Exercise · ZW-RESIDUE-009 · ★★★ Advanced
Compute \(\displaystyle\oint_{|z|=1}\frac{e^z}{z^3}\,dz\), with the circle traversed once counterclockwise.
Choose one option.
Show answer
- A. \(\pi i\)
Explanation
The integral equals \(2\pi i\) times the residue of \(\frac{e^z}{z^3}\) at \(0\), which is the coefficient of \(z^2\) in \(e^z\): \(\frac12\). The result is \(2\pi i\cdot\frac12=\pi i\).
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-008 What is the residue of (1)/(z^3(z-2)) at z=0?