Taking the Natural Log of e^(ki)

Date: 05/18/2000 at 00:54:37
From: Jim Thomas
Subject: Taking the natural log of e^xi

Hello!

I have a question about taking the natural log of e^(an imaginary 
number). Is the natural log defined for such numbers and, if so, how?

From

     e^(i*2pi) = cos(2pi) + i*sin(2pi) 
     e^(i*2pi) = 1
     
with real exponents ln(e^u) = u, but if I attempt to apply this to 
both sides of my equation I end up with:

     ln [e^(i*2pi)] = ln[1]
              i*2pi = 0

which doesn't seem possible. Any help you can give would be greatly 
appreciated.

Thank you,
Jim Thomas

Date: 05/18/2000 at 05:52:22
From: Doctor Mitteldorf
Subject: Re: Taking the natural log of e^xi

Dear Jim,

Good point! It looks like you've uncovered a paradox here.

On the one hand, certainly it must be true that the ln of e^x is x, no 
matter whether x is real or imaginary or a more distant algebraic 
construction. This is because when mathematicians make up meanings for 
these far-out constructions, all they have to go on is what works for 
real numbers; so it wouldn't be right to define the ln of e^x as 
anything else but x, when extending the definition into new fields.

The answer to this paradox is that ln is a "multi-valued function." 
That is to say, it's not a function at all. For every x, you can find 
a unique e^x; but that doesn't mean that for every e^x you can find a 
unique x. As you see, adding any multiple of 2pi to x gives you back 
the same e^x, so there is an infinite number of values of any ln, all 
differing by 2pi from each other.

This situation is familiar in other contexts. The easiest is the 
following paradox (which we actually see a good deal of, in one form 
or another, here at Dr Math's HQ):

     (-1)^2 = 1

Take the square root of both sides:

         -1 = 1?

The "root" of this paradox is that the square root is a two-valued 
function. You can't take the positive root on the right side and the 
negative root on the left.

More to the point,

     sin(pi/3) = sin(2pi/3)

Taking arcsin of both sides, you'd find:

          pi/3 = 2pi/3.

Again - the answer is that while sin is a good, single-valued 
function, arcsin has many values. If you take arcsin of both sides, 
you must be sure you are choosing the same branch in each case.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/