ON THE LOGARITHMS OF IMAGINARY QUANTITIES. 
18 
[386 
or the other direction) any closed curve surrounding the point z = 0 for which the 
function - under the integral sign becomes infinite: but in obtaining the value as 
Z 
above, no use is made of the principles relating to the integration of functions which 
thus become infinite. 
The equation 
gives 
log P = log r + id 
pm _ gin log P — pm Qimd 
or say 
(x + iy) m = r m e' 
where, m being any real quantity whatever, r m denotes the positive real value of r m . 
We have thus a definition of the value of (x + iy) m , and the value so defined may 
be called the selected value. And similarly, for an imaginary exponent m=p 4-qi, we 
have 
(¿g _j_ itjy+Qi — e (p+qi) dogr+ifl) 
_ gp]o%r—q6+i (p8+q\o%r) 
which is the selected value of (x + iy) p+ Q l . 
It may be remarked, in illustration of the advantage (or rather the necessity) of 
having a selected value, that in an integral jzdz, taken between given limits along 
a given path, it is necessary that we know, for the real or imaginary value of z 
corresponding to each point of the path, the value of the function Z; and consequently, 
if Z is a function involving log 2 or z m , the indeterminateness which presents itself 
in these symbols (considered as belonging to a single value of z) is, so to speak, 
indefinitely multiplied, and jzdz is really an unmeaning combination of symbols, unless 
by selecting, as above or otherwise, a unique value of log z or z m , we render the 
function to be integrated a determinate function of the variable.