I'm a physics student and learning complex analysis (to do definite integrals)
I think I lack knowledge in complex analysis because I cannot follow the logic in the following example.
https://en.wikipedia.org/wiki/Contour_integration#Example_6_.E2.80.93_logarithms_and_the_residue_at_infinity
In this example 6 about contour integral in Wikipedia, I have a question. I cannot see why we should set
$-\pi<\mathrm{arg}(z)<\pi$ and $0<\mathrm{arg}(3-z)<2\pi$
I think it would be more natural to set
$-\pi<\mathrm{arg}(z)<\pi$ and $-\pi <\mathrm{arg}(3-z)<\pi$
It doesn't change the branch cut, just branch themselves (maybe).
But if I change them, I cannot get answer. Answer becomes imaginary.
I think there are 2 possibilities.
1. Residue also changes when I change the branch so the final answer remains same.
2. I cannot set the branches like that.
But I cannot sure what is right, and why.
Can someone explain me where I did wrong?