no BELL SYSTEM TECHNICAL JOURNAL 



differential equation (2-3) gives rise to a set of ordinary differential equations 

 having v as the independent variable and the coefficients as the dependent 

 variables. The integral equation method is used in this paper. The 

 second method is discussed in the companion paper. 



3. Conversion of Differential Equation into an Integral Equation 



The differential equation (2-3) may be converted into an integral equation 



by using the appropriate Green's function in the conventional manner. 



The only modifications necessary are essentially those given by Poritsky 



and Blewett^ in a similar procedure. 



The conversion is based upon Green's theorem in the form 



where the integration on the right extends over the rectangular region Vi<v< i% 

 < < X (inside the straight guide associated with (v, 6), i.e. the guide of 

 Fig. 2) except for a very small circle surrounding the point (vo , do). 

 Q = Q{y^ , do ■,v, e) is the Green's function corresponding to 



^ + ^ + ;feV = (3-2) 



in the region — oo<t;<oo,O<0<7r subject to the boundary condition 

 dV/dn = on the walls (37/00 = at = and = tt). G becomes 

 infinite as —log r when r -^ 0, r being the distance between the variable 

 point {v, 6) and the fixed point {vo , do). Poritsky and Blewett* have shown 

 that, in the notation (2-1), 



G = E e„7m' cos mOo cos ^06-'"-'°'^- (3-3) 



TO = 



(3-4) 



eu ^ 1, e,„ = 2 for w = 1, 2, 3 • • • 

 Equation (3-1) leads to 



= k' [ ' dv I dd g{v, e)QG 



Jvi Jo 



from which the required integral equation for Q is found to be 

 Q{vo , 0o) = 



e'''"" -{- — Tdv fdd giv, d)Q{v, d) J2^r.y7: cos mdo cos md g-'^-'"''^"' (3-5) 



* We have replaced their t by - i since here we assume the time to enter through the 

 factor e'"' instead of e~'"'. 



