View source for Relationship between logarithm (multivalued) and logarithm

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==Theorem==
The following formula holds:
$$\mathrm{Log}\left(re^{i\theta}\right) = \left\{ \log(re^{i\theta}+2k\pi i \colon k \in \mathbb{Z} \right\},$$
where $\mathrm{Log}$ denotes the [[logarithm (multivalued)]], $\log$ denotes the [[logarithm]], $i$ denotes the [[imaginary number]], and $\pi$ denotes [[pi]]. 

==Proof==

==Note==
Sometimes this formula is written as 
$$\mathrm{Log}\left(re^{i\theta}\right) = \log(re^{i\theta})+2k\pi i, \quad k \in \mathbb{Z}.$$
Writing it this way emphasizes the multi-valued nature of $\mathrm{Log}$. For our purposes, this is not sufficient since we defined the [[codomain]] of the multivalued logarithm to be the [[power set]] of $\mathbb{C}$. 

==References==
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithm (multivalued)|next=Logarithm (multivalued) of product is a sum of logarithms (multivalued)}}: $4.1.5$

[[Category:Theorem]]
[[Category:Unproven]]