The complex function \( e^{-\left(\frac{2}{z-1}\right)} \) has:
- a simple pole at \( z = 1 \)
- an essential singularity at \( z = 1 \)
- a residue equal to \( -2 \) at \( z = 1 \)
- a branch point at \( z = 1 \)
The Correct Option is B, C
Solution and Explanation
Top Questions on Complex Functions
Let \( \Omega \) be a non-empty open connected subset of \( \mathbb{C} \) and \( f: \Omega \to \mathbb{C} \) be a non-constant function. Let the functions \( f^2: \Omega \to \mathbb{C} \) and \( f^3: \Omega \to \mathbb{C} \) be defined by \[ f^2(z) = (f(z))^2 \quad {and} \quad f^3(z) = (f(z))^3, \quad z \in \Omega. \]
Consider the following two statements:
S1: If \( f \) is continuous in \( \Omega \) and \( f^2 \) is analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \).
S2: If \( f^2 \) and \( f^3 \) are analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \). Then, which one of the following is correct?- GATE MA - 2025
- Mathematics
- Complex Functions
Let \( U = \{z \in \mathbb{C}: \operatorname{Im}(z) > 0\} \) and \( D = \{z \in \mathbb{C}: |z| < 1\} \), where \( \operatorname{Im}(z) \) denotes the imaginary part of \( z \).
Let \( S \) be the set of all bijective analytic functions \( f: U \to D \) such that \( f(i) = 0 \).
Then, the value of \( \sup_{f \in S} |f(4i)| \) is:- GATE MA - 2025
- Mathematics
- Complex Functions
- The only function from the following that is analytic is:
- TANCET - 2024
- Engineering Mathematics
- Complex Functions
- The values of function \(y(x)\) at discrete values of \(x\) are given in the table. The value of \(∫_0^4y(x)dx\), using Trapezoidal rule is _____ . (Rounded off to 1 decimal place)
x 0 1 2 3 4 y(x) 1 3 6 9 12 - GATE PI - 2024
- Mathematics
- Complex Functions
- In a complex function
\(f(x, y) = u(x, y) + i v(x, y),\)
\(i\) is the imaginary unit, and \(x, y, u(x, y)\) and \(v(x, y)\) are real.
If \(f(x, y)\) is analytic then which of the following equations is/are TRUE?- GATE PI - 2024
- Mathematics
- Complex Functions
Questions Asked in GATE PH exam
A wheel of mass \( 4M \) and radius \( R \) is made of a thin uniform distribution of mass \( 3M \) at the rim and a point mass \( M \) at the center. The spokes of the wheel are massless. The center of mass of the wheel is connected to a horizontal massless rod of length \( 2R \), with one end fixed at \( O \), as shown in the figure. The wheel rolls without slipping on horizontal ground with angular speed \( \Omega \). If \( \vec{L} \) is the total angular momentum of the wheel about \( O \), then the magnitude \( \left| \frac{d\vec{L}}{dt} \right| = N(MR^2 \Omega^2) \). The value of \( N \) (in integer) is:
- GATE PH - 2025
- Rotational Motion
- An electron with mass \( m \) and charge \( q \) is in the spin up state \[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \] at time \( t = 0 \). A constant magnetic field is applied along the \( y \)-axis, \( \mathbf{B} = B_0 \hat{j} \), where \( B_0 \) is a constant. The Hamiltonian of the system is \[ H = -\hbar \omega \sigma_y, \quad \text{where} \quad \omega = \frac{q B_0}{2m} < 0 \quad \text{and} \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}. \]
The minimum time after which the electron will be in the spin down state along the \( x \)-axis, i.e., \[ \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \] is:- GATE PH - 2025
- Quantum Mechanics
The figure shows an opamp circuit with a 5.1 V Zener diode in the feedback loop. The opamp runs from \( \pm 15 \, {V} \) supplies. If a \( +1 \, {V} \) signal is applied at the input, the output voltage (rounded off to one decimal place) is:
- GATE PH - 2025
- Electrical Circuits
- A system of five identical, non-interacting particles with mass \( m \) and spin \( \frac{3}{2} \) is confined to a one-dimensional potential well of length \( L \). If the lowest energy of the system is \[ E_{{min}} = \frac{N \pi^2 \hbar^2}{2m L^2}, \] the value of \( N \) (in integer) is:
- GATE PH - 2025
- Quantum Mechanics
- The Hamiltonian for a one-dimensional system with mass \( m \), position \( q \), and momentum \( p \) is: \[ H(p, q) = \frac{p^2}{2m} + q^2 A(q) \] where \( A(q) \) is a real function of \( q \). If \[ m \frac{d^2 q}{dt^2} = -5q A(q), \] then \[ \frac{d A(q)}{d q} = n \frac{A(q)}{q}. \] The value of \( n \) (in integer) is:
- GATE PH - 2025
- Mechanics