Complex Analysis: Theory and Applications · Problem 11.4.1

Fourier Series and the Laplace Transform — Howell & Matthews

Mathematics — Complex Analysis · fourier-transform

Problem

Exercises for Section 11.4: 1. Find

\mathfrak{F}\bigl(U(t)\bigr)

for each of the following: ((a))

U(t) = \begin{cases} 1 & \text{for } |t| < 1, \\ 0 & \text{for } |t| > 1. \end{cases}

((b))

U(t) = \begin{cases} 1 - |t| & \text{for } |t| \leq 1, \\ 0 & \text{for } |t| > 1. \end{cases}

2. Let

U(t) = \begin{cases} \sin t & \text{for } |t| \leq \pi, \\ 0 & \text{for } |t| > \pi. \end{cases}

Show that

\mathfrak{F}\bigl(U(t)\bigr) = \frac{i\sin\pi w}{\pi(1-w^2)}

. 3. Use the symmetry and linearity properties and the result of Exercise 1a to show that

\mathfrak{F}\left(\frac{\sin t}{t}\right) = \begin{cases} \frac{1}{2} & \text{for } |w| < 1, \\ 0 & \text{for } |w| > 1. \end{cases}

4. Let

U(t) = e^{-\frac{t^2}{2}}

. Show that

\mathfrak{F}\bigl(U(t)\bigr) = \frac{1}{\sqrt{2\pi}} e^{-\frac{w^2}{2}}

. Hint: Use the integral definition and combine the terms in the exponent, then complete the square and use the fact that

\int_{-\infty}^{\infty} e^{-\frac{r^2}{2}}\, dt = \sqrt{2\pi}

. 5. Use the time scaling property and the example in the text to show that

\mathfrak{F}(e^{-a|t|}) = \frac{|a|}{\pi(a^2 + w^2)}.

6. Use the symmetry and linearity properties and the result of Exercise 2 to show that

\mathfrak{F}\left(\frac{i\sin\pi t}{1 - t^2}\right) = \begin{cases} i\sin\frac{w}{2} & \text{for } |w| \leq \pi, \\ 0 & \text{for } |w| > \pi. \end{cases}

7. Use the time differentiation property and the result of Exercise 4 to show that

\mathfrak{F}\left(te^{-\frac{r^2}{2}}\right) = \frac{-iwe^{-\frac{w^2}{2}}}{\sqrt{2\pi}}.

8. Use the symmetry and linearity properties and the results of Exercise 1b to show that

F\left(\frac{\sin^2 \frac{t}{2}}{t^2}\right) = \begin{cases} \frac{1-|w|}{4\pi} & \text{for } |w| \leq 1, \\ 0 & \text{for } |w| > 1. \end{cases}

9. Write a report on the Fourier transform. Discuss some of the ideas you found in the literature that are not mentioned in the text.