OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
477 
If in this expression we put u — hz, and make h the independent variable, 
we have 
Q/ (a) = Le 0ll - n)m . 2 m . ^ ( ° + ’ 3 ~’ ° ^ 
^ v ' 7 47r sin (n + m) 7 r v 7 
In the particular case m — 0, we find 
JjJi+m 
(1 — 2/x/i + 
( 40 )- 
V 7 4 Sill 717T J 
h' 1 
(1 - 2 fill + h~y 
. . . . (41). 
Using the theorem (12), we deduce from (40) the formula 
Q/‘ (fi) = L e {m ~ n)ni . 2 ~ m 
n ( n + m ) n (- m - j) n (- i) , 3 __ r(i+.°+. 
IT (vi — m) 47r sin — ?n) it ] 
o—) h n ~ M 
(1 — 2 fill + 7i 2 ) 5_m 
cZ/i 
(42). 
It will be observed that in the formulm (40), (41), (42), the phases of the 
integrand are to be measured as follows :—Draw the figure in the A-plane corre¬ 
sponding to the figure we have drawn in the w-plane ; the points 2 , — correspond to 
the points z 2 , 1 respectively; the initial phase of h at A is to be the same as that 
of and will therefore be zero at the first passage through C; the phase of 1 — hz 
in the product 1 — 2 [xh + h z , which equals (1 — hz) ( 1 — — j, will be initially zero 
h \ 
at A, and that of 1-— will be zero at B. When the figure is displaced in any 
manner the phases can be found from the foregoing specifications by means of 
the principle of continuity. 
23. If the real parts of n + m + 1 and \ — m are positive, the integral in (40) 
can be reduced to the form 
(1 
, (n + m) 2 m 
) (1 - e~ (m+i)2n ) (* 
J n (1 
h n+m 
(1 — 2 fx,h -f A 2 ) 
m+f 
~dh, 
