What is the Laplace Transform of f(t)?

What is the Laplace Transform of f(t)?

• A. The Fourier transform of f(t)
• B. The derivative of f(t) with respect to t
• C. The Laplace transform of f(-t)
• D. The Laplace transform of f(t) equals ∫_0^∞ e^{-st} f(t) dt

Answer: (D)

The Laplace transform converts a time-domain function into a complex-frequency function by weighting f(t) with e^{-st} and integrating from 0 to ∞. The defining form is L{f(t)} = ∫_0^∞ e^{-st} f(t) dt, where s is a complex number. This integral expression is exactly what the Laplace transform is, so it is the correct choice. The Fourier transform, by contrast, uses ∫_{-∞}^{∞} e^{-iωt} f(t) dt and focuses on whole time history with a purely imaginary exponent, so it’s related but not the same as the Laplace transform. Taking the derivative with respect to t isn’t a transform itself; it’s an operation that corresponds to multiplying by s in the Laplace domain. Reversing time by substituting f(-t) changes the signal’s time axis and leads to a different relation in the s-domain, not the standard Laplace transform of f(t).

The Laplace transform converts a time-domain function into a complex-frequency function by weighting f(t) with e^{-st} and integrating from 0 to ∞. The defining form is L{f(t)} = ∫_0^∞ e^{-st} f(t) dt, where s is a complex number. This integral expression is exactly what the Laplace transform is, so it is the correct choice.

The Fourier transform, by contrast, uses ∫_{-∞}^{∞} e^{-iωt} f(t) dt and focuses on whole time history with a purely imaginary exponent, so it’s related but not the same as the Laplace transform. Taking the derivative with respect to t isn’t a transform itself; it’s an operation that corresponds to multiplying by s in the Laplace domain. Reversing time by substituting f(-t) changes the signal’s time axis and leads to a different relation in the s-domain, not the standard Laplace transform of f(t).