Exercise · ZW-CONTOUR-INTEGRAL-010 · ★★★ Advanced
For an integer \(n\), for which value is \(\displaystyle\oint_{|z|=1}z^n\,dz\) (counterclockwise) nonzero?
Choose one option.
Show answer
- C. \(n=-1\)
Explanation
\(\int_0^{2\pi}e^{int}\,ie^{it}\,dt=i\int_0^{2\pi}e^{i(n+1)t}\,dt\), which is \(0\) unless \(n+1=0\); then it equals \(2\pi i\). Hence \(n=-1\) is the only case.
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-007 A contour C is made of C_1 and C_2. If ∫_C_1f dz=2i and ∫_C_2f dz=1-i, what is ∫_C f dz?
- ZW-CONTOUR-INTEGRAL-008 Compute ∫_CRe(z) dz, with C the segment from 0 to 1+i.