Exercise · ZW-ISOLATED-SINGULARITY-010 · ★★★ Advanced
What is the order of the pole of \(\frac{z^2-1}{(z-1)^3}\) at \(z=1\)?
Choose one option.
Show answer
- A. \(2\)
Explanation
Simplify: \(\frac{z^2-1}{(z-1)^3}=\frac{z+1}{(z-1)^2}\), with numerator \(2\neq0\) at \(z=1\). The order is \(2\), not \(3\), because the numerator also vanishes once.
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.
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