Question

A function \( f(t) \) is defined only for \( t \geq 0 \). The Laplace transform of \( f(t) \) is \[ \mathcal{L}(f; s) = \int_0^\infty e^{-st} f(t) \, dt \] whereas the Fourier transform of \( f(t) \) is \[ \tilde{f}(\omega) = \int_0^\infty f(t) e^{-i\omega t} \, dt. \] The correct statement(s) is(are)

• A. The variable s is always real.
• B. The variable s can be complex.
• C. \( \mathcal{L}(f; s) \) and \( \tilde{f}(\omega) \) can never be made connected.
• D. \( \mathcal{L}(f; s) \) and \( \tilde{f}(\omega) \) can be made connected.

Hint:

The Laplace transform can be generalized to a complex domain, and when the real part of \( s \) is zero, the Laplace and Fourier transforms become equivalent.

Answer 1

The Correct Option is B, D

The Laplace transform is typically used to analyze functions defined for \( t \geq 0 \). The variable \( s \) in the Laplace transform is a complex variable, \( s = \sigma + i\omega \), where \( \sigma \) is the real part and \( \omega \) is the imaginary part. This makes the variable \( s \) complex, allowing for more general applications in system analysis, including stability and frequency response. On the other hand, the Fourier transform is a special case of the Laplace transform when \( \text{Re}(s) = 0 \). The Fourier transform uses the variable \( \omega \), which is purely imaginary in the context of the Laplace transform. Step 1: Option (A) The variable \( s \) is not always real. As explained, \( s \) is complex and can have both real and imaginary components. Hence, Option (A) is incorrect. Step 2: Option (B) The variable \( s \) can be complex, as it involves both real and imaginary components in general. Hence, Option (B) is correct. Step 3: Option (C) The Laplace transform and Fourier transform can be connected by evaluating the Laplace transform at \( s = i\omega \), which makes them equivalent in certain cases (when \( \text{Re}(s) = 0 \)). Hence, Option (C) is incorrect. Step 4: Option (D) As explained, \( \mathcal{L}(f; s) \) and \( \tilde{f}(\omega) \) can be made connected by setting \( s = i\omega \) in the Laplace transform. Therefore, Option (D) is correct. Final Answer: (B), (D)