List of top Mathematics Questions on Complex Analysis asked in CUET (PG)

Match the LIST-I with LIST-II
Choose the correct answer from the options given below:
Which of the following holds for complex variable $z$:
A. Let $C$ be the unit circle $z = e^{i\theta}, (-\pi \le \theta \le \pi)$. Then $\int_C \frac{e^{iz}}{z} dz = 2\pi i$ B. The $\{z \mid 1 < |z| < 2\}$ set is simply connected. C. $\frac{1}{1+z} = \sum_{n=0}^{\infty} z^n$, whenever $|z| < 1$ D. The unit disk $\{z \mid |z| < 1\}$ is simply connected. E. If $f$ is analytic at $z_0$, then $f$ is continuous at $z_0$.
Choose the correct answer from the options given below:
Which of the following are correct
A. $(e^z)^n = e^{nz}$, $(n = 0, \pm 1, \pm 2, \dots)$ B. Let $f(z) = u(x,y) + i v(x,y)$ be analytic on some domain $D$. Then $T(x,y) = e^{u(x,y)} \cos v(x,y)$ is harmonic in $D$. C. $e^z \neq 0$ for all $z \in \mathbb{C}$. D. The principal value of $(i)^i$ is $\exp\left(\frac{\pi}{2}\right)$
E. $\cos z = \frac{e^{iz} - e^{-iz}}{2}$ Choose the correct answer from the options given below:
If $f(z) = \frac{z}{\bar{z}}$, then $\lim_{z \to 0} f(z)$
If $f'(z) = 0$ everywhere in a connected open set $G \subset \mathbb{C}$, then $f(z)$ is
Which of the following function satisfies Cauchy-Riemann equations:
Every bounded entire function is constant. This theorem is known as:

Match List-I with List-II and choose the correct option:

LIST-I (Function)LIST-II (Value)
(A) \( \int_{\gamma} \frac{1}{z-a} \, dz \), where \( \gamma: |z-a|=r, r > 0 \)(III) \( 2i\pi \)
(B) \( \int_{\gamma} \frac{z+2}{z} \, dz \), where \( \gamma: z = 2e^{it}, 0 \le t \le \pi \)(IV) \( i\pi \)
(C) \( \int_{\gamma} \frac{e^{2z}}{(z-1)(z-2)} \, dz \), where \( \gamma: |z|=3 \)(II) \( 2i\pi(e^4 - e^2) \)
(D) \( \int_{\gamma} \frac{z^2 - z + 1}{2(z-1)} \, dz \), where \( \gamma: |z|=2 \)(I) \( -4 + 2i\pi \)


Choose the correct answer from the options given below:

Consider the following: Let f(z) be a complex valued function defined on a subset \( S \subset \mathbb{C} \) of complex numbers. Then which of the following are correct?
A. The order of a zero of a polynomial equals to the order of its first non-vanishing derivative at that zero of the polynomial
B. Zeros of non-zero analytic function are isolated
C. Zeros of f(z) are obtained by equating the numerator to zero if there is no common factor in the numerator and the denominator of f(z)
D. Limit points of zeros of an analytic function is an isolated essential singularity
The function \( f(z) = |z|^2 \) is differentiable, at
If C is the positively oriented circle represented by \( |z|=2 \), then \( \int_C \frac{e^{2z}}{z-4} dz \) is:
If \( f(z) = (x^2-y^2-2xy) + i(x^2-y^2+2xy) \) and \( f'(z)=cz \), where c is a complex constant, then \( |c| \) is equals to:

The value of the integral:

\[ \oint_C \frac{z^3 - z}{(z - z_0)^3} \, dz \] where \( z_0 \) is outside the closed curve \( C \) described in the positive sense, is:

The value of \( \int_0^{1+i} (x^2 - iy) dz \), along the path \( y = x^2 \) is:
If, \( u=y^3-3x^2y \) be a harmonic function then its corresponding analytic function \( f(z) \), where \( z=x+iy \), is: