In order to justify the invention of the complex frequency variable s - here are some additional comments:
The pole location can be marked in the complex plane. And now one can define two very important filter parameters:
1.) The value identical to the magnitude of the vector from 0/0 to the pole location is the so called "pole frequency" ωp.
2.) The angle δ between the negative-real axis and this vector is an indication of the so called "pole quality factor" Qp. The exact relation is: Qp=1/(2cosδ)
3.) Both parameters characterize the filter response and are given in relevant textbooks. More than that, both values can be measured with a frequency generator and an oscilloscop (better: network analyzer)
4.) These definitions - together with the 3D-picture as shown - reveal the relationship between pole location and magnitude response of a filter (with amplitude peaking corresponding to the pole location).
Added after 1 hours 41 minutes:
Sorry, I forgot to say someting to the zeros.
No, it is not correct that at system zeros the output "tends to zero" . This is only true for "real zeros".
In general, the slope of the magnitude function will change in the neighbourhood of zeros - nothing else.
For example, the slope of the magnitude function (BODE- diagram) goes from 0 to -20dB/dec (or from -20 to -40 dB/Dec) at the pole frequency .
In the same way the slope increases from -20 dB/Dec to 0 (or from -40 to -20 dB/Dec) caused by a complex zero.
LvW
Added after 2 hours 57 minutes:
Another addendum - perhaps helpful for somebody:
The end of the passband (corner frequency, 3-dB-frequency) is NOT identical to the pole frequency. However, both frequencies normally are not very far from each other.
(in practice: 5...10 %) .Exception: For a BUTTERWORTH response both frequencies are identical.