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We are learning about the principal vs. non-principal values of the natural logarithm. There are 2 problems:

  1. $\ln(-7)$

  2. $\operatorname{Ln}(-7)$

I solved $\ln(-7)$ for: $$=\ln|-7|+i\pi+2\pi k$$

Is the difference in notation for the other problem really just removing the $2\pi k$? Or maybe it would be better if I define the argument like $-\pi<\theta<\pi$?

whatwhatwhat
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    Which branches are these, i.e., what are your definitions of $\ln$ and $\operatorname{Ln}$? – Travis Willse Feb 12 '16 at 23:51
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    As a multivalue function, $\ln{z}$ has, as you mentioned has the periodic $2\pi k$. the $Ln{z}$ is the principal branch. – Eleven-Eleven Feb 12 '16 at 23:51
  • @Travis So there isn't a standard range then? Should I explicitly write that $-\pi<\theta<\pi$ ? – whatwhatwhat Feb 12 '16 at 23:56
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    There's no "canonical" choice for the branch cut of logarithm. I'd say that choosing a branch to take values in $(-\pi, \pi]$ is most common, but this is not universal, and not the right choice for all situations. For example, when using contour integration to evaluate certain real integrals involving the logarithm over $[0, \infty)$, choosing the branch with values in $[0, 2\pi)$ is easiest. – Travis Willse Feb 13 '16 at 00:00
  • I see, I see. Ok. Good to know, that's all I was wondering. Thanks!! – whatwhatwhat Feb 13 '16 at 00:01

1 Answers1

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It is a convention in complex analysis to denote the principle branch of a multivalued function by capitalizing the first letter.

In this case, $\operatorname{Ln}(z) = \operatorname{Ln}|z|+i \operatorname{Arg}(z)$

As for $\operatorname{Ln}$, $\operatorname{Arg}$ is the principle branch of the the argument function, taking values in $(-\pi, \pi]$.

MathMajor
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