Exercise · ZW-CAUCHY-GOURSAT-THEOREM-010 · ★★★ Advanced
Compute \(\displaystyle\oint_{|z|=2}\Bigl(\frac1z+\frac1{z-3}+z^3\Bigr)dz\).
Choose one option.
Show answer
- B. \(2\pi i\)
Explanation
By linearity: \(z^3\) and \(\frac1{z-3}\) are holomorphic on the disk \(|z|\le 2\) and contribute \(0\); only \(\frac1z\) contributes \(2\pi i\).
Review the concept: Cauchy-Goursat theorem
The Cauchy-Goursat theorem says that, if \(f\) is holomorphic on a simply connected open set \(D\), its integral over any closed contour \(C\) contained in \(D\) is zero.
\[\oint_C f(z)\,dz=0\]
More practice problems: cauchy-goursat theorem
- ZW-CAUCHY-GOURSAT-THEOREM-001 Compute oint_|z|=1(3z^2-2z+5) dz, with the circle traversed once.
- ZW-CAUCHY-GOURSAT-THEOREM-002 Compute oint_|z|=1e^z dz.
- ZW-CAUCHY-GOURSAT-THEOREM-003 Compute oint_|z|=2(dz)/(z^2+9).
- ZW-CAUCHY-GOURSAT-THEOREM-004 In which case can the Cauchy-Goursat theorem NOT be applied directly?
- ZW-CAUCHY-GOURSAT-THEOREM-005 Compute oint_|z-2|=1(dz)/(z).
- ZW-CAUCHY-GOURSAT-THEOREM-006 Compute oint_|z|=1bar z dz.
- ZW-CAUCHY-GOURSAT-THEOREM-007 Let D be simply connected and oint_C f dz≠ 0 for some closed contour C in D. What follows?
- ZW-CAUCHY-GOURSAT-THEOREM-008 It is known that oint_|z|=1(dz)/(z)=2π i. What is oint_|z|=3(dz)/(z)?