Richards' theorem

Due to Paul I. Richards, 1947.

As used in Bott-Duffin synthesis, Richards' theorem states for,

$R(s)={\frac {kZ(s)-sZ(k)}{kZ(k)-sZ(s)}}$

if $Z(s)$ is a PRF then $R(s)$ is a PRF for all real, positive values of $k$.

Is this an extension of Richards' original statement?

Hubbard (2010, p. 33) says that Richards theorem "is really Schwarz's lemma in light disguise" but that Bott and Duffin were not aware of this

Bibliography

• Bott, Raoul; Duffin, Richard, "Impedance synthesis without use of transformers", Journal of Applied Physics, vol. 20, iss. 8, p. 816, August 1949.
• Hubbard, John H., "The Bott-Duffin synthesis of electrical circuits", pp. 33–40 in, Kotiuga, P. Robert (ed), A Celebration of the Mathematical Legacy of Raoul Bott, American Mathematical Society, 2010 ISBN 9780821883815.
• Richards, Paul I., "A special class of functions with positive real part in a half-plane", Duke Mathematical Journal, vol. 14, no. 3, 777–786, 1947.
• Wing, Omar, Classical Circuit Theory, Springer, 2008 ISBN 0387097406.