Complex analysis isolated singularities

  • How do you classify isolated singularities?

    Isolated singularities are classified as one of 3 types: f has a removable singularity at z0 if f(z) is bounded on some punctured disc about z0: f(z) ≤ M when 0 \x26lt; z − z0 \x26lt; r , some M, r \x26gt; 0. f has a pole at z0 if limz→z0 f(z) = ∞.
    Everything else: f has an essential singularity at z0..

  • What are the types of singularities in complex analysis?

    In complex analysis, there are several classes of singularities.
    These include the isolated singularities, the nonisolated singularities, and the branch points..

  • What condition should a point satisfy to be called isolated singularity?

    A point p is an isolated singular point of a vector field V if V is nonvanishing and differentiable on some neighborhood of p—except at the point p itself..

  • What is an isolated singularity complex analysis?

    In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it..

  • Definition 1. f has an isolated singularity at z = a if there is a punctured disk B(a, R)\\{a} such that f is defined and analytic on this set, but not on the full disk. a is called removable singularity if there is an analytic g : B(a, R) → C such that g(z) = f(z) for 0 \x26lt; z − a \x26lt; R.
  • 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.
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Duration: 13:31
Posted: Apr 18, 2022
An isolated singularity of a function f(z) is a point z0 such that f(z) is analytic on the punctured disc 0 < |z − z0| < r but is undefined at z = z0. We usually call isolated singularities poles. An example is z = i for the function z/(z − i).
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number 

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