- #1
Amer
- 259
- 0
Can you check my work please,
Compute
$\displaystyle \int_{|z|=1} |z-1| . |dz| $
$ z(t) = e^{it} , 0 \leq t < 2 \pi $
$ |dz| =| ie^{it} dt | = dt $
$\displaystyle \int_{0}^{2\pi} |\cos(t) + i\sin(t) - 1 | dt $
$\displaystyle \int_{0}^{2 \pi} \sqrt{(\cos(t) -1)^2 + \sin ^2( t)} \, dt = \int_{0}^{2 \pi} \sqrt{\cos^2(t) - 2\cos(t) +1 + \sin^2(t) } \, dt $
$\displaystyle \int_{0}^{2 \pi} 2\sin \left( \frac{t}{2} \right) dt = 8 $
Compute
$\displaystyle \int_{|z|=1} |z-1| . |dz| $
$ z(t) = e^{it} , 0 \leq t < 2 \pi $
$ |dz| =| ie^{it} dt | = dt $
$\displaystyle \int_{0}^{2\pi} |\cos(t) + i\sin(t) - 1 | dt $
$\displaystyle \int_{0}^{2 \pi} \sqrt{(\cos(t) -1)^2 + \sin ^2( t)} \, dt = \int_{0}^{2 \pi} \sqrt{\cos^2(t) - 2\cos(t) +1 + \sin^2(t) } \, dt $
$\displaystyle \int_{0}^{2 \pi} 2\sin \left( \frac{t}{2} \right) dt = 8 $
