Natural log of complex numbers

Date: 8/31/96 at 12:28:13
From: Anonymous
Subject: Natural log of complex numbers

   e^i*Pi=isin(Pi) + cos (Pi) = -1
     
If I square both sides:
     
   e^2i*Pi= 1
     
Taking the natural log,
     
   2i*Pi=0 !
     
But how can the product of three constants = 0? 
     
Thanks in advance,
   David

Date: 9/13/96 at 21:28:1
From: Doctor Ceeks
Subject: Re: Natural log of complex numbers

Hi,

The natural logarithm is not well-defined on the complex plane for
the same reason that the exponention function is not one to one.

Thus, a=b does not imply ln(a)=ln(b).

Your argument is similar to the following:

4 = 4.

Take square roots.

-2 = 2 !

-Doctor Ceeks,  The Math Forum
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