Exercise · ZW-CAUCHY-GOURSAT-THEOREM-008 · ★★☆ Intermediate
It is known that \(\oint_{|z|=1}\frac{dz}{z}=2\pi i\). What is \(\oint_{|z|=3}\frac{dz}{z}\)?
Choose one option.
Show answer
- D. \(2\pi i\)
Explanation
On the annulus \(1\le|z|\le3\) the function \(1/z\) is holomorphic, so the theorem (version for regions with a hole) equates both integrals: \(2\pi i\).
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-009 Compute oint_Ttan z dz, with T the triangle with vertices 0, 1, i traversed once.
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