A Note on Positive Real Functions

Abstract

In this note we investigate properties of certain special classes of analytic functions—those which map the right half-plane into itself and the real axis into the real axis. These so-called “positive real” functions play a significant role in the synthesis of two-terminal, passive networks. Our concern is with the Taylor coefficients in the expansion about the point $z_0 = x_0 + iy_0 $, $x_0 > 0$. If \[f(z) = u + iv = \sum {\alpha _n (z - z_0 )^n } ,\] then it is well known that $| {\alpha _n } | \leqq u_0 x_0^{ - n} $, where $f(z_0 ) = u_0 + iv_0 $. We derive results on $\arg \alpha _n $, one such being: iffis a nonlinear positive real function andxis a real, positive number, then for every integer$N \geqq 1$there exist integers$N \geqq 1$and$m \geqq N$such that$f^{(n)} (x)$ and $f^{(m)} (x)$have opposite signs.