Real and imaginary parts of log
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Theorem
Write $z \in \mathbb{C}$ using polar coordinates: $z=x+iy=re^{i\theta}$. The following formula holds for $-\pi < \mathrm{arg}(z) \leq \pi$: $$\log(z)=\log(r)+i\theta,$$ where $\mathrm{arg}$ denotes the argument and $\log$ denotes the logarithm.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.1
