Some Mapping Properties of RC and RL Driving-Point Impedance Functions

We show that an
RC
or
RL
driving-point impedance function
Z(s)
maps the closed right-half s plane onto a region in the
Z
plane which is covered by the closed circular disk with the straight line segment joining
Z(0)
and
Z(\infty)
as a diameter; and which covers the open circular disk with center at
Z(u)
and radius
|{1/2}uZ'(u)|
, and the straight line segment
Z = Z(u) + jtuZ'(u)
,
-1 < t < 1
, for any real positive number
u
.