OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
511 
A particular case of (102), or (103) is Mehler’s form of one of Dirichlet’s 
expressions for P ;i (cos 6), 
P. (cos 0 ) = — f - 
7T J 0 2 
COS (ft. + I) 0 
(cos (f) — 2 cos 0)* ‘ 
The formula (103) holds for all values of n and m, real or complex, provided the 
real part of m + \ is positive. When m is a real integer, we have for unrestricted 
values of n, 
, . II (ft. — m) 
(— l) m —) -f 
' 7 II (n + m) 
47. Next let us suppose the real parts of n — m + 1, and of m + ^ to be positive ; 
the path of the integral 
h n ~ m (1 — 2 fih + Wy-^dh 
l/z 
can be taken as in the figure to consist of two circular arcs of unit radius, two 
straight portions along the real axis, and a circle of indefinitely small radius 
round the point h — 0 ; under the above conditions as to m and n, the circle con¬ 
tributes nothing to the value of the integral. 
In the integral taken along the arc joining the points — and — 1, the phase 
of 1 — 2fxh Id is 377 — <f), where h = e~^ ; in the integral, from — 1 to z, it is 
7r + where h = ; the two integrals together make up 
( (_ x e-^) — e G-»)^+('«-i)G+'(>)‘( te ^)j (2 cos 6 — 2 cos d(p, 
J e 
or 
e 2 n( m -iy j - 2 L cos [in -f ^) (f) — (m -f 77-] (2 cos 6 — 2 cos <f>) m ~'- d<f>. 
J e 
In the integrals from h = 1 to h = 0, the phase of 1 — 2 [xli -j- A 2 is 27? ; let 
A = e~ LW e~ v , for the lower path, and A = & w e~ v , for the upper path ; these portions of 
the integral give us 
