List of practice Questions

If \( w = \log_e z = \log_e(x+iy) \), where \( i = \sqrt{-1} \), then which one of the following statements is correct?

Let $C$ be the circle $|z| = \frac{3}{2}$ in the complex plane that is oriented in the counter-clockwise direction. The value of $a$ for which $\int_C \left(\frac{z+1}{z^2 - 3z + 2} + \frac{a}{z-1}\right) dz = 0$ is _______

The value of the integral \( \int_C \frac{e^z}{z^3} \, dz \), where \( C : |z| = 1 \) is \dots\dots.