garnentas3m

2022-01-03

Find a branch of $f\left(z\right)=\mathrm{log}({z}^{3}-2)$ that is analytic at $z=0$ .

redhotdevil13l3

Beginner2022-01-04Added 30 answers

Note that, $z=0$ is not a branch point of $f\left(z\right)$ . To find the branch points of $f\left(z\right)$ , solve the equation

${z}^{3}-2=0\Rightarrow {z}^{3}=2{e}^{2k\pi i}\Rightarrow z={2}^{\frac{1}{3}}{e}^{\frac{2k\pi i}{3}},k=0,1,2$

Anzante2m

Beginner2022-01-05Added 34 answers

Or without integration, just take $\mathrm{log}$ to be the natural branch, i.e. the one with a branch cut along the positive real axis. Or any branch cut that avoids $-2$ for that matter.

Vasquez

Expert2022-01-11Added 669 answers

This branch can be defined (at least, in the open unit disk centered at 0) as follows.

where the integration is taken over the interval [0,z] and

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

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