EXPANSION PROBLEMS. 405 



and p = re** which remains finite uniformly in x and 6 when r becomes 

 infinite. We shall use the symbol with this meaning from now on; 

 the functions \{/ and E will be different in different cases, of course, and 

 are to be replaced on occasion by quantities independent of the vari- 

 able X. The solutions described in (31), regarded as functions of x, 

 are continuous with their derivatives of all orders. Their analytic 

 character as functions of p need not be further specified, for the deter- 

 mination of the characteristic values does not have to be repeated 

 for the s.ystem (29), (30). It follows from a general theorem ^^ 

 that the characteristic values for this system are the same as for the 

 system (15), (16), and that the number of linearly independent char- 

 acteristic solutions corresponding to any one characteristic number pn 

 is the same for both systems. From a certain point on, the system 

 (29), (30), has just one characteristic solution vn(x) for each number 

 pn, and the corresponding coefficient in the formal expansion of a 

 function f{z) is 



/^ f{x)Vn{x)dx 



(32) -^ ^ 



/ Un{x)Vnix)dx 



We shall show how to define analytic functions /(.r) for which the 

 general term anun{x) does not remain finite. 



Except for an arbitrary factor independent of x, which will be chosen 

 later, the characteristic solution V7i{x) may be represented by a de- 

 terminant made up as follows: The t-th element in the first row is 

 Zt (x, pn), which, by (31), has the form e-"""""" [1]. The ^th element 

 in the 5-th row, s = 2, 3, . . . , v, is Vs{zt), or 



{-PnWtf''e-p"'^t''{l]. 



On division of the s-th. row by (—pn)^', s = 2, 3,. . ., v, and multi- 

 plication of the t-th. column by e''"^t'^, t = 1, 2, . . ., v, this determinant 

 is reduced to the form 



gPnWliw—x) Ml gp„W2 (v— X) Ml ... gPnWpi-K— X) Ml 



Wi^'-[1] wo^-'^l] ••• w/"-[l] 



31 See Birkhoff II, pp. 375-376; Bocher, loc. cit., p. 407. 



