THE SUMS OF PERIODIC SERIES. 56.5 



TT 



and since « evidently vanishes when A = co , we have C = — , whence 



TT A TT 



« = tan ' - , u = ~ 



2 n 2 



TT 



2 



.t sin ax 



Also M = when a = 0, and m = when a is negative, since u changes sign with a. By the 



value of II for a = 0, which is asserted to be 0, is of course meant the limit of f' — da: when 



Jo ^ 



a is first made to vanish and then X made infinite. 



It is easily proved that the convergency of the integral n. becomes infinitely slow when a 

 vanishes. In fact if 



, r" sin ax 

 u = \ dx, 



J-f X 



we get by changing the independent variable 



, /- " sin cT , 



ti = / dx • 



Jax •■^ 



Jr * sin X 

 ' d 



I 



an integral which might have been very easily proved to be greater than zero even had we been 

 unable to find its value. It readily follows from the above that if m' iias to be less than e the value 

 of X increases indefinitely as a approaches to zero. 



42. Consider next the integrals 



r'^ cosaxdx /•« , cosaxdx 

 «= / r , «= / e-"'- ^- (72). 



^ 1 + x' J„ 1 + x^ 



It is easily proved that the convergency of the integral v does not become infinitely slow when 

 h vanishes, whatever be the value of a. Consequently u is in all cases the limit of v for A = 0. 

 Now V satisfies the equation 



d?v /■« . h 

 - -1. = -/ e-'"coiiaxdx = - ; (73). 



da^ J„ li + a 



It is not however necessary to find the general value of r ; for if we put A = we see that u 

 satisfies the equation 



:r^-M = 0, (74), 



da' 



so long as a is kept always positive or always negative : but we cannot pass from the value of u 



j found for positive values of a to the value which belongs to negative values of a by merely writing 



- n for ffl in the algebraical expression obtained. For although u is a continuous function of a, it 



readily follows from (73) that -— is discontinuous. In fact, we have from this equation 

 •^ ^ da 



_ - = / wda -2 tan-'-. 



\dal„^^ Vrfa/„..* J^K '' 



Now let A first vanish and then X. Then v becomes u, and f vda vanishes, since v does not 



Vol.. VIII. Pabt V. 4D 



