Strange Result with Euler's Formula--Is There an Error?

Date: 11/29/2005 at 12:04:45
From: druide
Subject: Euler's formula weird result

Could you tell me where the error is in this work:

  Take f real but not integer.

  exp(2*pi*i*f) = exp(2*pi*i)^f = 1^f = 1

I guess the rule exp(a*b) = exp(a)^b isn't true in that case?  Maybe
that rule isn't correct when a is a complex expression, but I have
read on a lot of websites that this rule is valid for any a or b.

Date: 11/29/2005 at 14:57:43
From: Doctor Pete
Subject: Re: Euler's formula weird result

Hi,

The error is in the rightmost equality,

     1^f = 1.

Since f is not an integer, the value of 1^f is not necessarily 1.  For
instance,

     1^(1/2) = {-1, 1},

since

     (-1)^2 = (1)^2 = 1.

Similarly,

     1^(1/3) = {(-1+Sqrt[3]i)/2, (-1-Sqrt[3]i)/2, 1}
          = {Exp[2 Pi i/3], Exp[-2 Pi i/3], 1}.

One must be careful to observe the multivaluedness of
complex-valued functions.  For instance,

     F : C -> C
     F[z] = z^(1/n),

where n is a positive integer greater than 1, is a one-to-n
mapping, and the n values of F[1] are called the n(th) roots
of unity.

In your original problem, if f is irrational, then there are
infinitely many values of 1^f.  Furthermore there is another issue to
contend with, which is the periodicity of the complex exponential
function Exp[z].  Since

     Exp[it] = Cos[t] + i Sin[t],

it follows that Exp is periodic with period 2 Pi i.  This means that

     Exp[z + 2 Pi i k] = Exp[z]

for any integer k.  Thus the second-to-last equality in your
original problem,

     Exp[2 Pi i]^f = 1^f

is also misleading, since while it is true that Exp[2 Pi i] = 1, it is
also true that

     1 = Exp[4 Pi i] = Exp[6 Pi i] = Exp[-10 Pi i] = ... etc.

Consequently, raising each of these values to the power of f yields
an infinitude of distinct roots of unity when f is an irrational real.

I hope this discussion helps to clarify the issues presented in your
question.  Much of this information is covered in a course in complex
analysis.

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

Date: 11/30/2005 at 03:08:17
From: druide
Subject: Thank you (Euler's formula weird result)

Thank you for this answer!  So in fact, there were not only one but
"nearly" two errors.  I didn't think that manipulating complex
exponentials needed so much care!  I think the main error I made was
considering 1^f as a real function, so that its results could only be
1.  Thank you again, Doctor Pete, for reminding me about roots of 
unity.