ON THE CALCULATION OF MATHEMATICAL TABLES. 87 
Bessel Functions of Half-integral Order. 
The solution of the ae ee 
Wz 3 +t 3 1— 7D = Un — 0 
being taken in the symbolical form 
uu. = a” of ( - 1 =) ae Sk Be* £. 
a xe dz xe 
yields as standard functions of real quantities 
: A Set 18 it d\" sin 
er 2 [= x an) Pit 
eer 1 d\" cos a 
Ol) =a ( 3 7) Pe, 
x eg ae Piagt nines elie 
with E,(%) = « ( i ss) aaa C" (x) — 78, (2) 
as an important associated function. 
The functions (KE, (x) )? = (S, (x) )?-+ (C, (#) )? 
(H,’ (2) )? = (Sy! (2) )? + (Cy! (2) )? 
are of importance, and have been tabulated with S, (x), C, (x), and their 
derivatives 8,’ (x), C,’ (a). 
The connection with Bessel Functions is apparent from the differential 
equation, giving 
8, (2) = V4raJ,,;(2) 
C, (w) = (—1)"/E x(x) J_,_, (2). 
From the differential equation, we obtain 
S.! (a) = Eo? 8, (@) S3 (2) 
8," («) = 8, -1(2) — 28, (@) 
with corresponding formule for C,/ (a), E,’ (2). 
By elimination of §,’ (x), we get the recurrence formula 
8, +1 (x) = eS S, (a) rae S., -1 (x). 
