THE SUMS OF PERIODIC SERIES. 559 



will be more convergent than j f{x') sin (ix'dx, and the latter integral will be convergent, and 



■'(I 

 its convergency will remain finite* when /3 vanishes. In this case also we may put a = 05 . 



Thus if /(.t) = sin Lv{h- + .r^)"', we may put a = eo because /(.r) has its integral essentially 

 convergent: if /(.r) = {b + x)-h, we may put a = 00 because /(«) is always decreasing to zero 

 as its limit. But if /(x) = sin Iw {b + x)-k, the preceding rules will not apply, because /(.t), 

 though it has zero for its limit, is sometimes increasing and sometimes decreasing. And in fact in 

 this case the integral in equation (51) will be divergent when /3 = /, and 0(/3) will become infinite 

 for that value of /3. It is true that /(a) is still the limit to which the integral (53) tends when 

 h vanishes; but I do not intend to enter into the consideration of such cases in this paper. 



33. When m may be put for n, and /(.r) is continuous, we get from (55) 



(- 1)=<^.(/3) = (i'-cpi^) - -/S'-'Ao) + -/s-'-T'co) - ... + (- ly-fir-'io) (57). 



In this case <p{fi) will admit of expansion, at least to a certain number of terms, according to 

 odd negative powers of /3. If we suppose d)((3) known, and the expansion performed, so that 



<p O) = //„i8-> + i/./3-^ + i/,/3-= + ... 

 and compare the result with (49), we shall get 



/(0) = ^i/„; /"(O) = - |/f,; riO) = ^//,; &c (58). 



34. The integral 

 " \// (/3) cos /3a; d/3, (59), 



/ 



I 



where '^((i) = ^ f f(x') cos jix'dx', (60), 



which is analogous to the series (22), is another in which it is sometimes useful to develope a function 

 or conceive it developed. For positive values of x the value of (59) is the same as that of (50). 

 When X = the value is f(0) ; and for negative values of x it is the same as for positive. It is 

 supposed here that the integral (59) is convergent, which it may be proved to be in the same 

 manner as the integral (50) was proved to be convergent. 



Suppose that we wish to find, in terms of \//(/3), the developement of /''(,?;) in a definite 

 integral of the form (50) or (59), according as fi is odd or even. We cannot differentiate under 

 the integral sign, because the resulting integral would be divergent. We may however obtain the 

 required developement by transforming the expression \//(/3) by integration by parts, just as before. 

 We thus get for the case in which /i is odd 



(- iy~<pA0) = /3''>/'(/3) + -/S"-' SQ sin /3a + - /S""^ SQ, cos /3a - ... 



+ (-1) ^ - SQ^_,sin/3a, (fil), 



7r 



where 0^(/3) is the value of <p((i) in the direct developement of/''(.r) in the integral (50). In the 

 same way we may get the value of \|/^(/3) when fx is even, >|'^(/3) being the value of \J/(/3) in the 

 direct developement of /''(a?) by the formulae (59), (60). 



Sec- ni^xt Section. 

 ■tc2 



