Exercise · ZW-RESIDUE-THEOREM-007 · ★★☆ Intermediate
Compute \(\oint_{|z|=1}\frac{\cos z}{z^{3}}\,dz\) counterclockwise.
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- D. \(-\pi i\)
Explanation
\(\cos z=1-z^2/2+\dots\), so \(\cos z/z^3=z^{-3}-\tfrac12 z^{-1}+\dots\). The residue is \(-1/2\) and the integral is \(2\pi i\cdot(-1/2)=-\pi i\).
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-008 Compute oint_|z-i|=1(dz)/(z^2+1) counterclockwise.
- ZW-RESIDUE-THEOREM-009 Compute oint_|z|=2(e^z)/(z(z-1)) dz counterclockwise.