78 BIRKHOFF AND LANGER. 



as are desired may be used. These series are not in general con- 

 vergent. Nevertheless the formal matrix 



S(x)= Po(.r)+^ Pi(x)+ Ie(z) 



(i) 



c xry O) J p g) + Pi + | 



which is found in this case, formally satisfies the equation (26). 



With the convention that the term in -r is to be omitted if k = » 



X* 



formula (32) holds in all cases. S(x) will be called a formal matrix 



solution of equation (2(3) regardless of whether k is, for the case in hand, 



finite or infinite. 



It should be observed that each plj is not wholly determined but 



contains a single arbitrary constant of integration, independent of x. 



This arbitrariness corresponds to the fact that any convergent power 



series Cj in ( - ) with constant coefficients may be multiplied into each 



column of S(x) without thereby destroying its property of being a 

 formal solution in the sense (32). 



Using the notation [ipa]k, or [^ ], for an expression of the form 



where \p is bounded for | X | large, we have 



X 



S b.-dx 



S(x) 



It is clear that an alternative form is 

 (33) 



)£(.r). 



j(x) = J bji 



where Bj(x) = J bjj dx . 



a 



By similar considerations the equation Z' = — Z{A\ + B) may 

 be transformed and a formal matrix solution T(x) for the resulting 

 equation 



