Exercise · ZW-CONTOUR-INTEGRAL-007 · ★★☆ Intermediate
A contour \(C\) is made of \(C_1\) and \(C_2\). If \(\int_{C_1}f\,dz=2i\) and \(\int_{C_2}f\,dz=1-i\), what is \(\int_C f\,dz\)?
Choose one option.
Show answer
- A. \(1+i\)
Explanation
The integral is additive over the pieces of the contour: \(2i+(1-i)=1+i\). Multiplying the integrals makes no sense here.
Review the concept: Contour integral
The contour integral of a continuous function \(f\) over a contour \(C\) with parametrization \(\gamma\) is the limit of sums of \(f\) evaluated at points of \(C\) times the small complex displacements.
\[\int_C f(z)\,dz=\int_a^b f(\gamma(t))\,\gamma'(t)\,dt\]
More practice problems: contour integral
- ZW-CONTOUR-INTEGRAL-001 Compute ∫_C z^2 dz, with C the segment from 0 to 1+i.
- ZW-CONTOUR-INTEGRAL-002 Compute oint_|z|=1(dz)/(z), with the circle traversed once counterclockwise.
- ZW-CONTOUR-INTEGRAL-003 Compute oint_|z|=1z dz counterclockwise.
- ZW-CONTOUR-INTEGRAL-004 Compute oint_|z|=1bar z dz counterclockwise.
- ZW-CONTOUR-INTEGRAL-005 Compute ∫_Cbar z dz, with C the segment from 0 to 1+i.
- ZW-CONTOUR-INTEGRAL-006 If ∫_C f dz=3-i, what is the integral over the opposite contour -C?
- ZW-CONTOUR-INTEGRAL-008 Compute ∫_CRe(z) dz, with C the segment from 0 to 1+i.
- ZW-CONTOUR-INTEGRAL-009 Let C be the circle |z|=2 traversed once. Use |∫_C f dz|≤ ML with f(z)=1/z^2 to bound |∫_C f dz|.