List of top Mathematics Questions on Analytic functions

Which of the following statements are correct:
A. $e^{-x}(x \sin y - y \cos y)$ is harmonic B. $f(z) = u + iv$ is analytic, where $u = x, v = y$.
C. $f(z) = z^{1/2}$ is analytic on $\mathbb{C}$. D. $f(z) = \frac{1}{z}$ is not analytic on $\mathbb{C}$. E. $f(z) = \sin(z^3)$ is bounded.
Choose the correct answer from the options given below:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : The function $f : \mathbb{C} \to \mathbb{C}$ defined by $f(z) = z - \bar{z}$ is analytic at every point in $\mathbb{C}$ Reason R : $f$ does not satisfy the Cauchy - Riemann equations at any point.
In the light of the above statements, choose the correct answer from the options given below

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : The map $f : \mathbb{C} \to \mathbb{C}$ defined by $f(z) = \sin z$ is bounded. Reason R : The function $f(z) = \sin z$ is an entire map.
In the light of the above statements, choose the correct answer from the options given below

Which of the following function is analytic on $\mathbb{C}$