A periodic function ݂(ݔ)of period 2ߨis defined in the interval (-ߨ< ݔ< ߨ)as:
݂(ݔ)= ቄ-1, -ߨ< ݔ< 0
1, 0 <ݔ< ߨ
T
he appropriate Fourier series expansion for ݂(ݔ)is
Options:
(A) ...] 5 / ) 5 (sin 3 / ) 3 (sin [sin / 4 ) ( xx xx f
(B) ...] 5 / ) 5 (sin 3 / ) 3 (sin [sin / 4 ) ( xx xx f
(C) ...] 5 / ) 5 (cos 3 / ) 3 (cos[cos / 4 ) ( xx xx f
(D) ...] 5 / ) 5 (cos 3 / ) 3 (cos[cos / 4 ) ( xx xx f
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2016 Q.36)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Mathematical Physics** within *Section 1: Mathematical Physics* (2 Marks, 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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