In many reference materials i have come across fourier transform of a function $x(t)$ referred as $X(\omega)$ and $X(j\omega)$. But what is the difference between both the representations. Are they both one and the same, if not what is the relation between them. The following are definition for $X(\omega)$
$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{j\omega t}dt$
Thanks in advanced
Both notations refer to one and the same thing. One would use $X(j\omega)$ when transients are also dealt with in the text (e.g., switching in electric circuits, LTI systems in control theory) where one would also consider the Laplace transform $X(s)$. Then $s = j\omega$ is the stationary / steady state, where all exponentials have sufficiently decayed and the system can be described with just harmonics. Besides that, the $X(j\omega)$ notation is often used in electrical engineering texts just because the $j\omega$ is such a common term when dealing with electrical circuits involving capacitive or inductive components (cf. the impedance of an RLC circuit $Z = R + j\omega L + \frac{1}{j\omega C}$). If I wrote a text where transients are not of interest, the text is unrelated to the other concepts mentioned above and/or targeted to a non-specialist audience, I would definitely prefer $X(\omega)$.
