# Holomorphic function in Taylor series of principal Logarithm

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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In your specific example, $z_0 = -1 + i$, the power series about that point would agree with the branch of logarithm given by
$$\log{z} = \log{r} + i\theta: z = re^{i\theta}, -\pi/4 < \theta < 7 \pi/4$$