List of practice Questions

The residue of $f(z) = e^{-1/z}$ at $z = 0$ is}

Which one of the following functions is analytic over the entire complex plane?

The coefficient of \( \frac{1}{z} \) in the Laurent's series expansion of the function \( f(z) = \frac{1{z^2(1-z)} \) about \( z = 0 \), is \dots\dots.

If a complex function \( f(z) \) is analytic everywhere inside a closed contour \( C \) with anti-clockwise direction, then which of the following statements are correct?

If \( f(z) \) is analytic at \( z_0 \), then it is

If \(f(z) = \frac{1}{2} \log_e(x^2 + y^2) + i \tan^{-1} \left( \frac{y}{x} \right)\) be an analytic function, then \( \alpha \) is ___ .

The values of \( a \) and \( b \) for the function \( f(z) = (x^2 + a y^2 - 2xy) + i (b x^2 - y^2 + 2xy) \) to be analytic are ___ .

Let $p(z) = z^3 + (1 + j)z^2 + (2 + j)z + 3$, where $z$ is a complex number. Which one of the following is true?