DOUBLE GAMMA FUNCTION. 
325 
We jiroceed to show that 
Wo 
where ix' = 0, unless the axes L and 1/wo do not enclose the axis of — 1, in which 
case 
/a' = d: 1, according as I(o; 2 ) is positive or negative. 
For take the integral along the contour emhracing the axis L, and suppose the 
contour to expand so that it embraces also the axis of 1/wo. 
Then since a is positive with respect to and Wo, and since the angle between 
the axes of L and 1/w^ is less than Ftt, the value of the integral will be unaltered, 
for its value along the part of the great circle at infinity between the axes of L and 
1 /ojo is zero. 
Suppose now that the contour is taken to lie on the infinite-sheeted Neumann 
sphere, whose sheets intersect in the cross-cut from 0 to oo, on which the subject of 
integration of the integral is uniform. We may, without altering the value of the 
integral, deform the cross-cut so as to take up a position along the axis of I/oj^, 
instead of along the axis L, provided that in doing so we do not give a new specifica¬ 
tion to the logarithm. The latter phenomenon will occur when l/co,j and L embrace 
the axis of — 1, in which case we take the first contour in a sheet in which 2 can 
assume real values, while the second is taken in one in which log (— z) for real 
negative values of z is equal to 27rt. 
After deformation of the cross-cut we may comj)ress the contour so that it 
embraces the axis of l/w^. It is easy to see by this repetition of the argument 
previously employed in the “Theory of the Gamma Function,” that we have 
L r {Fg(-- ~) + 7} j., 
27r J L 1 — c"“2= 
i [ ^ [k'g(— 2) -t 2A7r( -F 7} 
1 - C-“2- 
i f z)-'^ {log(-z) + ryj 
= 2ih 
<02 
dz 
1 — c 
dz — 2p.'7nS\ 
where [x' has the value previously given. 
§ 45. The assumption made in § 43 for the values of the constants Xj and \.i will 
therefore lead to the relation 
1 ^2 1 I l(C I 0 )^) Q/ / I \ I I ' I 
log „ . — log S 1 (a I {M -F m -f- m + 
G ^ (^) Pi ("2) 
for (“ Theory of the Gamma Function,” § 37) 
f I O 
[X f ^TTt, 
log 
_ GG|co) _t r e “G— ^{iQg(— + 7 } 
Pi H 
/I J to 
dz. 
1 — 
