List of top Mathematics Questions on Analytic functions asked in CUET (PG)

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 $\mathbb{C}$

If \(u = x^2 - y^2\) is real part of an analytic function f(z), then f(z) is:

Match List I with List II

Choose the correct answer from the options given below:

If\(ƒ(x + iy) = x^3 − 3.xy^2 + ¡\Psi(x,y) \space where \space i= \sqrt{-1}\) and ƒ (x+iy) is an analytic function, then \(\Psi (x, y)\) is: 

Given below are two statements
Statement I: If f(z) = u + iv is an analytic function, then u and v are both harmonic function.
Statement II: If f (z) is analytic within and on a closed curve C, and if a is any point within C, then \(f(a)=\frac{1}{2πi}\int_c\frac{f(z)}{z-a}dz\)
In the light of the above statements, choose the most appropriate answer from the options given below