794 
MR. J. TV. L. GLAISHER ON RICCATI’S 
r x *»-2 e - x '- b i dx=a?“~ 1 ^ a J dx ; 
Jo x/ttU (a 2 + x*f 
2 b 
whence, replacing — by b, 
f cos , x — ri - 2»+1 
J o (a 2 +£ 2 )® T(n) 
3 o t „ a.-b- 
2 a—2 7 
x e ix dx . 
■ ■ (15) 
\/ 1T _ 
=rw a 
M ir(»-i)]i + ~ab- 
(2n-2)(2n-4) (abf 
{<zb\% n ^ 
+m-n+mT 
i+ 
■ab-\- 
(2n-2)(2n-3) 21 
(2n — 2)(2n—4)(2n—Q) (abf 
^ (2n-2)(2n-3)(2n-4) 3! 
2n(2n + 2) (abf 
_ 1 \Ar 
~^T(n) 
2n(2n + l) 2! 
2n(2n + 2)(2n + 4) (abf 
+ 2?i(2» + l)(2?i + 2) 3! 
+ &c. 
,—ab 
(2n—2)(2n — 3) 2! 
-b&c. 
+ r(_n+i)(i6f'- 1 {l+ f^+ 
2n(2n + 2) (abf 
2n(2n + l) 2! 
+ &c. \ e ab 
which represents the value of the integral for all values of n greater than unity. If 
n is a positive integer the first series terminates through the presence of a zero factor 
in a numerator, and this finite series is the value of the integral, the second series 
being ignored. 
If n is a positive integer, then, writing the terms of the series in the reverse order, 
r cos bx 7 . \/ir „, ,. „ ,, 
——— dx=\~rT(n—\)cC* n+1 
l 0 (a~+x~) n 2 T(n) v 2/ 
(n — 1)! (2 ab) 
(2 n — 2) . . . n (n — 1)! 
(,, ~ 1> --- 2 ^ • +~^) + l}e- 
(278.-2) . . . (71 + 1) 0-2)! 
x/tt b n ~ l (n— 1)0—t) ■ • • i-\Z 7r 9/l -: 
T(?i) a n 
(2n—2) . . . n 
0 14 -n(n — 
+ o+iQQ- 1)0-2 ) /yy 
2 ! 
(2n-2)\( 1 V t-1 l 
ab 
\2abJ ’ (n — 1)! \2abj 
IT 
1 b n ~ l \ n(n — Y)(l\ (n+V)n(n—Y)(n— / 1 \ 3 
TO) 2” a n 
which agrees with (14). 
1 + 
2 \ab 
+ 
2.4 
ab) + &C -^ * 
