84 BIRKHOFF AND LANGER. 



/ 





u 



is less numerically than fe ( * r) if. However, if R{\{y h (£) - y r {$}\ > 

 for X within the sector and any £, it is so for all £, and, provided that 

 t ^. x, the integral 



f Yi n ®-T r <*>}« (Ar)/7 . 



b 



is similarly bounded. Consequently if K/, r (X) is defined by the rela- 

 tions 



K hr = a if R{\{y h (£) - y r (M)}} ^ 0, K hr = b otherwise, 

 the numerical value of each element of the matrix 



+(x, X) = ( I J e* o^ dt\ 



n 



is clearly less than KM, where K = (b — a) 2 fc ( , for X within the 



sector and [ X | > N. Q. E. D. 



Assuming the limits chosen in the manner above, a $(x, X) corre- 

 sponding to each sector and to each U is uniquely determined. More- 

 over, we have 



(44) U(x) = S(x) CT(x) - ± *(z, X). 



A 



Consider now the particular solution, Yo(x), of equation (26) which 

 satisfies the relation Y (a) = S(a). Since S(x) as well as the coefficients 

 of equation (26) are analytic in X, Yo(x) is likewise analytic in X. 

 Moreover we know from page 56 that every solution of the equation is 

 of the form Y(x) = Y (.v) D, where the elements of D are constants 

 with respect to x. Substituting this form of Y(x) into equation (41) 

 and fixing x and solving for C we obtain the relation 



1 To . 



C = r(.r„) | l'„(.ro) + ^ S(.r„) T(t) *(/, x) r»(0 tit { D, 



