98 BIRKHOFF AND LANGER. 



Section VIII. 

 The formal expansion of an arbitrary vector. 



It was shown in the preceding section that both under regular 

 conditions and under the type of irregular conditions discussed the 

 characteristic values for system (46) are numerable and cluster about 

 the point X = °° . Denoting these values by X], X 2 , X 3 , . . . it is possible, 

 therefore, to assign the subscripts in such manner that | \ m | ^ | X m+ i |. 

 Moreover then lim | |\„, | = oo . 



TO=0O 



Assuming that system (46) is simply compatible at the characteristic 

 values there exists for each of these values just one solution of the 

 system in question and just one solution of its adjoint system. These 

 solutions for X = Xa will be designated by Y (x) • and -Z (k) {x) respec- 

 tively. 



Now if X = is a characteristic value for system (46) let the para- 

 meter be changed by setting X = X + c, c being a constant. Equation 

 (46) then becomes 



Y'(x)- = {R(x)\+B(x)} Y(x)-, 



where B(x) = c R(x) + B(x). 



The characteristic values of the system thus modified are X = X&— c 

 and it is clearly always possible to choose c so that X = is not a 

 characteristic value. No loss of generality is entailed, therefore, by 

 the assumption, which will be made, that Xa- =£ for any k. 

 Writing the equation (46) for X = Xa- in the form 



Y w \x)< = B(x) Y (k \x)- +\ k R(x) Y (k) (.r)- 



and considering this as a non-homogeneous equation we have from 

 page 67 



6 



(73) Y (k \x)- = \ k Ja(x, t) R(t) Y (h \t)-dt, 



a 



where G(x, t) is the Green's function for the system 



