BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 531 



If now X be made to approach a point .tq of C, and if y be taken as the 

 foot of the normal from .r to C, this term approaches a Hmit which is 

 continuous along C. In fact the factor {t — yY/it - x)^ remains finite 

 for t along C, and approaches the limit 1 uniformly save in the vicinity 

 of xq. Thus if X (and y also of course) approaches xq, 6?^^-^$ {x)/dx'^~^ 

 approaches a limit along C continuous for an arbitrary .tq. 



A similar proof shows that ^(.r) has derivatives of all orders, con- 

 tinuous outside of and along C. This proof is based on the fact that 

 the elements of A~'^{x) satisfy the restrictions placed on the function 

 a{x) in § 1. 



The above results also lead to the conclusion that at any point 7 of C 

 at which one or more elements of A{x) fails to be analytic, the ele- 

 ments of $(.r) and ^ {x) admit of asymptotic expansion in a series in 

 positive integral powers of x — 7. This is an immediate consequence 

 of an expansion like (18) for $(.r) or ^{x) in which now t and ?/ can 

 be any points within or without C respectively, and B {t, y) is contin- 

 uous in t and y.^ 



§ 6. ^ Normal Form for <J>(.r) and ^{x). 



By a series of simple normalizations it is always possible to obtain 

 a solution 4>(.r), ^(x) of (17) such that$(.r) is (as before) a matrix of 

 regular inner functions, and ^(.r) is a matrix of function analytic, 

 without C in the extended plane except for a possible pole at x = 00 , 

 and furthermore such that | $(.r) | does not vanish icithin or on C, and 

 I ^(.r) 1 does not vanish without C. 



This solution may be directly obtained from that found in § 4. 

 If \^{x) I vanishes a,tx = c within C say, we can determine a matrix M 

 of constants such that all the elements of the first row of M <l>(.r) 

 vanish at x = c, while \M \ 9^ 0. Now il/$ (x), M^ (x) yield a new 

 solution of (17), which has the properties given for $(.r), ^(.r) in § § 4, 5. 

 If we divide the elements of the first row of M $(.r) by x — c, we obtain 

 a matrix $(.r) of functions analytic within C and continuous along C; 

 if the same operation be applied to "^{x), we obtain ^(.r), a matrix 

 of functions analytic without C in the extended plane save for a pos- 

 sible pole atx = 00 . Moreover $(.r) and '^(.r) yield a solution of (17), 

 since the effect of this operation is to alter the matrix on either side 

 only in the removal of a factor x — c from the first row. By this 

 device we have diminished the multiplicity of the zero of | 4>(.r) | at 



8 For the relation between the existence of derivatives and of asymptotic 

 series see W. B. Ford, Bull. Soc. Math. France, 39, 347-352 (1911). 



