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# 1.10: Concise summary of branches and branch cuts

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We discussed branches and branch cuts for $$\text{arg} (z)$$. Before talking about $$\text{log} (z)$$ and its branches and branch cuts we will give a short review of what these terms mean. You should probably scan this section now and then come back to it after reading about $$\text{log} (z)$$.

Consider the function $$w = f(z)$$. Suppose that $$z = x + iy$$ and $$w = u + iv$$.

Domain. The domain of $$f$$ is the set of $$z$$ where we are allowed to compute $$f(z)$$.

Range. The range (image) of $$f$$ is the set of all $$f(z)$$ for $$z$$ in the domain, i.e. the set of all $$w$$ reached by $$f$$.

Branch. For a multiple-valued function, a branch is a choice of range for the function. We choose the range to exclude all but one possible value for each element of the domain.

Branch cut. A branch cut removes (cuts) points out of the domain. This is done to remove points where the function is discontinuous.