List of top Mathematics Questions on Complex Analysis

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:
Which one of the options given below is false?
For an entire function \(f\), which one of the following statements is false?
If \(\displaystyle\sum_{n=-\infty}^{\infty}a_nz^n\) is the Laurent series expansion of the function \(f(z)=\dfrac{1}{2z^2-13z+15}\) in the annulus \(\{z\in\mathbb{C}:3/2<|z|<5\}\), then \(a_1/a_2=\) ____.
Which of the following statements is/are true?
(I) The function \(f(z)=\sin\left(\frac{1}{\cos(1/z)}\right)\) has an isolated singularity at \(z=0\).
(II) The value of the integral \(\oint_{|z|=1}\frac{dz}{z\sin(z)}\) is zero.
Consider the following statements:
(I) Residue of \(f(z)=e^z\) at \(z=\infty\) is \(-1\).
(II) If a meromorphic function \(f\) has 5 simple zeros and 2 simple poles inside \(|z|=1\), then \(\oint_{|z|=1}\frac{f'(z)}{f(z)}dz = 6i\pi\).
Pick out the correct option.
Which of the statements given below is/are false?
(I) \(f(z)=x^2+y^2+2ixy\) (\(z=x+iy\)) is differentiable only at the points that lie on the x-axis.
(II) \(f(z)=|z-1|^2\) is differentiable at \(z=1\), but not analytic at \(z=1\).
(III) There exists an analytic function in \(\mathbb{C}\) whose imaginary part is \(x^2\).
Which one of the following options is true?
(A) If \(\operatorname{Log}(z)\) denotes the principal value of the logarithm, then \(\operatorname{Log}(zw)=\operatorname{Log}(z)+\operatorname{Log}(w)\) for all \(z,w\in\mathbb{C}\setminus\{0\}\).
(B) The solution set of the equation \(e^{iz}=-1\) is \(\{(2k+1)\pi:k\in\mathbb{Z}\}\).
(C) \(\cos(z)\) is bounded in the whole complex plane.
(D) A Mobius transformation with three fixed points is a constant.
The radius of convergence of the power series \(\sum_{n=1}^\infty\left(\frac{2^n}{n}+\frac{3^n}{n^2}\right)z^n\) is ____.
If \(C\) is a simple closed curve not passing through the points \(-1\), \(0\) and \(1\) in the complex plane, then the set of all possible value(s) of the integral \(\int_C \dfrac{dz}{z(1-z^2)}\) is ____.
Consider the following statements:
(I) The function \(f(z)=\dfrac{x^2-y^2+2ixy}{x^2+y^2}\) (\(z=x+iy\)) has a limit as \(z\to0\).
(II) The function \(f(z)=\begin{cases}\operatorname{Re}(z)/|z|, & z\ne0\\ 1, & z=0\end{cases}\) is continuous at \(z=0\).
Choose the correct answer:

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: