Exercise · ZW-RESIDUE-008 · ★★★ Advanced
What is the residue of \(\frac{1}{z^3(z-2)}\) at \(z=0\)?
Choose one option.
Show answer
- C. \(-\frac18\)
Explanation
Near \(0\), \(\frac{1}{z-2}=-\frac12\left(1+\frac z2+\frac{z^2}{4}+\cdots\right)\). Dividing by \(z^3\) puts the \(z^2\) term at \(z^{-1}\): the coefficient is \(-\frac12\cdot\frac14=-\frac18\).
Review the concept: Residue
The residue of a function \(f\) at an isolated singularity \(z_0\) is the coefficient \(a_{-1}\) of \((z-z_0)^{-1}\) in its Laurent series on a punctured disc of radius \(r\).
\[\operatorname{Res}(f,z_0)=a_{-1}=\frac{1}{2\pi i}\oint_{|z-z_0|=r/2}f(z)\,dz\]
More practice problems: residue
- ZW-RESIDUE-001 What is the residue of (1)/(z^2) at z=0?
- ZW-RESIDUE-002 What is the residue of e^1/z at z=0?
- ZW-RESIDUE-003 What is the residue of (1)/(z(z-1)) at z=0?
- ZW-RESIDUE-004 What is the residue of (1)/(z^2+1) at z=i?
- ZW-RESIDUE-005 What is the residue of (e^z)/((z-1)^2) at z=1?
- ZW-RESIDUE-006 What is the residue of (sin z)/(z^4) at z=0?
- ZW-RESIDUE-007 What is the residue of (1)/(sin z) at z=π?
- ZW-RESIDUE-009 Compute oint_|z|=1(e^z)/(z^3) dz, with the circle traversed once counterclockwise.