I have a question about the notation adopted for the Fourier Transform of both continuous-time signals and discrete-time sequences in Signal Processing.
In the textbook I am using (Signals and Systems by Alan Oppenheim), the Fourier Transform of the signal $x(t)$ is denoted by $X(j\omega)$ and is defined as $$X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} \mathrm{d}t.$$ Its discrete-time counterpart (for a sequence $x[n]$ is denoted by $X(e^{j\omega})$ and is given by $$X(e^{j\omega}) = \sum_{n = -\infty}^{\infty} x[n] e^{-j\omega n}.$$
By virtue of the inherent periodicity in frequency of discrete-time periodic signals, the discrete-time Fourier Transform is periodic, but the continuous-time Fourier Transform is not periodic in general. Now, although we denoted the continuous-time Fourier Transform by $X(j\omega)$, couldn't have we just as well used $X(e^{j\omega})$? Except, the problem with this notation is that it implies that $X(e^{j(\omega + 2\pi)}) = X(e^{j\omega})$, which is not true in the continuous-time case. I know that the notation doesn't work, but could someone explain to me mathematically why it is invalid to do so?
Apologies for using $j$ to denote the imaginary unit.
The continuous time Fourier Transform's result can be thought of as having its independent variable being on the $j\omega$ axis of the s-plane. (The same s-plane that the Laplace transform maps to.)
The discrete time Fourier Transform's result can be thought of as having its independent variable being on the unit circle of the z-plane. (The same z-plane that the Z Transform maps to.)
Hence the difference in notation of the independent variable.
The $e^{j\omega}$ notation indicates the discrete time Fourier transform result wraps around the unit circle and appears periodic when unwrapped.
The continuous time Fourier transform maps to an infinitely long axis and doesn't wrap around and is not periodic. Hence the $j\omega$ notation for the independent variable.