BOUNDARY PROBLEMS AND DEVELOPMENTS. 109 



Section IX. 

 The Convergence of the expansion. 



From the distribution of characteristic values as found in section 

 VII, for both the cases of regular and irregular conditions there dis- 

 cussed, it becomes readily apparent that given any constant iV suffi- 

 ciently large it is always possible to choose a circle C whose center 

 is at X = 0, and whose radius both exceeds the value N and is such that 

 the distance from any point of C to a characteristic value is greater 

 than some fixed constant 8 > 0. 



We shall consider the convergence of the contour integral in formula 

 (95) as the contour of integration is taken successively as a larger and 

 larger circle of the type C above. With the size of the circle the num- 

 ber of characteristic values which it includes and hence the number of 

 terms of the series which are summed by the integration may be in- 

 creased indefinitely, the limit of the integral for | X | = °° , being, if it 

 exists, the sum of the corresponding series. 



Let us recall the hypotheses already made concerning the functions 

 7i(x). We have 



(96) 



(i) 7i(x) * 



(ii) 7i0*0 continuous a ^ x ^ b (page 72) 



(iii) if n ^ 2 y { (x) =j= y,{x) for i =f= j 



(iv) if 7i ^ 2 arg{7i(.r) — y,{x)} = // l7 (page 83) 



To these we shall add 



(97) arg 7»(.r) = Hi (a constant), i = 1, 2,. . .n. 

 Concerning the vector to be expanded we shall assume 



that the elements fi(x), i = 1, 2, . . .n, consist in the interval 



(98) a ^ x ^ b, of only a finite number of pieces each of which is 

 real, continuous, and has a continuous derivative. 



It will readily be seen from (96) and (97) that we have restricted 

 each yj{x) to vary along a ray in the complex plane. Moreover, if 

 n ^ 2 the dependence of y 3 (x) upon x must be such that the slope of 

 the line joining any two of these points remains constant. This 



