OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
501 
If rti = 0, we have 
pf0 _ 
Q„ (/x) = (/x — A/x 2 — 1 cosh w)" dw .(88). 
J 0 
It is interesting to compare (87) with the formula obtained by changing m into 
— m, in (85), 
Q ’ M 
TT/ Jf\ /.CO 
_ e m m . 2 m . - ni 2 cos mir. (/x 2 — 1 )“*“ {^u, —(— \/p, 2 — 1 coshw}® - ” -1 sinh -2,/ Or dw (89), 
II( f) Jo 
which holds under the same conditions as (87). 
In (87) change m into — m, we have then 
Q-" (/*) 
= e 
7)lTTL O 'ill 
n (n+m) n(-i) 
II ( n—m) 
. 0<•' — !)-“ j (p, — \//x 3 — 1 cosh w) n “ sinh 2 ' ; Oc dw (90), 
which holds when the real parts of n — m -f- 1, ^ + m are positive. 
41. In the formula (9), change n into — n — 1, we then have 
Q_._ 1 *W = 
n (») 
,.. + , j 
(- 1 +, 1 -) 
where 
4( sin (n — in) 7r ’ IT (« — m) 
X = (/x 2 - If ' 1 (£ 2 - I )'”" 1 (f - /x)' ! 
Xefc, 
Place the path as in the figure; starting from A, a semicircle of centre /x is first 
described, then a straight line from E to oo, a semicircle of infinite radius, then a 
straight path from oo' to A, followed by a similar path taken negatively round the 
point + 1. 
oo 
