248 REPORTS ON THE STATE OF SCIENCE, ETC. 



The Sine and Cosine Integrals, Si (x) and Ci (x). 



For large values of the argument x, the Sine and Cosine Integrals were calculated 

 from the asymptotic expansions of these functions. 



HW -5-^(i-3+2-S+ ) 



2 x \ x 2 x* x J I 



_sina;/l!_3! + 5!_.7! \ 



X \X X K X r > X 1 J 



, rr/ . sina;/, 2! , 4! 6! , \ 



andCi.r= (1- + -_4- 



x \ x 2 x* x a I 



_coaa!/l!_3! Sj_7! \ 



X \ X ' X ' X r ' X 1 I 



For the series, 1— —A — \ 'A- when 2e=2»4- a, the 'converging factor' is 



X 2 x i x b 



l+j (l-2a)-JL(l_6«)--i 3 (34>20<x-a 2 -2a*) 



2 4w 8n 2 16n 3 



and for the series -— _"4-4— _" + , when 2*=2» + ot f -— , (1 4- 2a) 



* x* x' x 1 2 4m 



+ Q 3 2 (1+a)- 1 (l3+4«-7a' 2 -2a 3 ) 



8w 2 16?r 



The following tables of Si (x) and Ci (x-) from a:=5-0 to 20 - have been employed in 

 tabulating derivatives of Bessel functions,* viz., 



[§v Jv(X) l=i =Ji(a:) Ci {2X) ~ J " i{x) S ' (2X) 



and [^ • Jv(^)] v= _ i =J-lW Ci (2*) + Ji(*) Si (2*). 



Schafheitlinf also gives an interesting relation between these derivatives and the 

 Sine and Cosine Integrals. 



* Ansell and Fisher. " Note on the numerical evaluation of a Bessel function 

 derivative." Proc. Lond. Math. Soc, vol. 24, 1926. 



t Schafheitlin. Sitzungsber. Berlin. Math. Oes., vol. 8, 1909, pp. 62-67. 



