76 C. V. L. Charlier 



so tliat 



so that 



1 ■ 12 14 16 



This series is divergent. 1 shall return to this question below. 

 The general integral is obtained through adding 



Ä COSX -\- B s\n X. 



It is, however, ^ = 5 = 0. To prove that we deduce the series (183) directly 

 from the integral 



00 



r~~^^ dt 

 ^1 I ,2 ■ 



Put 



so that 



1 + 



0 



xt = z : . xdt = dz 



(184) = X 



e dz 

 x^ -\- z"' 



0 



For large values of a; — with which we are here concerned — we may (at 

 least formally) develop into powers of z and put 



00 



(185) '!^[x) = -\d,e + 



/y I 1 /yi^ ^yi* /ytV 



\Åj I ' lA/ lA/ 



0 



or 



1( 2 4 6 

 (185*) '1>(.^)-- + 



1 --y»'-' -y»^ ^ytw 



t/y I (Ay lA/ lA^ 



which is the expression (183). 



This series is evidently divergent for all values of x. It may, nevertheless, 

 be used for numerical computations, if x is rather large. We have, indeed, 



1 le dz 



x J X 



o 



