Use Theorem 7.4.3 to find the Laplace transform F(s) of the given periodic function. -9s 1-e F(s) = s(1+e=9s) f(t) 4 1 | a 2a 3a 4aj meander function t
Use Theorem 7.4.3 to find the Laplace transform F(s) of the given periodic function. -9s 1-e F(s) = s(1+e=9s) f(t) 4 1 | a 2a 3a 4aj meander function t
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
![### Periodic Function and Its Laplace Transform
**Objective:** Use Theorem 7.4.3 to find the Laplace transform \( F(s) \) of the given periodic function.
#### Given Function and Incorrect Transform
The given function is a meander function, a type of periodic function that alternates between 1 and -1 at regular intervals.
#### Provided (Incorrect) Laplace Transform
\[ F(s) = \frac{1 - e^{-9s}}{s \left(1 + e^{-9s}\right)} \]
The provided transform is indicated to be incorrect with a red cross.
#### Graphical Representation
- **x-axis (t):** Represents time.
- **y-axis (f(t)):** Represents the function values which are 1 and -1.
- Intervals:
- At \( t = a \) the function jumps from 1 to -1.
- At \( t = 2a \) the function rises from -1 to 1.
- At \( t = 3a \) it drops from 1 to -1.
- The cycle repeats periodically.
This visual description illustrates the periodic nature of the function:
1. The function \( f(t) \) starts at 1 for time t=0.
2. When \( t \) reaches \( a \), \( f(t) \) shifts to -1 and remains there until \( t \) reaches \( 2a \).
3. This pattern continues, creating a rectangular waveform with period \( 2a \).
Understanding and applying Theorem 7.4.3 correctly can help derive the accurate Laplace transform for such periodic functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96023f4d-6281-4c81-803e-adfe3e4197d6%2F7ff8e596-4a6a-4c61-bffb-27e29b01c14a%2Fehhjlz8_processed.png&w=3840&q=75)
Transcribed Image Text:### Periodic Function and Its Laplace Transform
**Objective:** Use Theorem 7.4.3 to find the Laplace transform \( F(s) \) of the given periodic function.
#### Given Function and Incorrect Transform
The given function is a meander function, a type of periodic function that alternates between 1 and -1 at regular intervals.
#### Provided (Incorrect) Laplace Transform
\[ F(s) = \frac{1 - e^{-9s}}{s \left(1 + e^{-9s}\right)} \]
The provided transform is indicated to be incorrect with a red cross.
#### Graphical Representation
- **x-axis (t):** Represents time.
- **y-axis (f(t)):** Represents the function values which are 1 and -1.
- Intervals:
- At \( t = a \) the function jumps from 1 to -1.
- At \( t = 2a \) the function rises from -1 to 1.
- At \( t = 3a \) it drops from 1 to -1.
- The cycle repeats periodically.
This visual description illustrates the periodic nature of the function:
1. The function \( f(t) \) starts at 1 for time t=0.
2. When \( t \) reaches \( a \), \( f(t) \) shifts to -1 and remains there until \( t \) reaches \( 2a \).
3. This pattern continues, creating a rectangular waveform with period \( 2a \).
Understanding and applying Theorem 7.4.3 correctly can help derive the accurate Laplace transform for such periodic functions.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
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,
