Exercise · ZW-COMPLEX-PRIMITIVE-010 · ★★★ Advanced
Compute \(\displaystyle\int_C\frac{dz}{z}\), with \(C\) the upper semicircle \(|z|=1\) from \(1\) to \(-1\).
Choose one option.
Show answer
- A. \(i\pi\)
Explanation
With \(z=e^{it}\), \(0\le t\le\pi\): \(\frac{dz}{z}=i\,dt\) integrates to \(i\pi\). Along the lower semicircle it would give \(-i\pi\): the value depends on the path.
Review the concept: Complex primitive
A primitive of a function \(f\) on an open set \(D\) is a holomorphic function \(F\) on \(D\) whose derivative is \(f\) at every point.
\[\int_{z_0}^{z_1} f(z)\,dz=F(z_1)-F(z_0)\]
More practice problems: complex primitive
- ZW-COMPLEX-PRIMITIVE-001 Which is a primitive of f(z)=3z^2 on ℂ?
- ZW-COMPLEX-PRIMITIVE-002 Compute ∫_0^1+iz^2 dz along any path.
- ZW-COMPLEX-PRIMITIVE-003 Compute ∫_0^iπe^z dz.
- ZW-COMPLEX-PRIMITIVE-004 Compute ∫_i^2iz dz.
- ZW-COMPLEX-PRIMITIVE-005 Compute oint_|z|=1z^3 dz.
- ZW-COMPLEX-PRIMITIVE-006 Compute oint_|z|=1(dz)/(z^2).
- ZW-COMPLEX-PRIMITIVE-007 Why is oint_|z|=1(dz)/(z)≠ 0 even though 1/z is holomorphic on ℂsetminus\0\?
- ZW-COMPLEX-PRIMITIVE-008 Compute ∫_0^iπsin z dz.