Question

Find the Fourier transform of x(t) = e⁽⁻⁵ᵗ⁾ for t > 0.

Answer 1

To find the Fourier transform of x(t) = e⁽⁻⁵ᵗ⁾ for t > 0, one must solve an integral that represents the transformation from time domain to frequency domain, resulting in a function of ω.

Step-by-step explanation:

The Fourier transform of x(t) = e⁽⁻⁵ᵗ⁾ for t > 0 is typically found by applying the definition of the Fourier transform, which involves an integral that transforms the function from the time domain to the frequency domain. The Fourier transform F(ω) of x(t) is given by:

F(ω) = ∫0∞ e⁽⁻⁵ᵗ⁾ e⁽⁻iωt⁾ dt for t > 0

This is a standard calculus integral that involves exponential functions. Solving this integral requires integration techniques for exponential functions. The result will be a function of the angular frequency ω, representing the frequency content of the original time domain function x(t).