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Is there an extension for Euler's identity $e^{i\theta}=-1$ for 3 dimensions, expressed in terms of $\theta$ and $\phi$ ? The formula above can only be used in 2 dimensions, and can only be expressed in terms of $\theta$.

Calcudev
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    Euler's identity is $e^{i\pi} = -1$, and has no variables, so what exactly is your question? – Jacob R Jul 22 '19 at 19:17
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    Euler's identity is a set of constants $e^{i\pi}=-1$ there are no variables in it. If you are talking about Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ you could always parameterize $\theta$ as the sum of two other variables – wjmccann Jul 22 '19 at 19:17
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    If you are talking about the fact that Euler's identity is a fact about complex analysis, and $C$ is a 2 dimensional space, there is a 4 dimensional complex space known as the quaternions, and this post talks about an extension to Euler's formula for quaternions https://math.stackexchange.com/questions/41574/can-eulers-identity-be-extended-to-quaternions – Jacob R Jul 22 '19 at 19:20
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    Euler's formula type identities work for division algebras. Since there is no 3 dimensional division algebra there can be no "3D Euler's identity". As remarked by Jacob Raymond the closest you can get is a 4D version using quaternions. – Theo Diamantakis Jul 22 '19 at 19:32

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What you really mean is $r\mathrm e^{\mathrm i\theta} = r\cos \theta + r \mathrm i \sin \theta$.

Hint: This is derived from Taylor series. If polar coordinates use $r$ and $\theta$, then how could you extend this to spherical coordinates?

gen-ℤ ready to perish
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