GATE PH 2024 • Q31
MSQ • 1 Mark
Section 1: Mathematical Physics
Complex analysis
Subtopic: Residue Theorem & Contour Integration
Question Statement:
The complex function
e-( 2
z-1)
has __________
Options:
-
(A) a simple pole at z = 1
-
(B) an essential singularity at z = 1
-
(C) a residue equal to -2 at z = 1
-
(D) a branch point at z = 1
Official Answer Key:
B; C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.31)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 1: Mathematical Physics* | **Topic**: *Complex analysis* (`Residue Theorem & Contour Integration`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **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): `an essential singularity at z = 1`**.
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