Note: this is homework.
Say we assume a time-varying momentum is of the form
$p(t) = \Re \{p(w) e^{-iwt} \}$
Now we would like to know the solution of the following equation, in frequency space:
$ \tfrac{d}{dt} p(t) = -\tfrac{1}{\tau}p(t) - eE(t) $
The part where I get confused is the fourier transform of $ \tfrac{d}{dt} p(t) $. I thought the Fourier transform of the derivative, ie $\tfrac{d}{dt} p(t)$, is $ (iw) p(w) $. However, in the solutions to the problem, it is $(-iw)p(w)$. Further more, Wolfram Alpha outputs the same.
Is this because of how we assume $p(t)$ to look like? Or does my class and Wolfram Alpha define the Fourier Transformation differently than what I'm used to?
This is most likely due to a sign convention mismatch in the definition of your Fourier transform and your reference's definition. Some like to define the Fourier transform as $$ \mathcal{F}[f](w)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R}e^{-i\omega t} f(t)dt $$ and others as $$ \mathcal{F}[f](w)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R}e^{i\omega t} f(t)dt, $$ which will change the sign of the formula for the transform of the derivative of a function.