List of top Mathematics Questions on Analytic Functions

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