Online Proof of Fourier Transform
The link above shows the full proof of the Fourier Transform using the limiting case of the Fourier Series. I don't understand why this was done in the proof however:
$$ x(t) = \sum\limits_{n = - \infty }^{ + \infty } {{c_n}{e^{jn{\omega _o}t}}} $$ $$ c_n = {1 \over T}\int_T {x(t){e^{ - jn\omega_0t}}dt} $$ $$ T{c_n} = \int_T {x(t){e^{ - jn\omega_0t}}dt} $$
As $T \rightarrow \infty$ the fundamental frequency, $\omega_0=2\pi/T$, becomes extremely small and the quantity $n\omega_0$ becomes a continuous quantity that can take on any value (since $n$ has a range of $\pm \infty$) so we define a new variable ω=nω0; we also let $X(\omega)=Tc_n$. Making these substitutions in the previous equation yields the analysis equation for the Fourier Transform (also called the Forward Fourier Transform).
$$ X(\omega ) = \int\limits_{ - \infty }^{ + \infty } {x(t){e^{ - j\omega t}}dt} $$
Wouldn't multiplying coefficients by T, result in the Fourier Transform being scaled by T according to the proof? If so, the Fourier transform does not accurately describe the 'amount' of frequencies in a function as they too are scaled by T.