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Yesterday, I made a series expansion (Taylor's) of principal Logarithm i.e $Log: \mathbb C$ \ $(-\infty,0] \to \mathbb C$ at a point $z_0$, The radius of convergence comes out to be $|z_0|$, Now when i take $z_0=-1+i$ or anything near that my circle of convergence crosses negative axis where my function is not actually defined, which holomorphic function is represented by this series at that region?

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On your given disc of convergence, your power series equals some branch of the complex logarithm that avoids that disc. It just so happens that this branch of log agrees with the principal Log away from the negative axis.

In your specific example, $z_0 = -1 + i$, the power series about that point would agree with the branch of logarithm given by

$$ \log{z} = \log{r} + i\theta: z = re^{i\theta}, -\pi/4 < \theta < 7 \pi/4$$

Of course, there are infinitely many branches that work, so long as the branch cut avoids the relevant disc of convergence.

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