The Fourier transform and its inverse transform are respectively defined as
f ̃ (𝜔) =
1
√2𝜋 ∫ f(x)ei𝜔xdx
+∞
-∞ and f(x) =
1
√2𝜋 ∫ f ̃ (𝜔)e-i𝜔xd𝜔
+∞
-∞ . Consider two
functions f and g. Another function f ∗ g is defined as
(f ∗ g)(x) = 1
√2𝜋
∫ f(y)g(x - y)dy
+∞
-∞
Which of the following relation is/are true?
Note: Tilde (~) denotes the Fourier transform.
Options:
(A) f ∗ g = g ∗ f
(B) f ∗ g̃ = g ∗ f̃
(C) f ∗ g̃ = fg̃
(D) f ∗ g̃ = f̃g̃
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.56)**
**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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