GATE PH 2025 • Q21
MCQ • 1 Mark
Section 1: Mathematical Physics
Mathematical Physics
Subtopic: Complex Analysis, Transforms & Differential Equations
Question Statement:
Consider the function f(z) =
1
z2(z-2)3 of a complex variable z. The residues of
the function at z = 0 and z = 2, respectively, are
Options:
-
(A) -
3
8 and
3
8
-
(B) 3
8 and -
3
16
-
(C) -
3
16 and
3
16
-
(D) -
3
8 and
3
16
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.21)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Mathematical Physics** within *Section 1: Mathematical Physics* (1 Mark, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Mathematical Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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