Exercise · ZW-ISOLATED-SINGULARITY-005 · ★★☆ Intermediate
What is the order of the pole of \(\frac1{z^2(z-1)}\) at \(z=0\)?
Choose one option.
Show answer
- A. \(2\)
Explanation
It is \(\frac{g(z)}{z^2}\) with \(g(z)=\frac1{z-1}\) holomorphic and \(g(0)=-1\neq0\). The exponent of \(z\) in the denominator is the order: \(2\).
Review the concept: Isolated singularity
An isolated singularity of a function \(f\) is a point \(z_0\) where \(f\) is not holomorphic but is holomorphic on a punctured disc \(0<|z-z_0|<r\) around it.
More practice problems: isolated singularity
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- ZW-ISOLATED-SINGULARITY-004 The singularity of (cos z-1)/(z^2) at z=0 is removable. What value must f(0) take to remove it?
- ZW-ISOLATED-SINGULARITY-006 If |f| is bounded on a punctured disc 0<|z-z_0|<r, what kind of isolated singularity does f have at…
- ZW-ISOLATED-SINGULARITY-007 Which of these functions has at z=0 a singular point that is NOT an isolated singularity?
- ZW-ISOLATED-SINGULARITY-008 Classify the singularity of (z)/(sin z) at z=π.
- ZW-ISOLATED-SINGULARITY-009 Classify the singularity of zsin 1z at z=0.