The angle property of positive real functions simply derived

The angle property of positive real (rational) functions , namely, that in the right half of the -plane, can be demonstrated very simply by an examination of the imaginary parts of the functions and , i.e., . In particular, on a contour enclosing the entire first quadrant, can rather easily be shown to be nonnegative. The extremum theorem of analytic functions then assures that 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.