The angle property of positive real functions simply derived

The angle property of positive real (rational) functionsZ(s), namely, that|arg s | geqq |arg Z(s)|in the right half of thes-plane, can be demonstrated very simply by an examination of the imaginary parts of the functionsln(s/Z(s))andln (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 thatarg 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.