8.5: Singularities
- Page ID
- 6520
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A function \(f(z)\) is singular at a point \(z_0\) if it is not analytic at \(z_0\)
For a function \(f(z)\), the singularity \(z_0\) is an isolated singularity if \(f\) is analytic on the deleted disk \(0 < |z - z_0| < r\) for some \(r > 0\).
\(f(z) = \dfrac{}{}\) has isolated singularities at \(z = 0\), \(\pm i\).
\(f(z) = e^{1/z}\) has an isolated singularity at \(z = 0\).
\(f(z) = \log (z)\) has a singularity at \(z = 0\), but it is not isolated because a branch cut, starting at \(z = 0\), is needed to have a region where \(f\) is analytic.
\(f(z) = \dfrac{1}{\sin (\pi /z)}\) has singularities at \(z = 0\) and \(z = 1/n\) for \(n = \pm, \pm 2, ...\) The singularities at \(\pm 1 /n\) are isolated, but the one at \(z = 0\) is not isolated.
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