77, 78.] COMPLEX VARIABLES. 



and therefore 



or logo*,*,) &quot;log *t+log, .................. (21). 



Putting z 2 = 0, we have 



log = log z l + log 0, 

 whence log = oo, 



or rather, since it appears that the real part of the logarithm 

 of a small quantity is essentially negative, 



logO = - oo ......................... (22). 



In the same way 



log 00= oo ......................... (23). 



Let us examine the properties of the function inverse to the 

 logarithm ; viz. writing 



w = log z, 



let us investigate the properties of z as a function of w. This 

 function we denote by e w . It follows at once from (21) that 



the fundamental property of the exponential function. Hence, 

 and from (22), (23), 



Also since log z is cyclic, the constant being 2? , e w is a periodic 

 function, the period being 2?, viz. 



where n is any integer. 



Let us map out, in the manner explained in Art. 76, the 

 relation between the two functions z and w. It appears from (19) 



that w = logz=l [-id (25)*. 



J i r 



The first term on the right-hand side of (25) is essentially real ; 

 we denote it by log r. We have then log r and 6 as the rect- 



* Putting r = l in (18) and (25) we see that 



e i9 cos 6 + i sin 0. 



