GATE PH 2023 ā¢ Q56
MSQ ā¢ 2 Marks
Section 1: Mathematical Physics
Complex analysis
Subtopic: Residue Theorem & Contour Integration
Question Statement:
Consider the complex function
f(z) = z2 sin z
(z - š)4 .
At z = š , which of the following options is(are) CORRECT?
Options:
-
(A) The order of the pole is 4
-
(B) The order of the pole is 3
-
(C) The residue at the pole is
š
6
-
(D) The residue at the pole is
2š
3
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.56)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 1: Mathematical Physics* | **Topic**: *Complex analysis* (`Residue Theorem & Contour Integration`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Complex analysis** (`Residue Theorem & Contour Integration`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `The order of the pole is 3`**.
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