OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
451 
e un II (n + m) 
so that in this case c n m = —.- A.. . — 
47 T Sill 111T 11 (71) 
We shall choose this value of c„ m for 
all values of n and m, thus obtaining a definition of the function V n m (/ 1 ) for all values 
of n and m real or complex ; P n m (p) is accordingly defined by the expression 
P/(p) = 
1 II (n + to) 
47t sin 7iir 2" II (n) 
- 1)<* | 
+, i +, fi —, l -,) 
(t 2 —l) u dt (4) 
for unrestricted values of n and m, the phases of the expressions in the integrand 
being assigned as in Art 2. In order that this function P„“ (p) may be a single-valued 
function of p we must suppose that a cross-cut is made along the real axis from the 
point 1 to—oo , so that the phases of p — 1, p + 1 in (p 3 — l) iw are restricted to lie 
between i it, the function is then, when we take into account the remarks which 
we have to make in the next article, a single-valued function over the whole plane so 
cut, the values at points indefinitely close to one another on opposite sides of the 
cross-cut being in general different. It should be observed that the integrand in the 
integral for a given value of p varies continuously in crossing the cross-cut which has 
no reference to the variable t, but applies to p only. 
When p in such that mod. (1 — p) < 2, we have 
-r. , , sm in it „ , . / 
P n l (p) = ■■ - n (m — 1 ) ( 
p 4- 1 
\p — 1 
\bl 
— n, n + 1, 1 — m, 
_ i _p +jv* F 
n ( - m) v - V 
— 7i, n -f 1, 1 - m, 
(5). 
The formula (5) represents the function P/ d (p) over that part of the plane which is 
contained within a circle of radius 2 with its centre at the point p = 1 ; this function 
can be analytically continued over the whole plane and (with the cross-cut) the 
function so continued is uniform, and is given by the definite integral formula (4) 
which affords a general definition of the function. 
When m is an integer positive or negative, the expression (4) can be simplified ; in 
this case the integrand returns to its initial value after a Dosit.ive turn round each of 
the points p and 1, denoting the parts of the integral taken round Ca/3C, CySC 
(fig. 1, Art. 2) by P and Q respectively, the complete integral is 
P _}_ Q _ P e 2>m _ Qgfn+ ),«,+ 1)2*^ 
or 
(i - e 2nm ) (P + Q); 
now P + Q is the integral taken along a curve which encloses both the points 
p and + 1, and is described positively, hence, in the case in which m is an integer, 
the formula (4) becomes 
3 m 2 
