Question

Fourier transform of a compound function t²e⁻ᵃᵗ and am not sure how to employ the transform tables. I know that the derivative of e⁻ᵃᵗ is 1/jω+1 and anything in the form of tⁿf(t) translates into (−dF(s)/ds) but am not sure whether to employ the product rule or generally how to proceed with even solving te⁻ᵃᵗ.

Answer 1

To find the Fourier transform of t²e⁻ᵃᵗ, you can employ the product rule for derivatives. Start by finding the derivative of t²e⁻ᵃᵗ using the product rule. Then, use the property that the Fourier transform of tⁿf(t) is equal to (-dF(s)/ds), where F(s) is the Fourier transform of f(t). Simplify the resulting expression to obtain the final answer.

Step-by-step explanation:

To find the Fourier transform of the function t²e⁻ᵃᵗ, we can employ the product rule for derivatives. Let's start by finding the derivative of t²e⁻ᵃᵗ.

Using the product rule, we have (t²e⁻ᵃᵗ)' = (t²)'e⁻ᵃᵗ + t²(e⁻ᵃᵗ)'. The derivative of t² is 2t, and the derivative of e⁻ᵃᵗ is -ᵃe⁻ᵃᵗ. So the derivative of t²e⁻ᵃᵗ is 2t*e⁻ᵃᵗ - ᵃt²e⁻ᵃᵗ.

Now, to find the Fourier transform of t²e⁻ᵃᵗ, we can use the property that the Fourier transform of tⁿf(t) is equal to (-dF(s)/ds), where F(s) is the Fourier transform of f(t).

Applying this property, the Fourier transform of t²e⁻ᵃᵗ is equal to (-d(2t*e⁻ᵃᵗ - ᵃt²e⁻ᵃᵗ)/ds). You can simplify this expression further to obtain the final result.