Determine where a complex function is analytic

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Homework Statement


Determine the region of analyticity of ln|z| + i*arg(z) and justify your answer.


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The Attempt at a Solution


I said that if z=x+iy, the function has a singularity when both x and y are equal to 0 since ln(0) is undefined and arg(z) = arg(x+iy) = -i*log(z+iy/sqrt(x^2+y^2)) which is also undefined.
Thus, I said, the function is analytic everywhere but when |z| = 0.

However, the back of the book claims it is analytic for all z not on the non-positive real axis, but I don't see this. For example, if I pick a value of z = -5 + i*0, ln(-5) is ln(5) + pi*i and arg(z) = pi/ln(10)

The only things I can thing of is that my equation for arg(z) is wrong (I went on wolfram alpha to try and visualize it and I saw that they used it). And the other thing is that arctan(z) would be arctan(y/x) but when y=0, arctan is just equal to zero. My answer makes sense to me, but the books answer does not. What am I missing/doing wrong?

Thank you!
 

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