The title says it all: what branch cut (like which value of $k$ in $\ln(z)=\ln(r)+i(\theta+2\pi k))$ does Wolfram Alpha use by default in calculating the complex logarithm. I would say "principal branch"; however, this does not give me the same as this, even though I used the principal branch. It is entirely possible I made a mistake (in which so, some can tell me where), but even in that case, I still want to know.


1 Answer 1


W|A uses the same branch as Mathematica: the principal branch.

We can see this by expanding the complex log, assuming both x and y are real:

ComplexExpand[Log[x + I y]]
1/2 Log[x^2 + y^2] + I Arg[x + I y]

Or by specifically asking:

FunctionRange[Log[z], z, w, Complexes]
-π < Im[w] <= π
  • $\begingroup$ If so, why is there the difference in answers in the links? $\endgroup$
    – DUO Labs
    Apr 10, 2020 at 2:22
  • 1
    $\begingroup$ It looks like you evaluated the log over a different cut. Yours evaluated to $-61 i \pi/5$, whereas the principal value is $-i \pi/5$. $\endgroup$
    – Greg Hurst
    Apr 10, 2020 at 2:31
  • $\begingroup$ Oh, ok. Thank you. $\endgroup$
    – DUO Labs
    Apr 10, 2020 at 2:39

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