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. |
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