When the maximally flat optimality criterion is applied to the general (analog) squared amplitude response , a surprisingly simple result is obtained :
The analytic continuation (§D.2) of to the whole -plane may be obtained by substituting to obtain
The poles of this expression are simply the roots of unity when is odd, and the roots of when is even. Half of these poles are in the left-half -plane ( re ) and thus belong to (which must be stable). The other half belong to . In summary, the poles of an th-order Butterworth lowpass prototype are located in the -plane at , where [64, p. 168]
for . These poles may be quickly found graphically by placing poles uniformly distributed around the unit circle (in the plane, not the plane--this is not a frequency axis) in such a way that each complex pole has a complex-conjugate counterpart.
A Butterworth lowpass filter additionally has zeros at . Under the bilinear transform , these all map to the point , which determines the numerator of the digital filter as .
Given the poles and zeros of the analog prototype, it is straightforward to convert to digital form by means of the bilinear transformation.