CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 



171 



where A, B, C... a, /3, 7... are constants which have to be determined. Differentiating, and 

 substituting in (11), we get 



a (a - 1) An"' 2 + 8 ()3 - l) B^" 2 + ... 



2^/3 



{(4a + 1) An a ~i + (4/3 + 1) BnP-l + ...} = 0. 



As we want a series according to descending powers of n, we must put 



4a + 1=0, /3 = a - §, 7 = /3 - 



4. 



a(a-l) . „ , 8(13-1) T 



B=2<\/-3 ~ -A, C=2\/-3 ^ — '- B. 



whence 



4/3+ 1 



t V--7 *•< 



47 + 1 



V^T 



l6.v/(3rc 3 ) 

 1.5.7.11. 13. 17 / \/ - 1 



1^5 .7.11 / </-! \ 

 + 1.2 \l6y/(3n 3 )) 



1.2.3 



1 16^/(3^)) + "T 



(14) 



>l6y/(3n z \ 



By changing the sign of v — 1 D °th m the index of e and in the series, writing B for A, and 

 adding together the results, we shall obtain the complete integral of (ll) with its two arbitrary 

 constants. The integral will have different forms according as n is positive or negative. 



First, suppose n positive. Putting the function of n of which A is the coefficient at the 



second side of (14) under the form P + \/ — 1 Q, and observing that an expression of the 

 form 



A(P + y/^lQ) +i(P_v'~lQ), 



where A and B are imaginary arbitrary constants, and which is supposed to be real, is equiva- 

 lent to AP + BQ, where A and B are real arbitrary constants, we get 



1 / 2 /n 3 2 /n?\ ,/ .2 /ri> „ 2 /w 3 \ , „ 



U=An~i LRcos-V - + ^sin-\/— + Bn'l LB sin- V Scos- \/ -\ , . (15) 



\ o v 00] \ o o o o / 



where 



B = l 



S = 



1.5.7.11 1.5.7.H.13.17.19-23 

 + 



1.2. lo^ra 3 ' 1.2.3.4.16 4 . 3 2 « 

 1.5 1.5.7.11 .13.17 



. (16) 



l.l6(3ra 3 )* 1 .2.3.16? (3^)1 



seeing in Moigno's Repertoire d'optique moderne, p. 189, the 

 following formulae which M. Cauchy has given for the calcula- 

 tion of Fresnel's integrals for large, or moderately large, values 

 of the superior limit : 



y o cos-H« 2 rf» = 2- JVcos^m 2 + M sin^m*; 



f 



r 



sin ■* a 2 da = s - JWcos -5 m 2 - iVsin ■= m 2 ; 



where 



1 



1.3 , 1.3.5.7 



M= „— , H 0—5— 



1.3.5 



jf= : . h — 



m 3 ir a m 7 it* 



The demonstration of these formulae will be found in the 

 15th Volume of the Comptes Rendus, pp. 554 and 573. They 

 may be readily obtained by putting irx'—2x, and integrating 

 by parts between the limits \ irm 2 and co of x. 



22—2 



