Exercise · ZW-COMPLEX-PRIMITIVE-008 · ★★☆ Intermediate
Compute \(\displaystyle\int_0^{i\pi}\sin z\,dz\).
Choose one option.
Show answer
- B. \(1-\cosh\pi\)
Explanation
With \(F=-\cos z\): \(-\cos(i\pi)+\cos 0=1-\cosh\pi\), since \(\cos(iy)=\cosh y\). Watch the sign of the primitive.
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-009 Which of these functions has NO primitive on ℂ?