Bandwidth from a transfer function Currently I'm having problem wrapping my head around the following.
Suppose you have a dynamical system described by the transfer function
$$
G(s)=\frac{as}{(s+b)(s+c)}
$$
depending on the variables $a$, $b$ and $c$. In order to calculate the frequency response of the system $s=i\omega$. With that one is now able to draw the Bode plot wherein the magnitude specified by
$$
M(\omega)=20\log_{10}\lvert G(i\omega)\rvert
$$
is plotted over $\omega$.
The bandwidth of this frequency response should now be defined (as far as I understood it) as the differences of frequencies for which the maximal magnitude is lowered by 3 dB.
Is this the correct definition? Because people mainly show example in which the maximal magnitude is 0 dB and the bandwidth then is calculated as the difference of frequencies for which the magnitude crosses 3 dB.
Would it then be correct to solve
$$
20\log_{10}\lvert G(i\omega)\rvert = 20\log_{10}G_\text{max}-3 \text{ dB}
$$
for $\omega$ with 
$$
\begin{align}
20\log_{10}G_\text{max}&=\frac{d}{d\omega}\left(20\log_{10}\lvert G(i\omega)\rvert\right)\\
G_\text{max}&=\frac{d}{d\omega}\lvert G(i\omega)\rvert
\end{align}
$$
If so, could someone help me with that? I should be able to do that but seem to be slow on the uptake.
Thanks
 A: 
The bandwidth of this frequency response should now be defined (as far
  as I understood it) as the differences of frequencies for which the
  maximal magnitude is lowered by 3 dB.

Both definitions are used. But I believe this also depends a bit on the field of study. For control engineering we use the cross-over frequency, the bandwidth is the frequency at which the magnitude is equal to 1, i.e., 0 [dB]. This is done because it is related to the stability at the system. Because if the magnitude response $|G(jw)|$ crosses at 0 [dB] and the phase $\angle G(jw)$ is larger then -180 degrees, you know your system is stable, assuming the magnitude does not cross 0 [dB] afterwards anymore. This has to do with Nyquist stability criterion. 
However in electrical engineering, specific to filter design, The frequency defined at -3 [dB] is being used as the bandwidth. Because this is the magnitude at which the filter is not active anymore. From wikipedia
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)#X-dB_bandwidth

In basic electric circuit theory, when studying band-pass and
  band-reject filters, the bandwidth represents the distance between the
  two points in the frequency domain where the signal is
  $\frac{1}{\sqrt{2}}$ of the maximum signal amplitude (half power).

