List of practice Questions

The residue of \(\dfrac{z}{(z-a)(z-b)}\) at infinity is:

Consider the integral \[ I = \frac{1}{2 \pi i} \oint \frac{1}{(z^4 - 1)(z - \frac{a}{b})(z - \frac{b}{a})} \, dz, \] where \( z \) is a complex variable and \( a, b \) are positive real numbers. The integral is taken over a unit circle with center at the origin. Which of the following option(s) is/are correct?

The value of the integral \[ \int_{C} \frac{z^{100}}{z^{101} + 1} \, dz \] where C is the circle of radius 2 centered at the origin taken in the anti-clockwise direction is

Consider the function \( f(z) = \frac{1}{(z+1)(z+2)(z+3)} \). The residue of \( f(z) \) at \( z = -1 \) is _________.

Let \( R = \{ z = x + iy \in \mathbb{C} : 0 < x < 1 \text{ and } - 11 \pi < y < 11 \pi \} \) and \( r \) be the positively oriented boundary of \( R \). Then the value of the integral \[ \frac{1}{2 \pi i} \int_r \frac{e^z}{e^z - 2} \, dz \] \text{is \(\underline{\hspace{1cm}}\) .}

Evaluate $\displaystyle \oint_C \frac{dz}{z^2(z-4)}$ where $C$ is the rectangle with vertices $(-1-j), (3-j), (3+j), (-1+j)$ traversed counter-clockwise.