THE SUMS OF PERIODIC SERIES. 56l 



When f(x) is continuous, and eo may be put for a, we have from (61) 

 (- 1) ' %ili) =/3'*x^(m) + -/B'-V'Co) - -/3''-V"'(o) + ... + (- 1) ^ - /3r-^(o). ... {63). 



IT TT TT 



If \|/(;3) be given we can find the values of /'(O), /'"(O) ... just as before. 



36. The integral 



- r f° cos (i (a;' - x)f(w')d(idx', (64), 



T ■'o ''-II, 



in which the integration with respect to ,t' is supposed to be performed before that with respect 

 to /3, so that the integral has the form 



/'"x(j8)cos/3a;rf/3 + r"o-(/3) sin/3a;d/i, (65), 



may be treated just as the integral (59) ; and it may be shown that in the same circumstances we 

 may replace the limits — a^ and « by - co , + co respectively. If we suppose \{(i) and (t(/3) 

 known, we may find as before the values of ,t? for which /(.r), /'(«) ... are discontinuous, and the 

 quantities by which those functions are suddenly increased. We may also find the direct develope- 

 ment of/'(.r), f"(x) ... in two integrals of the form (65); and we may if we please clear the 

 integrals (65) of the part which renders /(.v), f'(x) ... discontinuous. 



37. In the developement of f(a) in an integral of the form (50) or (59), or in two integrals 

 of the form (65), it has hitherto been supposed that f{a;) is not infinite. It may be observed 

 however that it is allowable to suppose /(.r) to become infinite any finite number of times, provided 

 Jf{x) dx be essentially convergent about the values of x which render f{x) infinite. This may be 

 shown just as in the case of series. Hence, the formulse such as (55) which give the develope- 

 ment of /''(a;) are true even when/*'(.T) is infinite, f'^''(x) being finite. 



SECTION III. 



On the discontinuity of the sums of infinite series, and of the values of integrals 



taken between infinite limits. 



38. Let u, + u.. ... + u„ + (66), 



be a convergent infinite series having U for its sum. Let 



M, + W2 ... + v„+ (67), 



be another infinite series of which the general term v„ is a function of the positive variable A, and 

 becomes equal to u„ when h vanishes. Suppose that for a sufficiently small value of A and all 

 inferior values the series (67) is convergent, and has V for its sum. It might at first sight be 

 supjjosed that the limit of V for h = was necessarily equal to U. This however is not true. For 

 let the sum to n terms of the series (67) be denoted by /(n, /«) : then the limit of V is the limit 

 of fin, h) when n first Ixcomes infinite and then h vanishes, wliereas U is the limit of f(n, h) when 

 h first vanishes and then n becomes infinite, and these limits may be dift'erent. Whenever a dis- 

 continuous function is developed in a periodic series like (15) or (30) we have an instance of this; 

 but it is easy to form two series, having nothing to do with periodic scries, in which the same 



