Find a branch of f(z)=\log(z^3−2) that is analytic at z=0.

garnentas3m

garnentas3m

Answered question

2022-01-03

Find a branch of f(z)=log(z32) that is analytic at z=0.

Answer & Explanation

redhotdevil13l3

redhotdevil13l3

Beginner2022-01-04Added 30 answers

Note that, z=0 is not a branch point of f(z). To find the branch points of f(z), solve the equation
z32=0z3=2e2kπiz=213e2kπi3,k=0,1,2
Anzante2m

Anzante2m

Beginner2022-01-05Added 34 answers

Or without integration, just take 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

Vasquez

Expert2022-01-11Added 669 answers

This branch can be defined (at least, in the open unit disk centered at 0) as follows.
f(z):=0z3t2t32dt+log(2)
where the integration is taken over the interval [0,z] and log(2)=log2+πi.

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