Positive-real function

<h2 id="definition">Definition</h2>
The term positive-real function was originally defined by<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> <a href="/facts/Otto_Brune/1tkYVryc">Otto Brune</a> to describe any function Z(s) which<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ul><li>is <a href="/facts/Rational_function/2nJGu0Ko">rational</a> (the quotient of two <a href="/facts/Polynomials/Lzak8VVx">polynomials</a>),</li>
<li>is real when s is real</li>
<li>has positive real part when s has a positive real part</li></ul>
Many authors strictly adhere to this definition by explicitly requiring rationality,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> or by restricting attention to rational functions, at least in the first instance.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> However, a similar more general condition, not restricted to rational functions had earlier been considered by Cauer,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> and some authors ascribe the term positive-real to this type of condition, while others consider it to be a generalization of the basic definition.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="history">History</h2>
The condition was first proposed by <a href="/facts/Wilhelm_Cauer/TfLzRRAc">Wilhelm Cauer</a> (1926)<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> who determined that it was a necessary condition. <a href="/facts/Otto_Brune/1tkYVryc">Otto Brune</a> (1931)<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> coined the term positive-real for the condition and proved that it was both necessary and sufficient for realisability.

<h2 id="properties">Properties</h2>
<ul><li>The sum of two PR functions is PR.</li>
<li>The <a href="/facts/Function_composition/r5Pvnmth">composition</a> of two PR functions is PR. In particular, if Z(s) is PR, then so are 1/Z(s) and Z(1/s).</li>
<li>All the <a href="/facts/Zeros_and_poles/gnA3U8VP">zeros and poles</a> of a PR function are in the left half plane or on its boundary of the imaginary axis.</li>
<li>Any poles and zeroes on the imaginary axis are <a href="/facts/Zero_(complex_analysis)/gnA3U8VP">simple</a> (have a <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a> of one).</li>
<li>Any poles on the imaginary axis have real strictly positive <a href="/facts/Residue_(complex_analysis)/GllQJk32">residues</a>, and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative.</li>
<li>Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a <a href="/facts/Harmonic_function/H6pg4dMT">harmonic function</a> over the plane, and therefore satisfies the <a href="/facts/Maximum_principle/TcomDNRa">maximum principle</a>).</li>
<li>For a <a href="/facts/Rational_function/2nJGu0Ko">rational</a> PR function, the number of poles and number of zeroes differ by at most one.</li></ul>
<h2 id="generalizations">Generalizations</h2>
A couple of generalizations are sometimes made, with intention of characterizing the <a href="/facts/Immittance/FcbsAsvf">immittance</a> functions of a wider class of passive linear electrical networks.

<h3>Irrational functions</h3>
The impedance Z(s) of a network consisting of an infinite number of components (such as a semi-infinite <a href="/facts/Ladder_network/B1lMjhKB">ladder</a>), need not be a rational function of s, and in particular may have <a href="/facts/Branch_points/lJ7Y6Iht">branch points</a> in the left half s-plane. To accommodate such functions in the definition of PR, it is therefore necessary to relax the condition that the function be real for all real s, and only require this when s is positive. Thus, a possibly irrational function Z(s) is PR if and only if

<ul><li>Z(s) is analytic in the open right half s-plane (Re[s] > 0)</li>
<li>Z(s) is real when s is positive and real</li>
<li>Re[Z(s)] ≥ 0 when Re[s] ≥ 0</li></ul>
Some authors start from this more general definition, and then particularize it to the rational case.

<h3>Matrix-valued functions</h3>
Linear electrical networks with more than one <a href="/facts/Port_(circuit_theory)/g2mokqv0">port</a> may be described by <a href="/facts/Impedance_parameters/4ocIpsWI">impedance or</a> <a href="/facts/Admittance_parameters/0PJ6u6IH">admittance matrices</a>. So by extending the definition of PR to matrix-valued functions, linear multi-port networks which are passive may be distinguished from those that are not. A possibly irrational matrix-valued function Z(s) is PR if and only if

<ul><li>Each element of Z(s) is analytic in the open right half s-plane (Re[s] > 0)</li>
<li>Each element of Z(s) is real when s is positive and real</li>
<li>The <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian</a> part (Z(s) + Z†(s))/2 of Z(s) is <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive semi-definite</a> when Re[s] ≥ 0</li></ul>

<ul><li><a href="/facts/Wilhelm_Cauer/TfLzRRAc">Wilhelm Cauer</a> (1932) <a href="https://projecteuclid.org/euclid.bams/1183496197">The Poisson Integral for Functions with Positive Real Part</a>, <a href="/facts/Bulletin_of_the_American_Mathematical_Society/8ER5KY5z">Bulletin of the American Mathematical Society</a> 38:713–7, link from <a href="/facts/Project_Euclid/UXdRpPcW">Project Euclid</a>.</li>
<li>W. Cauer (1932) <a href="https://link.springer.com/article/10.1007/BF01455892">"Über Funktionen mit positivem Realteil"</a>, <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a> 106: 369–94.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June, 2000. Retrieved online 19 September 2008. <a href="http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf" target="_blank">http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June, 2000. Retrieved online 19 September 2008. <a href="http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf" target="_blank">http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", Doctoral thesis, MIT, 1931. Retrieved online 3 June 2010. <a href="http://dspace.mit.edu/bitstream/handle/1721.1/10661/36311006.pdf?sequence=1" target="_blank">http://dspace.mit.edu/bitstream/handle/1721.1/10661/36311006.pdf?sequence=1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Bakshi, Uday; Bakshi, Ajay (2008). Network Theory. Pune: Technical Publications. ISBN 978-81-8431-402-1. <a href="978-81-8431-402-1" target="_blank">978-81-8431-402-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Wing, Omar (2008). Classical Circuit Theory. Springer. ISBN 978-0-387-09739-8. <a href="978-0-387-09739-8" target="_blank">978-0-387-09739-8</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June, 2000. Retrieved online 19 September 2008. <a href="http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf" target="_blank">http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Wing, Omar (2008). Classical Circuit Theory. Springer. ISBN 978-0-387-09739-8. <a href="978-0-387-09739-8" target="_blank">978-0-387-09739-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Cauer, W, "Die Verwirklichung der Wechselstromwiderst ände vorgeschriebener Frequenzabh ängigkeit", Archiv für Elektrotechnik, vol 17, pp355–388, 1926. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", Doctoral thesis, MIT, 1931. Retrieved online 3 June 2010. <a href="http://dspace.mit.edu/bitstream/handle/1721.1/10661/36311006.pdf?sequence=1" target="_blank">http://dspace.mit.edu/bitstream/handle/1721.1/10661/36311006.pdf?sequence=1</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", J. Math. and Phys., vol 10, pp191–236, 1931. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Positive-real function open-in-new

Positive-real function