On the extension of Euler's identity in 3 dimensions

Question

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 $\theta$.

Answer 1

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?