I'm learning about signals from the textbook Signals and Systems Second Edition, by Oppenheim and Willsky in order to learn more about Fourier Analysis and I'm having difficulty with the Energy equation that I found for a complex periodic signal. For this type of signal, we have the following basic form for the energy of the signal over 1 period:
$$E_{period} = \int_{0}^{T_0} |e^{j\omega_0t}|^2dt$$
which then simplifies to just:
$$E_{period} = \int_{0}^{T_0} 1 \cdot dt$$
I'm assuming that the text is saying $e^{j\omega_0t} = e^{j\omega_0T} = 1$ but I don't understand how it arrives to this conclusion. Based on earlier information in the text, I know that for any complex periodic signal:
$$x(t) = e^{i\omega_0t} = e^{i\omega_0(t+T)}$$
and this equation is used to obtain :
$$x(t) = e^{i\omega_0t} = e^{i\omega_0t}e^{i\omega_0T} \Rightarrow e^{i\omega_0T} = 1$$
However, I do not see how this can also be used to deduce that $e^{j\omega_0t}=1$ as well. So, could someone please help clear up why the signal inside the integral simplifies to 1?
Note: at this point in the book, it is just going over the different types of signals and has not yet touched Fourier Analysis.
Parseval's theorem for the Fourier Transform is defined as follows: $$E_x = \int_{-\infty}^{\infty}|x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty}|X(\omega)|^2 d\omega $$
To answer your equation, please take note of your notation. Given the complex signal: $$ x(t) = e^{j\omega_0 t} $$ This implies: $$ |x(t)| = |e^{j\omega_0 t}| = 1 $$ Comments: Recall the $|\cdot|$ function means the length of the complex number. $e^{j\omega_0 t}$ is a unit circle in the complex plane, with the angular frequency of $\omega_0$. Even if the complex signal is periodic over $T$, i.e. $e^{j\omega_0 (t+T)}$, the length remains the same, i.e. $|e^{j\omega_0 t}|=|e^{j\omega_0 (t+T)}|=1$. Hope it clears your doubts.