1. Show that the Laplace transform of f is F(s) = r- Fi(s). Hint: Split the domain [0, 0] as [0, T]U [T, ∞0]; for the second integral, do a change of variable to make the integral look like the standard Laplace integral. %3D 1–e-
1. Show that the Laplace transform of f is F(s) = r- Fi(s). Hint: Split the domain [0, 0] as [0, T]U [T, ∞0]; for the second integral, do a change of variable to make the integral look like the standard Laplace integral. %3D 1–e-
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Transcription of Laplace Transform Educational Material**
---
**III. Periodic Function and Laplace Transform**
Let function \( f \) be periodic with period \( T > 0 \), that is, \( f(t) = f(t + T) \), for \( -\infty < t < \infty \). Define
\[
F_1(s) = \int_0^T e^{-st} f(t) dt.
\]
1. **Show that the Laplace Transform of \( f \) is \( F(s) = \frac{1}{1 - e^{-sT}} F_1(s) \).**
*Hint*: Split the domain \([0, \infty)\) as \([0, T] \cup [T, \infty)\); for the second integral, do a change of variable to make the integral look like the standard Laplace integral.
2. **Use the result to compute the Laplace transform of the half-wave rectified sine signal given by:**
\[
f(t) =
\begin{cases}
\sin \omega t & \text{if } \frac{2n\pi}{\omega} < t < \frac{(2n+1)\pi}{\omega} \\
0 & \text{if } \frac{(2n+1)\pi}{\omega} < t < \frac{(2n+2)\pi}{\omega}, \quad n = 0, 1, 2, \ldots
\end{cases}
\]
*You may use:*
\[
\int e^{\alpha t} \sin(\omega t) dt = \frac{e^{\alpha t}}{\alpha^2 + \omega^2} (-\omega \cos(\omega t) + \alpha \sin(\omega t)).
\]
---
This transcription outlines the steps to demonstrate and apply the Laplace transform for periodic functions and specifically for a half-wave rectified sine signal. It includes definitions, a mathematical hint for splitting domains, and the integral identities used to perform the calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f92a2-1c4d-41a3-af19-0cdf00d27604%2F1900e1e3-a596-493c-9417-171959203d4a%2Fwxbooyd_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription of Laplace Transform Educational Material**
---
**III. Periodic Function and Laplace Transform**
Let function \( f \) be periodic with period \( T > 0 \), that is, \( f(t) = f(t + T) \), for \( -\infty < t < \infty \). Define
\[
F_1(s) = \int_0^T e^{-st} f(t) dt.
\]
1. **Show that the Laplace Transform of \( f \) is \( F(s) = \frac{1}{1 - e^{-sT}} F_1(s) \).**
*Hint*: Split the domain \([0, \infty)\) as \([0, T] \cup [T, \infty)\); for the second integral, do a change of variable to make the integral look like the standard Laplace integral.
2. **Use the result to compute the Laplace transform of the half-wave rectified sine signal given by:**
\[
f(t) =
\begin{cases}
\sin \omega t & \text{if } \frac{2n\pi}{\omega} < t < \frac{(2n+1)\pi}{\omega} \\
0 & \text{if } \frac{(2n+1)\pi}{\omega} < t < \frac{(2n+2)\pi}{\omega}, \quad n = 0, 1, 2, \ldots
\end{cases}
\]
*You may use:*
\[
\int e^{\alpha t} \sin(\omega t) dt = \frac{e^{\alpha t}}{\alpha^2 + \omega^2} (-\omega \cos(\omega t) + \alpha \sin(\omega t)).
\]
---
This transcription outlines the steps to demonstrate and apply the Laplace transform for periodic functions and specifically for a half-wave rectified sine signal. It includes definitions, a mathematical hint for splitting domains, and the integral identities used to perform the calculations.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
