Transfer function of amplifier (exercise)
At first, I will try to answer your main question (concerning the numerator) based on a basic RC-Bandpass. 

The transfer function is H(s)=sR1C1/[1+s(R1C1+R2C2+R1C2)+S²R1R2C1C2

As you can see, it is the TIME CONSTANT of the first RC element which appears in the numerator. But this is not the inverse of one of the two pole frequencies. Instead, we have two REAL pole frequencies (on the negative real axis  in the s-plane). These pole frequencies are computet by setting the denominator equal to zero. 

(And, of course, the result will be NOT simply w1=1/R1C1 and w2=1/R2C2. )

The midband gain can be found by dividing numerator and the s-element of the denominator Ao=R1C1/(R1C1+R2C2+R1C2). From this expression one can see which role the expression in the numerator plays (it determines the midband gain - together with the other two parts).







simulate this circuit &ndash; Schematic created using CircuitLab