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I have a problem interpreting 2 different representations of Fourier transform and proving their property. In some texts Fourier transform is represented as:
$$ \begin{align*} x(t) = \frac{1}{2\pi} & \int_{-\infty}^{+\infty}{X(j\omega)e^{j\omega t}\,d\omega} \\ X(j\omega) = & \int_{-\infty}^{+\infty}x(t)e^{-j\omega t}\,dt \end{align*} $$
While in other texts the complex number $j$ is omitted in $X(j\omega)$. The problem comes when we try to prove things where the two notations don't work likely; for example, $\frac{d}{d\omega}X(j\omega) = jX'(j\omega)$, and $\frac{d}{d\omega}X(\omega) = X'(\omega)$.
I faced this confusion while proving the Duality property of Fourier transform; namely:
$$ \text{if}\;\mathcal{F}[x(t)] = X(j\omega),\;\; \text{then}\; \mathcal{F}[X(t)] = 2\pi\,x(-j\omega)\, $$
in one representation and
$$ \begin{align*} x(t) & \leftrightarrow X(j\omega) \\ X(t) & \leftrightarrow 2\pi\ x(-\omega) \end{align*} $$
in other representation.
I believe that in duality result of the former representation $X(j\omega)$ should contain additional $j$, i.e. $X(t)\rightarrow 2\pi jx(-j\omega)$, when i tried to prove it. But now I am confused.
Here is my proof, can someone validate that this is correct or correct it? Notice the extra $j$ I am getting.
Proof
$$ x(t) = \frac{1}{2\pi} \int_{-\infty}^{+\infty}{X(j\Omega)e^{j\Omega t}\,d\Omega} $$
Replace $t$ with $-j\omega$, and $j\Omega$ with $t$ (just a change of variables),
$$ \begin{align*} \Longrightarrow\; & 2\pi x(-j\omega) = \int_{-\infty}^{+\infty}X(t)e^{-j\omega t}\,d(t/j)\,, \\ \Longrightarrow\; & 2\pi jw(-j\omega) = \int_{-\infty}^{+\infty}X(t)e^{-j\omega t}\,dt\,, \end{align*} $$
So, I can say that $\mathcal{F}\{X(t)\} = 2\pi jx(-j\omega)$. END
Lot of articles or tutorials on google search uses this $X(\omega)$ notation instead of $X(j\omega)$.
I have found $X(j\omega)$ notation intuitive lot of times. Can some one please explain?