \, F(\omega) \, e^{i \omega t} d\omega$$

Hence the Fourier transform of $f'(t)$ is $ i \omega \, F(\omega)$

leonbloy
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    A very good answer indeed. +1 – Swapnil Tripathi Nov 09 '14 at 18:59
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    @leonbloy Why exactly can we move the derivative inside the integral (apply Leibniz rule)? – Konstantin Mar 05 '17 at 12:17
  • @Konstantin why not? As you mentioned, we've used Leibniz rule here as the function inside the integral has a continuous partial derivative w.r.t. the variable t with respect to we're differentiating – Arkya Mar 16 '17 at 18:43
  • Could you elaborate on how that second inequality tells you that the Fourier transform of $f'(t)$ is $i\omega F(\omega)$? – Atsina Sep 09 '18 at 17:19
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    @Atsina In general, if you have $g(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} G(\omega) , e^{i \omega t} d\omega $, then you know that $G(\omega)$ is the Fourier transform of $g(t)$ – leonbloy Sep 10 '18 at 02:16
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    @leonbloy How 'w' came out of the integral as we are integrating with respect to 'dw'? Can anyone elaborate, please? – UJM Feb 13 '21 at 10:24
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    I hate to be that guy, but doesn't your answer rely on the fact that the Fourier inversion formula holds for $f$? Which means you are assuming extra conditions on $f$? Like you are saying that $F$ is integrable (which it doesn't have to be in general). – Smooth Alpert Frame Sep 16 '21 at 16:58
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    @UjjawalM. Note that the proof is not $f'(t) =i \omega F(\omega)$ but rather that the Fourier transform of $f'(t)$ is $i \omega F(\omega)$. I got confused too. – Bemipefe Sep 30 '21 at 20:08
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The Fourier transform of the derivative is (see, for instance, Wikipedia) $$ \mathcal{F}(f')(\xi)=2\pi i\xi\cdot\mathcal{F}(f)(\xi). $$

Why?

Use integration by parts: $$ \begin{align*} u&=e^{-2\pi i\xi t} & dv&=f'(t)\,dt\\ du&=-2\pi i\xi e^{-2\pi i\xi t}\,dt & v&=f(t) \end{align*} $$ This yields $$ \begin{align*} \mathcal{F}(f')(\xi)&=\int_{-\infty}^{\infty}e^{-2\pi i\xi t}f'(t)\,dt\\ &=e^{-2\pi i\xi t}f(t)\bigr\vert_{t=-\infty}^{\infty}-\int_{-\infty}^{\infty}-2\pi i\xi e^{-2\pi i \xi t}f(t)\,dt\\ &=2\pi i\xi\cdot\mathcal{F}(f)(\xi) \end{align*} $$ (The first term must vanish, as we assume $f$ is absolutely integrable on $\mathbb{R}$.)

Nick Peterson
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  • Thanks integration by parts was the trick. – NebulousReveal Jun 27 '13 at 14:57
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    Why the first term must vanish? I think we need the additional condition $\lim_{t\to\infty}f(t)=0$ to guarantee this. – Xiang Yu Feb 27 '16 at 03:51
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    Since $f \in L^1 \cap C^1$ f is continous and integrable, and must tend to zero when t tends to infinity...? – user202542 Jun 29 '16 at 17:48
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    The limit need not exist, although if it exists it must be zero. There are smooth, i.e., $C^\infty$, $L^1$ functions that do not tend to zero as $x \to \infty$. For an example, just make smooth "spikes" of height 1 at each integer $n$, such that the spike at $n$ has width $2^{-n}$. The limit must be zero, however, if you replace "continuous" with "uniformly continuous." – Zach Jan 12 '17 at 19:22
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    Where did we ever assume that $f$ is absolutely integrable, or was that an assumption appended as a bandage? – Your neighbor Todorovich Dec 06 '17 at 02:50
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    It seems that absolute integrability can NOT imply that $f$ vanishes at the infinity. See Did's answer here: https://math.stackexchange.com/questions/108191/prove-that-f-continuous-and-int-a-infty-fx-dx-finite-imply-lim-limi – Sam Wong Sep 18 '18 at 02:53
  • If $f$ is in the Schwartz space, then it does vanish at infinity, and then your argument works. – Sam Wong Sep 18 '18 at 03:06
  • I'm also not 100% following the "first term must be 0" argument. We're not actually evaluating that term at infinity. That doesn't make mathematical sense. – Pavel Komarov Jul 12 '23 at 22:53
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    How does being absolutely integrable imply that the evaluation of a function from $-\infty$ to $\infty$ is $0$? – Pavel Komarov Jul 12 '23 at 23:04
  • @PavelKomarov If $f, f' \in L^{1}(\mathbb{R})$, then $f$ is absolutely continuous on every bounded interval. In this case, we are justified to apply the FTC over $[a,x]$ for some fixed $a$, which guarantees that $f(x) \rightarrow c$ for some $c$ as $x \rightarrow \infty$. Now, assume that $c \neq 0$. Then you can argue that $f(x) \sim c$ for large enough $x$ and use some standard triangle-type inequalities to obtain that the integral of $|f|$ diverges, which contradicts $f \in L^{1}(\mathbb{R})$. The missing condition is probably having both $f$ and $f'$ in $L^1$. – huh Apr 13 '25 at 05:58
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One could derive the formula via dual numbers and using the time shift and linearity property of the Fourier transform. If $f$ is analytic around $x \in \mathbb{C}$, then (using $\varepsilon$ as the imaginary unit of dual numbers and assuming $y \in \mathbb{C} \setminus \left\{ 0 \right\}$): \begin{align*} f'\left( x \right) &= \frac{f\left( x \right) - f\left( x + y \cdot \varepsilon \right)}{y \cdot \varepsilon}\\ \mathcal{F}_{x}\left[ f'\left( x \right) \right]\left( x \right) &= \frac{F\left( x \right) - e^{2 \cdot \pi \cdot i \cdot y \cdot \varepsilon \cdot x} \cdot F\left( x \right)}{y \cdot \varepsilon} = \frac{F\left( x \right) - \left( 1 + 2 \cdot \pi \cdot i \cdot y \cdot \varepsilon \cdot x \right) \cdot F\left( x \right)}{y \cdot \varepsilon}\\ \end{align*} $$\fbox{$\mathcal{F}_{x}\left[ f'\left( x \right) \right]\left( x \right) = 2 \cdot \pi \cdot i \cdot x \cdot F\left( x \right)$}$$

Or more general: $$\fbox{$\mathcal{F}_{x}\left[ f^{\left( n \right)}\left( x \right) \right]\left( x \right) = \left( 2 \cdot \pi \cdot i \cdot x \right)^{n} \cdot F\left( x \right)$}$$

The Art Of Repetition
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