/sci/ - Science & Math » Thread #11113883

59KiB, 508x678, Euler's identity.jpg

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Anonymous Sun 03 Nov 07:46:56 2019 No.11113883 View Reply Original Report

Quoted By: >>11113891 >>11113903 >>11113939

Can someone please explain this Euler's identity fallacy.(see image)

Anonymous

Anonymous Sun 03 Nov 2019 07:59:36 No.11113890 Report

Quoted By:

what have you tried?

Anonymous

Anonymous Sun 03 Nov 2019 07:59:39 No.11113891 Report

Quoted By: >>11113997

>>11113883
exp is not injective in the complex plane, therefore "taking natural log" is not an equivalent operation.

Anonymous

Anonymous Sun 03 Nov 2019 08:08:16 No.11113903 Report

Quoted By:

>>11113883
The fallacy is that $e^{ix}$ is a periodic function.

Anonymous

Anonymous Sun 03 Nov 2019 08:51:26 No.11113939 Report

Quoted By:

>>11113883
You can make the same fallacy with any $\theta \in \mathbb{R}$ :

Let $y = e^{i \theta}$ . Then $y = e^{i (\theta + 2 \pi n)}$ for any $n \in \mathbb{Z}$ . Then from $y = y$ we have $e^{i \theta} = e^{i (\theta + 2 \pi n)}$ . Take the log of both sides to obtain $\theta = \theta + 2 \pi n$ for all $\theta \in \mathbb{R}$ , $n \in \mathbb{Z}$ .

This is precisely why complex logs (and related operations like sqrt) are performed with a branch cut specified.

Anonymous

Anonymous Sun 03 Nov 2019 09:57:18 No.11113997 Report

Quoted By:

>>11113891
This. log, sqrt etc are only defined on the cut complex plane. That excludes negative real numbers.

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