CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 169 



be convergent. But this is not sufficient in order that we may be at liberty to assert the 

 equality of the results obtained from (2), (4) by putting 9 = — before integration. It is more- 

 over necessary that the convergency of the integral (2) should not become infinitely slow when 9 



IT 



approaches indefinitely to — , in other words, that if X be the superior limit to which we must 



integrate in order to render the remainder, or rather its modulus, less than a given quantity 

 which may be as small as we please, X should not become infinite when 9 becomes equal to 



— *. This may be readily proved in the present case, since the integral (2) is even more con- 

 6 



vergent than the integral 



•'o 

 which may be readily proved to be convergent. 



7T 



Putting then 9 = — in (2) and (4), we get 



u 



where 



eo » 



= f cos (a? - nw) dso - %/ - 1 f sin (# 3 - nw) dw, . . (5) 



u= fcos^-v/~sinZn yV^-J^d*, ... (6) 



p = (cos - + •%/ - 1 sin -jw (7) 



Let 



u= U-V-l W, 

 and in the expression for U got from (5) put 



•■(f)* 1 ' n= (J) m; (8) 



then we get 



W 



-(!) * (9) 



4. By the transformation of u from the form (5) to the form (6), we are enabled to differ- 

 entiate it as often as we please with respect to n by merely differentiating under the integral 

 sign. By expanding the exponential e?* in (6) we should obtain u, and therefore U, in a 

 series according to ascending powers of n. This series is already given in Mr. Airy's Supple- 

 ment. It is always convergent, but is not convenient for numerical calculation when n is 

 large. 



• See Section in. of a paper, "On the Critical Values of the sums of Periodic Series." Camb. Phil. Trans. Vol. vm. p. 661. 

 Vol. IX. Part I. 22 



