136 BELL SYSTEM TECHNICAL JOURNAL 



If only e~''^ is required the following series may be used. 



e-" = t (f)" *4? (1-18) 



P=o Vt/ pl 



where R, y, and z have the same meaning as above and the coefficients are 

 computed from 



bo = e~' , 61(2) = —e~\ ^2(2) = e~M 1 + - 



6p+2(z) = bp{z) - -^ 6p+i(z) 



2 



/RxX 

 In the first term of the series ( y- ) denotes /, 



Second Method: F, e~^^ and Z,, may be regarded as functions of the square 

 matrix ZY. In order to express these functions in a form suitable for calcu- 

 lation we apply Sylvester's theorem . The characteristic matrix of ZY is 



7(72) = y/ _ ZY (1.19) 



where now 7- is regarded as a variable instead of a fixed number as in the 

 first method. We shall suppose that ZY is a square matrix of order m 

 and that the roots 71 , 72 , • • • 7™ of the characteristic function, i.e. of the 

 determinantal equation 



1/(^)1 = 0, (1.20) 



are distinct. Let the matrix ^(7-) be the adjoint oi f{y-) and denote the 

 derivative of the characteristic function by 



\f(y')r = /^,\fiy')\ (1.21) 



Since 71, 72, • • • 7m are all different |/(7r) i*^ is unequal to zero for r = 

 1, 2, • • • m. Sylvester's theorem says that if P{ZY) is any polynomial 

 in ZF then 



m 



P{ZY) = E N(yl)P{yl) (1.22) 



r = l 



where P{yr) is a scalar (and thus deviates from our convention that capital 

 letters denote square matrices). N(yr) is a square matrix: 



N(yl) = i^l^r, (1.23) 



When m = 2, Niyi) is equal to / — A' (71). 



* F.D.C. §3.9. The u and X of the reference are the ZY and y^ of the present section. 



