Confusion about filter transfer function
The problem has come from the substitution of s for 2*PI*f.

s is a substitution for j*w

thus the equation actually is:   1/(1+j)
If you resolve this back to magnitude:   1+j = sqrt(1^2 + 1^2)  = sqrt(2)

resulting in:  1/sqrt(2)  = 0.707
as you expect

So take a low-pass filter: RC







simulate this circuit &ndash; Schematic created using CircuitLab

$$
Vout = Vin*\frac{\frac{1}{j\omega C}}{R + \frac{1}{j\omega C}}
$$
$$
Vout = Vin*\frac{1}{j\omega RC + 1}
$$
$$
\frac{Vout}{Vin} = \frac{1}{j\omega RC + 1}
$$
$$
\frac{Vout}{Vin} = \frac{1}{\sqrt{(\omega RC)^{2} + 1^{2}}}
$$
From a magnitude point of view.