Final answer:
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.