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Using the complex integral

$z=\cos(x)+i\sin(x)$

$\frac{dz}{dx}=-\sin(x)+i\cos(x)$

$dz=i[\cos(x)+i\sin(x)]dx$

$dz=iz\cdot dx$

$\frac{1}{z}dz=i\cdot dx$

$\ln(z)=ix$

$z=e^{ix}$

$\cos(x)+i\sin(x)=e^{ix}$

$x=\pi$

$\cos(\pi)-i\sin(\pi)=e^{i\pi}$

$-1=e^{i\pi}$

$\ln(-1)=i\pi$

$\frac{\ln(-1)}{i}=\pi$

$i=\sqrt{-1}$

$\pi=\frac{\ln(-1)}{\sqrt{-1}}$

A friend told me that a possible problem with this is that the log function is only defined for real numbers and this is a complex natural log. I'm wondering if it's a valid equality, and if so what does it mean/indicate?

Jyrki Lahtonen
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Alec Rhea
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  • Yes, it is a valid equality. Just search up Euler's identity. – nathan.j.mcdougall Apr 17 '15 at 02:54
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    There's a lot of symbol-pushing here. I'm afraid it doesn't hold up as a formal proof. – MPW Apr 17 '15 at 02:54
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    It's worth noting that the complex logarithm is multivalued - sure, $e^0=1$, but so does $e^{2i\pi}=1$ (and any integer multiple of $2i\pi$) - so it's not as simple as it is when restricted to the positive real numbers. – Milo Brandt Apr 17 '15 at 02:58
  • I can find the general form $e^{i*pi}=-1$, but not the form rewritten to isolate pi nor it's meaning.

    Do you have any idea what it means that a transcendental number is equal to a ratio of two terms that have no real world context?

     – Alec Rhea Apr 17 '15 at 02:58
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    @AlecRhea It means exactly the same as the $e^{i\pi}=-1$ that you started with. It means that $\pi$ radians are required to rotate 1 on the Argand plane to the point -1, with respect to the origin. – nathan.j.mcdougall Apr 17 '15 at 03:05
  • @nathan.j.mcdougall I don't know how to upvote on here but thank you. – Alec Rhea Apr 17 '15 at 03:07

1 Answers1

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To answer your questions, Yes this is perfectly valid, however, is missing a bit of the full picture in that square roots and the complex natural log are multi-valued. As for what it means or indicates, it seems better to ask what we want out of a generalization. We have a system where exponentiation and logarithms make sense for real numbers and we want to extend this concept to complex numbers in a way that still makes sense. Euler gave us one way to do this. https://en.wikipedia.org/wiki/Complex_logarithm

DanG
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