I am working on the proof of the Fourier Transform of the derivative of a function. I am accompanied by some proof lines but having some issue in one of the integral evaluation. I searched out its proof in this forum, and I found out couple of proof, but there are some ambiguities in that which I want to clear.

I have defined the Fourier Transform by the following equation:

$\mathfrak{F}\{f(t)\} = \int_{-\infty}^{+\infty} {f(t)e^{-j \omega t}} dt$

where f is a vector fucntion. i.e., $f(R^1\rightarrow R^2)$

and I want to verify the Fourier of the derivative of function $f$. i.e.,

$\mathfrak{F}\{f'(t)\} = j\omega \mathfrak{F}\{f(t)\}$


Step.1: I took the left hand side of the property and apply the Fourier Transform defintion to obtain:

\begin{equation} \mathfrak{F}\{f'(t)\} = \int_{-\infty}^{+\infty} {f'(t)e^{-j \omega t}} dt \end{equation}

Step.2: I applied the integration by parts to obtain the following equation:

\begin{equation} \mathfrak{F}\{f'(t)\} = {e^{-j \omega t}f(t)}\mid_{-\infty}^{+\infty} - \int_{-\infty}^{+\infty} {f(t)e^{-j \omega t} {(- j \omega)}} dt \end{equation}


\begin{equation} \mathfrak{F}\{f'(t)\} = {e^{-j \omega t}f(t)}\mid_{-\infty}^{+\infty} + {j \omega} \int_{-\infty}^{+\infty} {f(t)e^{-j \omega t}} dt \end{equation}


\begin{equation} \mathfrak{F}\{f'(t)\} = {e^{-j \omega t}f(t)}\mid_{-\infty}^{+\infty} + j\omega \mathfrak{F}\{f(t)\} \end{equation}

This last equation is almost same as given in one of the answer to a question on the following link.

Fourier Transform Properties - Proving

Now to prove the derivative property the first term of the last equation should be equal to zero. i.e.,

\begin{equation} {e^{-j \omega t}f(t)}\mid_{-\infty}^{+\infty} = 0 \end{equation}

In another answer to one of the question regarding derivative property in this forum, it says "The first term must vanish, as we assume $f$ is absolutely integrable on $R$" which can be found on the following link:

Fourier Transform of Derivative

By the absolute integrability, it means that integral of norm of function would be convergent, but:

How would I prove it (i.e., the first term equal to zero).

I will highly appreciate the response to the question.

Thanks in advance and best regards.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.