GATE 2021 PH – Question 52
A function $f(t)$ is defined only for $t\geq0$. Its Laplace and Fourier transforms are $$\mathcal L(f;s)=\int_0^\infty e^{-st}f(t)\,dt,\qquad \widetilde f(\omega)=\int_0^\infty f(t)e^{-i\omega t}\,dt.$$ Which statements are correct?
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Correct answer: (B) The variable $s$ can be complex.; (D) $\mathcal L(f;s)$ and $\widetilde f(\omega)$ can be connected.
Explanation
The Laplace parameter is generally $s=\sigma+i\omega$, with real $\sigma,\omega$, so it need not be real. Setting $s=i\omega$ makes its kernel identical to the stated Fourier kernel. This connects the transforms whenever the imaginary axis lies in the convergence domain, or a suitable limiting transform exists. Thus B and D are correct; the connection is conditional on convergence, not forbidden.