Exercise · ZW-ISOLATED-SINGULARITY-008 · ★★☆ Intermediate
Classify the singularity of \(\frac{z}{\sin z}\) at \(z=\pi\).
Choose one option.
Show answer
- C. Simple pole
Explanation
Near \(\pi\), \(\sin z=-(z-\pi)+\cdots\) has a simple zero and the numerator equals \(\pi\neq0\). The limit of \((z-\pi)\frac{z}{\sin z}\) is \(-\pi\): simple pole.
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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