Evaluation of constants in conformal representation

Samuel I. Plotnick, Thomas C. Benton
1954 Quarterly of Applied Mathematics  
As 7 approaches one, j+(0) becomes infinite. This will be true not only for zero but for all x. The implication is plain. The Fokker-Planck process is a degenerate process in which the one sided current density of the system is infinite. A Fokker-Planck model for the velocity motion of a colloid particle would describe an infinite number of changes of direction of the particle per unit time. Such a model used to describe voltage fluctuations would imply an infinite number of polarity reversals
more » ... polarity reversals per second. Since a process Ay will afford the same correlation function and equilibrium distribution, and finite polarity reversal frequency, it is suggested that such a model may better describe noise, and that the number of zero crossings be regarded as an independent macroscopic physical quantity on an equal footing with t0 , E0 . In using the Schwarz-Christoffel transformation [1], dz = k n a -r.)(ai/T>-ftr = Km df * = i whereby the upper half f-plane is mapped into a simple connected polygon, the evaluation of the unknown constant K (if complex K = ce'x, c, X real), is oftentimes tedious. We shall show a simple method of evaluating the unknown constant K by examples, proving first a Theorem: By the Schwarz-Christoffel transformation if in the f-plane corresponds to two points P{ , Qi in the z-plane and f = f,-is a simple pole of /(f), then K dist (P.-, Qj) «7?(f = f.) R, denoting residue and dist (P<, Qi), denoting the distance between the two points P{ and Q,. *Received May 8, 1953.
doi:10.1090/qam/60594 fatcat:pfi6dh2ovbedhbbtxdkxtsezem