Chapter 17, Problem 5P

(a)

To determine

Calculate the Fourier transform of the function $f\left(t\right)=\left|t\right|e^{-a\left|t\right|}$, where a is a positive real constant and $-\infty \le t\le \infty$.

(b)

To determine

Calculate the Fourier transform of the function $f\left(t\right)=t^3{e}^{-a\left|t\right|}$, where a is a positive real constant and $-\infty \le t\le \infty$.

(c)

To determine

Calculate the Fourier transform of the function $f\left(t\right)=e^{-a\left|t\right|}\cos {\omega }_{0}t$, where a is a positive real constant and $-\infty \le t\le \infty$.

(d)

To determine

Calculate the Fourier transform of the function $f\left(t\right)=e^{-a\left|t\right|}\sin {\omega }_{0}t$, where a is a positive real constant and $-\infty \le t\le \infty$.

(e)

To determine

Calculate the Fourier transform of the function $f\left(t\right)=\Delta \left(t-t_0\right)$, where a is a positive real constant and $-\infty \le t\le \infty$.