Exercise · ZW-RESIDUE-THEOREM-009 · ★★★ Advanced
Compute \(\oint_{|z|=2}\frac{e^{z}}{z(z-1)}\,dz\) counterclockwise.
Choose one option.
Show answer
- B. \(2\pi i\,(e-1)\)
Explanation
The residues are \(e^0/(0-1)=-1\) at \(0\) and \(e^1/1=e\) at \(1\). They add to \(e-1\), so the integral is \(2\pi i(e-1)\).
Review the concept: Residue theorem
If \(f\) is holomorphic on a simple closed contour \(\gamma\) and inside it except at isolated singularities \(z_k\), its integral over \(\gamma\) is \(2\pi i\) times the sum of its residues there.
\[\oint_{\gamma} f(z)\,dz=2\pi i\sum_{k}\operatorname{Res}(f,z_k)\]
More practice problems: residue theorem
- ZW-RESIDUE-THEOREM-001 Compute oint_|z|=2(dz)/(z-1), with the circle traversed counterclockwise.
- ZW-RESIDUE-THEOREM-002 Compute oint_|z|=1(e^z)/(z^2) dz counterclockwise.
- ZW-RESIDUE-THEOREM-003 Compute oint_|z|=1(dz)/(z^2+4) counterclockwise.
- ZW-RESIDUE-THEOREM-004 Compute oint_|z|=2(z)/(z^2+1) dz counterclockwise.
- ZW-RESIDUE-THEOREM-005 Compute oint_|z|=2(dz)/(z-1) traversed clockwise (negatively).
- ZW-RESIDUE-THEOREM-006 Compute oint_|z|=3(dz)/(z^2-1) counterclockwise.
- ZW-RESIDUE-THEOREM-007 Compute oint_|z|=1(cos z)/(z^3) dz counterclockwise.
- ZW-RESIDUE-THEOREM-008 Compute oint_|z-i|=1(dz)/(z^2+1) counterclockwise.