Constant argument immittance realization by a distributed RC network

he driving point properties of a ladder type distributed RC network, characterized by the distributions
r (x) = r_{o} exp (-ax)
and
c(x) = c_{o} exp (- bx)
, are investigated. It is shown that the infinite length network exhibits, at one of the ports, a constant argument impedance of the form
s^{-a/(a+b)}
, in an asymptotic manner. With a finite length, the approximation is found to be valid over a finite frequency band only, while the asymptotic behavior is that of an impedance
s^{-1/2}
, irrespective of the values of
a
and
b
.