OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
491 
which the fourth element is --— ; when 2 = e*, this expression for P/ a (/ u) becomes 
-L 
approximately, when n is large, 
1 [IT (n + m) TI (%) sill (n + m) it 
~P„ m (cosh xp) — 
-n\Js 
J1T [ n ( n ) n (n + 4) cos mr y/ 2$ sink p 
1 — 4 to 2 e ^ 
4 n 2 sinli p 
+ 
IT (n) IT (n — \) 
^(n + l)\f/ 
II (n — to) IT (n) y/ 2e^ sink yfr 
1 + 
1 — 4m' 3 e * 
4n 2 sink p, 
except when n (supposed positive) is half an odd integer, the first term is very much 
less than the second, on account of the factor e~ n * ; hence 
y/ 
1 k- 1 
7 Til 
a 
8 n) \/l - e~W 
1 + 
1 — 4m 2 e~ 2 ^ 
4 n 1 — e -2 ^ 
or. 
II (n — to) t-, , , , x 1 
e n v 
-2i if 
3 to 2 v 1 — 4 to 3 1 
8 n n 4 n 1 — e~^ 
(65). 
The asymptotic value of 
II (n — to) 
n (n) 
1 e 
P,/" (cosh xb) is therefore - 7 = . 7 - 7 — . 7 , 
x r/ Vim■ \/l—e- z * 
except in the case in which n is equal to half an odd integer. 
From (61) we see that the approximate value of ^ Q/' (cosh xp), for large 
values of n is 
v 7 -71 
or 
nw 
n (w -f m) 
3 \ 
e -(ii+l)^ 
GO ' 
o* 
v/1 - c- 2 * 
/ 77 
g —(71+ l)if/ 
/ 71 
• yi-<r 2 * 
n (n + m) 
- 4to 2 e -2 ^ 
4?i 1 — e~ 2 ^ J ’ • 
. to 3 1 — 4m 2 1 
4:71 ' 1 —e -2 ^ 
( 66 ), 
It may be remarked that the semi-convergent expressions for Bessel’s functions 
J m ( x ), Y m ( x ) may be obtained from the series (59), (60), by putting 6 = x/n and 
proceeding the limit n = co . 
Expressions for P,'"‘ (g), as definite integrals taken along real paths. 
33. In (50) change m into — m, we have then 
7 r sec TO 7 r 
1 1 
27 ri' 2 » ! 'n(- 4 )n(TO-i) 
f(z + , 1 / 2 -) 
—- (g 3 ~ 1)"*™ h n ~ m (1 — 2g/( + h 2 ) m ~ 4 c7/j 
"9 J J 
3 R 
Q 
