[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 115 
we find that 
lee, Ji - Dis RU Re eu ES Tes ce (36) 
sin vr > Sie Se 

a result given in N. p. 68, but proved in an entirely different manner. 
"GC 
vers ta CCE 2 Sas Se with «simple poles at 
ee 
v=0, +2, +4, ..., and apply the same method as that just used, 
we find that 

? sin = > 


| © (—1) 2 
= 2 es 37 
Jnty Jar Pr. ane iv Ast Ja Je Ne ( ) 
(N. p. 68, ak the factor (—1)5 here inserted has been accidentally 
omitted in Nielsen’s book). The method by which Nielsen arrives at 
(36), (37) leads him to describe (37) as a very peculiar expansion when 
compared with the expansion (36). Our method has the advantage of 
showing that the two expansions are naturally and simply related. 
§ 15. The special case of (37) furnished by putting v=1, #n=0, 
namely 
ce il 
: i Ds fr > 2 
sin 2x =2x J+4x 2 ae Ie (38) 
can be made to lead to a companion formula for 1—cos 2x. 
Write (88) in the form 
sin 2x = 2x | (Jo JA) +4 (J%1— Je) +4 (PJ) + ...] 
and let v denote the expression within square brackets. 
It is easy to prove that 
| 6 (eee, cae i re ee ae (real part ofy>0) (39) 
For D, (2 J,.,J)=J,.,4422,_,J,4+25,., 7, 

JS, J x =) 
xX 
V5 x Le (Gee so a =) 
x 
= Ei J Sy SA 
Integrating both sides of (38) we infer that 
| Rudi x (Wo Intel la ty te s+.) 
o 
