Understanding the complex logarithm
The complex logarithm \operatorname{Log}(z) is the inverse function of the complex exponential for z \ne 0. Unlike the real case, the periodic property of the exponential introduces an essential phenomenon: \operatorname{Log}(z) is inherently multivalued, assigning infinitely many distinct values to each nonzero z.
Given z = x + iy, write z in polar form as z = r e^{i\theta}, where r = |z| and \theta = \operatorname{arg}(z). By definition, w = \operatorname{Log}(z) satisfies e^{w} = z. Comparing e^{w} and z in polar form,
\begin{aligned} & z = r e^{i\theta} = e^{u} e^{iv} \\ & r = e^{u} \\ & \theta = v + 2 n \pi,\ n \in \mathbb{Z} \end{aligned}
So the general expression for the complex logarithm is
\operatorname{Log}(z) = \ln |z| + i\left(\operatorname{Arg}(z) + 2 n \pi\right),\quad n \in \mathbb{Z}
This means the function admits infinitely many values, differing by integer multiples of 2\pi i.
To work with a single-valued and (locally) analytic version, the argument is restricted to a chosen interval, yielding a particular branch. The standard or principal branch enforces n = 0 and selects \operatorname{Arg}(z) \in (-\pi, \pi]:
\operatorname{Log}(z) = \ln|z| + i\operatorname{Arg}(z),\quad \operatorname{Arg}(z) \in (-\pi, \pi]
A consequence of this restriction is a discontinuity, the branch cut, typically placed along the negative real axis.
Approaching a negative real number from above gives \operatorname{Arg}(z) \to \pi, while from below gives \operatorname{Arg}(z) \to -\pi, so the imaginary part jumps by 2\pi i there.
The function \operatorname{Log}(z) is analytic on \mathbb{C} with the branch cut and the origin removed, i.e., \mathbb{C} \setminus (-\infty, 0].
Other branches arise by selecting different intervals for the argument. Each choice leads to a different branch cut placement and a distinct analytic domain.
The origin z = 0 and infinity z = \infty are identified as branch points, as encircling these leads to a transition between branches.
For more insights into this topic, you can find the details here.