524 PROCEEDINGS OF THE AMERICAN ACADEMY. 



provided that x lies between C and Co, since the function t^'(x) g'^{x) 

 is analytic in the ring C, Ci and continuous along its boundary. Thus 

 we may write 



w /-(■>■) = ,T^i>'«^^»^^T5^'* 



~*~27rV— iJci t — x 



The first integrand on the right-hand side is continuous in x and t, for 

 t on C and x in the ring C, C2 unless x = t, when the integrand is not 

 defined; in the neighborhood of a pair of values x = t, the integrand 

 remains finite by (1). Hence the first integral approaches a continu- 

 ous limit as x approaches the boundary. Inasmuch as t is restricted 

 to lie on Ci in the second integrand, the same statement is certainly 

 true of the second integral. Hence /~(.r) may be so defined as to be 

 continuous along C. 



Likewise by means of the relation 



(« .n.)r(.)=,-^/;-^ 



dt 



x 



i'-"'V)s*(f) 



2 IT V— 1 Jci t — X 



valid for x between C and Ci by Cauchy's integral formula, we obtain 



(6) r ix) - r%v)gXv) a{x) = ^ f r^' (t) g- (t) ^^^^-^^"^^ dt 



2 T V — Wc t — x 



2 IT V— 1 Jcl t — x 



From this equation we can at once infer that/+(.r) may be so defined 

 as to be continuous along C. 



Thus f^{x) and f~{x) are respectively regular inner and outer func- 

 tions. 



A comparison of the relations (4) and (6) which are both valid along 

 C gives us the fundamental equation 



(7) r (x) - f- (x) = r^ (.r) g^ (.r) a (x) along C. 



Let us now consider the maximum modulus of f~(x) outside of or 

 along C. This maximum modulus, and likewise that for (7+(.t) within 

 or along C, are attained on C of course. Suppose that we have along C 



(8) ■ \g'-(x)\gL. 



