148 BELL SYSTEM TECHNICAL JOURNAL 



Incidentally, when we put ZY — ly +i? in (1.51) and rearrange the 

 terms so as to get a power series in R we get the series (1.28). 

 1.10 Proof of the First Method 



The first method consists essentially of determining F from the series 

 expansion of (1.13): 



r = Ti:M^"^", (1.53) 



„=o n\ y^" 



where (— 1)„ = (— ^)(J)(|) • • • (« — I) when n > and ( — |)o = 1, and 

 then computing Zg and the required exponential and hyperbolic functions 

 of xT. From §1.7 and §1.8 it follows that the first method gives the correct 

 result provided that F as determined by (1.53) satisfies the conditions: 

 (a) its square is equal to ZY and (b) every element of e~^ approaches zero 

 as .T -^ 00 . 

 These two conditions are satisfied by the matrix 



F = PGP~' (1.54) 



where P and G are matrices defined by equations (Al.l) and (A1.3) of Ap- 

 pendix I, G being a diagonal matrix whose rth element is yr . For from 

 (A1.9) the square of F is 



Furthermore, 



e-r ^ J2 ^^^ (PGp-'r 

 n\ 



= pY ^~^^" c'p'^ ^^•'^^^ 



n\ 



= PM{x)P~' 



where M{x) is diagonal matrix (A1.5) whose rth element is e"'^'"'^. Since 

 the real part of y, is positive and the elements of P are independent of x 

 it follows that the second condition is satisfied. 



It will now be shown that PGP~ may be expanded in the series (1.53) 



'V 



provided that y may be chosen so as to make all of the points Tr = — , 



7 

 r = 1,2, • • • m, in the complex ^ plane lie within that loop of the lemniscate 

 I f ^ — 1 I = 1 which contains the point f = 1. For then we may write 

 the rth element in G as a convergent series: 



(2 i 

 1 + 5 



^' ' , (1.56) 



y (^)« (7^ - 7^)" 

 ^ n=o n\ 7^" 



