318 BELL SYSTEM TECHNICAL JOURNAL 



the element in the w* row andy* column of k\ i.e. if a root 7, is repeated, 

 say, 5 times there are 5 linearly independent columns {k/s) corresponding 

 to 7,-. Substitution of (2.1-2) in (2.1-1) gives 



m=l y=i 



00 



(2.1-3) 



where 



00 



^m{z) = T.kmiqe-''^ (2.1-4) 



Since dm{Xj y) is analogous to the product of the trigonometrical terms in 

 (1.3-1) or (1.3-11) these equations show that iUw(s) plays the same role as 

 aml{z) or ^ml{z). Therefore, in accordance with the discussion given at the 

 beginning of this section, we wish to show that the column matrix )u(z) 

 (which is similar to a{z) or /3(2)) whose rn' element is ixm{z) may be ex- 

 pressed as 



m(z) = e-'^t (2.1-5) 



In this equation F is a square matrix to be determined and /+ is a column 

 matrix of constants similar to ^ or J+. 



The rules of matrix multipUcation and equation (2.1-4) show that 



^,{z) = k[e~''']dc (2.1-6) 



in which [exp {—zy)]d is a diagonal matrix having exp (— S7y) as the j 

 element in its principal diagonal and c is the column matrix formed from the 

 c/s. We introduce the column /+ by defining it as /x(0) whence 



/+ = kc, c= k-'f+ (2.1-7) 



Incidentally, from (2.1-3), the value of $ at 2 = is 



00 

 ^z=o = JlfterrXx,y) (2.1-8) 



where /i is the m^^ element in/"*". 

 From (2.1-6) and (2.1-7) 



m(2) = k[e-''U k-' f"- (2.1-9) 



In this equation k [exp ( — zy)]d k~^ is a square matrix which may be expressed 

 as 



t ^-^ *w; r ' = z t^" {k[y], k-r 



