The angle property of positive real functions simply derived

The angle property of positive real (rational) functions
Z(s)
, namely, that
|\arg s | \geqq |\arg Z(s)|
in the right half of the
s
-plane, can be demonstrated very simply by an examination of the imaginary parts of the functions
\ln(s/Z(s))
and
\ln (sZ(s))
, i.e.,
\arg s \mp \arg Z(s)
. In particular, on a contour enclosing the entire first quadrant,
\arg s \mp \arg Z(s)
can rather easily be shown to be nonnegative. The extremum theorem of analytic functions then assures that
\arg s \mp \arg Z(s)
cannot be negative inside the first quadrant; thus the angle property is demonstrated in the first quadrant. The same result is obtained immediately in the fourth quadrant.