sin x 2 dx. [568 
no ground for doubting 
t and ultimately become 
+ ^7r. Writing herein 
f- hir ; or, putting herein 
n the theory of elliptic 
be symmetrical in regard 
dent on the form of the 
y = + bk, then the val ue 
te sides; and so if the 
^ral will depend on the 
“ On the inverse elliptic 
aginary parts might have been 
l that course. The like remark 
/“GO Ç oo 
568] NOTE ON THE INTEGRALS COS OC 2 clx AND I sin X 2 dx. 61 
Jo Jo 
functions,” Carnb. Math. Jour., t. iv. (1845), pp. 257—277, [24]; and “Mémoire sur les 
fonctions doublement périodiques,” Liouv. t. x. (1845), pp. 385—420, [25]. 
A like theory applies to series, viz. as remarked by Cauchy, although the series 
A 0 + A 1 +A 2 +... and B 0 + B 1 + B 2 + &c. are respectively convergent, then arranging the 
product in the form 
A 0 B 0 + A () B l + A 0 B 2 + ... 
+ AiB 0 + A 1 B 1 + A^B 2 + ... 
+ A. 2 B 0 + A 2 B l + A 2 B. 2 + ... 
+ ..., 
say the general term is G m>n , then if we sum this double series according to an 
assumed relation between the suffixes m, n (if, for instance, we include all those terms 
for which m 2 + n 2 < k 2 , making k to increase continually) it by no means follows that 
we approach a limit which is equal to the product of the sums of the original two 
series, or even that we approach a determinate limit. 
Mr Walton, agreeing with the rest of the foregoing Note, wrote that he was 
unable to satisfy himself that the value of /: e ix2 dx is correctly deduced from that of 
e (-a+U)yyti-x^y Writing n=-\, the question in fact is whether the formula 
roo Jirr) gl 1 '® ( b \ 
e (-a+bi)y v ~h dy _ y \ = tan -1 - , angle between i7r, — hir , 
Jo (a 2 + b 2 y\ a / 
which is true when a is an indefinitely small positive quantity, is true when a = 0; 
that is, taking b positive, whether we have 
f 00 .. , 7 f(ir)e iin 
j 0 e ' h ^ d y = ^m~‘ 
Write in general 
U = I e (~a+bi)y y-h dy } 
Jo 
then, differentiating with respect to b, we have 
^ = [ °° iif e ( ~ a+U)y dy, 
db Jo 
or, integrating by parts, 
— = e ( ~ a+U)y — —-—T-7T [ y~$e ( ~ a+bi)y dy, 
db -a + bi J 2(—a + bi)J 0 J