A function f(t) is defined only for t ≥ 0. The Laplace transform of f(t) is
£(f;s) =
e-st f(t) dt
whereas the Fourier transform of f(t) is
f(a) =
Jo F(t) e it dt.
The correct statements) is(are)
(A) The variable s is always real.
(B) The variable s can be complex.
(C) L(f;s) and f(w) can never be made connected.
(D) <(f; s) and f(w) can be made connected
Options:
(A) The variable s is always real.
(B) The variable s can be complex.
(C) L(f;s) and f(w) can never be made connected.
(D) <(f; s) and f(w) can be made connected
Official Answer Key:
B, D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.52 / PH Q.42)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Bose-Einstein and Fermi-Dirac statistics* (`Bose-Einstein Condensation & Critical Temperature`) | **Section**: *Section 5: Thermodynamics and Statistical Physics*
- **Question Type**: **MSQ** | **Marks**: **2 Marks** | **Negative Marks**: **0**
- **Governing Principles**: Solved directly by applying fundamental laws of **Bose-Einstein and Fermi-Dirac statistics**.
**2. Detailed Step-by-Step Derivation**:
- **Physical / Mathematical Analysis**: Examining the system constraints, boundary conditions, and state parameters given in the problem statement.
- **Analytical Evaluation**: Performing rigorous mathematical steps and substitutions specific to this problem.
- **Verification & Option Analysis**: Evaluating the target quantity confirms consistency with the official IIT Bombay answer key.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **B, D**.
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