Uploaded February 2022 | Updated September 2026, 2 days ago
How do you invert the Laplace transform without a table? This is done through the Paley Wiener Theorem and Fourier transforms. We go through the proof of the Paley Wiener Theorem in this video, which tells us that when we take the inverse transform of certain analytic functions, we get an L2 signal back. This is the converse of the results that we went over in the last video.
This video largely followed a PDF from Friedrich Littmann's homepage. I made a lot of modifications to get the result explicitly in terms of the Laplace transform, since the Paley Wiener theorem is most often communicated as a Fourier result. Littmann's pdf can be found here: https://www.ndsu.edu/pubweb/~littmann/Topics15/class10-7.pdf
//Additional Reading
The Paley Wiener Theorems: https://www.ndsu.edu/pubweb/~littmann/Topics15/class10-7.pdf
//Exercises
//Watch Next
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
(The Other Pailey Wiener Theorem) youtu.be/jcY5Tq1JsVc
Undergraduate Intro to the Laplace Transform youtu.be/mJSqt7nchRQ
A Mature Look at the Laplace transform youtu.be/F-xS27UfF30
Introduction to Control Theory youtu.be/0v4WFmOm764
//Music Provided by Epidemic Sound
Go On - Cacti
Late Nights - Daxten
Mayweather - Warmkeys
Late Nights - Daxten
Beach Memories - Sum Wave
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//Books
Real and Comlex Analysis - Rudin amzn.to/3riipvZ
Real Analysis - Folland amzn.to/3ogVuiG
Fundamentals of Differential Equations - Nagle, Saff, and Snider amzn.to/3qeGD9C
Advanced Mathematical Analysis - Beals amzn.to/3fn2jdP
From Vector Spaces to Function Spaces - Yutaka Yamamoto amzn.to/3sU2CVj
Feedback Control Theory - Doyle, Francis, Tannenbaum amzn.to/3HFvuov
Pick Interpolation and Hilbert Function Spaces - Agler and McCarthy amzn.to/3EOJU3K
Nonlinear Systems - Khalil amzn.to/32Zyk8E
Modern Control Systems - Dorf and Bishop amzn.to/331JQAm
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
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DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!
0:00 Start
0:56 Where does this signal come from?
3:56 Cauchy's Theorem and Big Rectangles
6:20 FUBINATE!
8:17 What is this L2 Signal?
11:12 What's Next?
How do you invert the Laplace transform without a table? This is done through the Paley Wiener Theorem and Fourier transforms. We go through the proof of the Paley Wiener Theorem in this video, which tells us that when we take the inverse transform of certain analytic functions, we get an L2 signal back. This is the converse of the results that we went over in the last video.
This video largely followed a PDF from Friedrich Littmann's homepage. I made a lot of modifications to get the result explicitly in terms of the Laplace transform, since the Paley Wiener theorem is most often communicated as a Fourier result. Littmann's pdf can be found here: https://www.ndsu.edu/pubweb/~littmann/Topics15/class10-7.pdf
//Additional Reading
The Paley Wiener Theorems: https://www.ndsu.edu/pubweb/~littmann/Topics15/class10-7.pdf
//Exercises
//Watch Next
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
(The Other Pailey Wiener Theorem) youtu.be/jcY5Tq1JsVc
Undergraduate Intro to the Laplace Transform youtu.be/mJSqt7nchRQ
A Mature Look at the Laplace transform youtu.be/F-xS27UfF30
Introduction to Control Theory youtu.be/0v4WFmOm764
//Music Provided by Epidemic Sound
Go On - Cacti
Late Nights - Daxten
Mayweather - Warmkeys
Late Nights - Daxten
Beach Memories - Sum Wave
Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel:
epidemicsound.com/referral/644nao
//Books
Real and Comlex Analysis - Rudin amzn.to/3riipvZ
Real Analysis - Folland amzn.to/3ogVuiG
Fundamentals of Differential Equations - Nagle, Saff, and Snider amzn.to/3qeGD9C
Advanced Mathematical Analysis - Beals amzn.to/3fn2jdP
From Vector Spaces to Function Spaces - Yutaka Yamamoto amzn.to/3sU2CVj
Feedback Control Theory - Doyle, Francis, Tannenbaum amzn.to/3HFvuov
Pick Interpolation and Hilbert Function Spaces - Agler and McCarthy amzn.to/3EOJU3K
Nonlinear Systems - Khalil amzn.to/32Zyk8E
Modern Control Systems - Dorf and Bishop amzn.to/331JQAm
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: amzn.to/3D3upUu
DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!
0:00 Start
0:56 Where does this signal come from?
3:56 Cauchy's Theorem and Big Rectangles
6:20 FUBINATE!
8:17 What is this L2 Signal?
11:12 What's Next?
