Difference between revisions of "Relationship between logarithm (multivalued) and logarithm"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
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| − | * {{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 | + | * {{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)}}: $4.1.5$ |
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| + | [[Category:Theorem]] | ||
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Latest revision as of 08:32, 18 December 2016
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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.5$
