Evaluate $\oint_c {4z - 1}\,dz$ along the circle $|z| = 1$

Question

Evaluate $\displaystyle\oint_c {4z - 1}\,dz$ along the circle $|z| = 1$ from the point $(0,-1)$ to $(1,0)$

My question is how to do a contour integration in the circle? I only know to do it in straight line.

Answer 1

You'll need to parameterize the variable in polar coordinates, i.e. with the change $$C=z_0 + re^{i\theta}, z=re^{i\theta}, dz=ire^{i\theta}$$. Your curve $C$ is along the unit circle (with center zero) in the fourth quadrant, whose points have angle $-\frac{\pi}{2}\leq \theta \leq 0$, so $z_0=0,$ and $ z=1*e^{i\theta}$ throughout your curve. Putting together all these substitutions, we have $$\int_C (4z-1)dz=\int_{-\frac{\pi}{2}}^0 (4e^{i\theta}-1)ie^{i\theta}d\theta. $$

Conclude.

Answer 2

A circle of radius $r$ centered at $z = a$ can be traversed CCW by $z = a+re^{i\theta}$ where $-\pi \le \theta \le \pi$, or any interval of length $2\pi$. In particular, the unit circle $|z| = 1$ can be parameterized by $z = e^{i \theta}$.

Here, the curve is only a quarter of the unit circle, so you'll have to restrict $\theta$ to a smaller range. Can you figure out what range that is?