OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 515 
the real axis, and a circle round the point O ; the circular paths contribute nothing 
to the value of the integral, and we have 
P n *(p) = r— 2 m ^ 
— 7 Ti 11 ( g/ J — oo 
_ e -(»+m'+l)nr+2Hr(m+J)j e (n+i)« ^ CO sh U + du 
2 m+1 II (m — 1) . , ,_, 7 
-pp-——sm (n — m)7r . cosh (w+t) w • (2 cosh w + 2p) “ s du, 
ll ( —o) Jo 
this holds for all values of p of which the real part is positive, provided m is half an 
integer, and is less than |, also provided the real part of n — m is negative and of 
n + w. + 1 is positive; the only value of m which satisfies these conditions is 
m — 0 ; we thus obtain the formula 
(/*) 
2 
77 
cos 77 
x cosh (n + J) u 
o (2 cosh u + 2p)*• 
(m), 
which holds, provided the real part of n is between 0 and — 1, and that of p is 
positive. 
Definite Integral Expressions for P,“ (p), when p real and greater than unity. 
50. In the formula 
?>r m (p) = 
i 
2 »n(- |)n (m- f) 
e -ur(m-\) (p2 _ 1 J-J» ^*-**(1 _ 2p/i + h*Y~ k dh 
i/-’ 
where the real part of m -f ^ is positive ; when p is real and greater than unity, put 
p = cosh xjj, then 2 = e*,. - = e^, thus putting h = e’h we obtain the formula 
3 u 2 
