1.10: Concise summary of branches and branch cuts
- Page ID
- 50403
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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.