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THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



8. Green's Formula 



The simple examples we have just discussed could have been solved with- 

 out recourse to the potential analogous method, since the charge distribu- 

 tions were easy to recognize. In general, this is not the case, and we now 

 turn to systematic methods of determining the charge distribution when the 

 gain, a, is given as an analytic function of w in a prescribed frequency 

 range, | w | < wo . The corresponding transmission function is obtained if 

 we replace co by p/i and regard /> as a complex variable. Then the mathe- 

 matical problem is to determine a charge distribution on some contour C 

 which will have this function F{p) as its complex potential. 



The contour C is to a large extent arbitrary. We shall assume that it is 

 a simple closed curve in the />-plane, enclosing the frequency band of in- 



REAL p 



REAL p 



(a) GAIN INVARIANT TRANSFORMATION ( b) PHASE INVARIANT TRANSFORMATION 



Fig. 10 — Illustrations for (a) gain invariant (b) phase invariant transformations. 



terest, and subject only to the imitations that it must be symmetric in the 

 real />-axis, and that F(p) must be analytic inside C. 



Then one very general solution of the charge distribution problem in 

 potential theory is given by Green's formula, f which has the form 



for the logarithmic potential in two dimensions. The integral expresses the 

 potential at any point P inside C in terms of the values of V and of its 

 normal derivative on C. The differential ds is an element of length on C 

 and n is the normal drawn out of the region we are considering. At points 



tSee e.g. A. G. Webster, Partial Differential Equations of Mathematical Physics, 

 G. E. Stechert and Co., New York, 1927, p. 210. 



