1.10: Concise summary of branches and branch cuts

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Page ID: 50403

Jeremy Orloff
Lecturer (Mathematics Education) at Massachusetts Institute of Technology
Publisher: MIT OpenCourseWare

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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.