[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 107 
1 
= | peta [-14 +940] 
1 1 
a ee — /2\a-1 : 
Da Ste [a Pja-1 (1+6) di 
a di 
7 2a ua ee le 
1 
= fan 
CIE 
This integral au cos? 1 @d6, and its value can be found at once 
o 



ue NÉS ER 
from the formula “cos odd= 7 , in which the real part 
2 T(E 
of vy is supposed to be greater than —1. Thus the value of the 
Vr (a) uC) 
integral micro MAN 
It should be observed that the formula (17) bears a close resem- 
blance to two formulae, of which the former at least has played a con- 
siderable part in modern analysis: 
1 1 = NOSE oe IGS 1) 
= = = ea SD) (Stirling’s series) 
© 71 AY _ 2 1 ylytt) yts-1) 
So NX VTS xs ts x(x+1)...(x+s—1) 
(Nielsen, Handbuch der Theorie der Gammafunktion, p. 83.) 
When a is a positive integer #, formula (17) becomes 
n(n—1) n(n—1) (n—2) (n—3) 

TE GC CENTER CE AR 
! ! 
= Q2n-1 = (18) 
(i) For even values of m an immediate proof is obtained by equat- 
ing the coefficients of x?” in the formula 
cosh 2x+1=2 cosh?x, 
thus getting 
eee 5 1 5. Nr ee 
on | Pre De On—2121 2n—4!4! ie 
1 ji : Ne 
+ Le + nl ‘ which, on multiplication by 
3n !n ! converts at once into (18). 
(ii) For odd values of # the same process must be applied to 
cosh 2x —1=2 sinh?x. 
