The Stereographic Product of Positive-Real Functions is Positive Real

Abstract The circle product, o, has recently been introduced as the key operation in the algebraic structure of stereographic trigonometry. Likewise, the positive-real functions have been found to be important in the design of optimal control theory while serving also as the basic functions of circuit synthesis. Here we extend the stereographic circle operation to complex-valued functions and show that z = y 1 o y 2 is positive real when y 1 and y 2 are. Several physical interpretations are given, including z being the input impedance when an admittance-based Richards section of parameter y 1 is loaded in the admittance y 2 . Zero, 0, is the identity, 0o x = x , so − x is the inverse of x under o—that is, (− x )o x = 0, − x will not be positive real when x is; that is, the inverse under o need not be positive real. However, y 2 = (− y 1 )o z will be if y 1 and z are positive real and normalized to give y 2 as a Richards function.