Exercise · ZW-ISOLATED-SINGULARITY-009 · ★★★ Advanced
Classify the singularity of \(z\sin\frac1z\) at \(z=0\).
Choose one option.
Show answer
- B. Essential
Explanation
\(z\sin\frac1z=\sum_{k\ge0}\frac{(-1)^k}{(2k+1)!}z^{-2k}\) has infinitely many negative powers. Tending to \(0\) along the real axis is not enough: along the imaginary axis, \(z=iy\), \(|f|=y\sinh(1/y)\to\infty\).
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
- ZW-ISOLATED-SINGULARITY-001 What kind of singularity does (sin z)/(z) have at z=0?
- ZW-ISOLATED-SINGULARITY-002 What kind of singularity does e^1/z have at z=0?
- ZW-ISOLATED-SINGULARITY-003 What kind of singularity does 1z^2+1 have at z=i?
- 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-005 What is the order of the pole of 1z^2(z-1) at z=0?
- 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.