138 BELL SYSTEM TECHNICAL JOURNAL 



First Method: Choose a number 7- and compute the matrices R, S, F, Z^ 

 i descril 

 given by 



as described in the first method for the infinite Hne. The matrix e' is 



e^^ 



where s = yx and /^ is computed from the series 



zS T I 20 ,20, 



' ^^ + n + "2r + -" 



If the elements of %S are so large that the series converges slowly it may be 



(zS\ 

 worthwhile to divide zS by 16, say, compute exp I — 1 from the series, and 



then obtain e^ by four matrix multiplications. When e' is known its 

 inverse eT"^ can be computed and e"^ obtained from (1.16). The hyper- 

 bolic functions are given by 



cosh xV — \ {e + c""^ ) 



(1.27) 

 sinh xY = \ {e — e "" ) 



which follow from the series definitions of the various matrices. 



If only the coefficients in (1.25) are required we may choose 7- and com- 

 pute R and powers of the matrix Rx/2y. Then the coefficients in (1.25) 

 are given by 



RxY Op+iiz) 



^V27/ ph 



p=0 \ 



smhxTZ.= T,{P) "J^Z (1.2«) 



P=o\2y/ ply 



where R' is the transposed of R, and the scalar coefficient a-iXz) is a function 

 oi z = yx given by 



00(2) = cosh z ai(z) = sinh s 



, . , sinh z 

 ^2(2) = cosh z — ■ /J 29) 



0^+2(2) = ap{z) — dp+iiz), 



2 



and it is understood that (i?.v/27)'^ = /. 



