Introduction to the Laplace Transform @JoelRosenfeld

Uploaded January 2022 | Updated September 2026, 1 week ago

This fast paced video gives an advanced introduction to the Laplace transform for delta functions and distributions. This necessitates the definition of a Schwartz space for the Laplace transform, and it defines the Laplace transform of a distribution through a limit of its action on a particular sequence of functions.

//Additional Reading
Zemanian - An Introduction to Generalized Functions and the Generalized Laplace and Legendre Transformations
(JSTOR) jstor.org/stable/2027808?seq=1#metadata_info_tab_contents
(SIAM) epubs.siam.org/doi/abs/10.1137/1010001?journalCode=siread

//Exercises
- Show that regular functionals defined through functions of exponential type with support in [M,infinity) for some M in R are continuous over L.
- Find an explicit function phi in L such that phi is 1 for t less than 1 and 0 for t greater than 2.
- Find the Laplace transform of the distribution that maps u in L to its derivative at zero from the definition of Laplace transforms.
- Show that the function g(t) = e^{-st} is not in L for any s.

//Watch Next
The Schwartz Space (Full Lecture) youtu.be/StzEO8B1pFY
Undergraduate Intro to the Laplace Transform youtu.be/mJSqt7nchRQ
Introduction to Control Theory youtu.be/0v4WFmOm764
What if I were to tell you that the Fourier Transform is not an integral? youtu.be/YaeGIgtE6R4
Fixed Point Iteration - Loki gets slapped 16 times for SCIENCE youtu.be/_I790SqNDjs

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Advanced Mathematical Analysis - Beals amzn.to/3fn2jdP
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Modern Control Systems - Dorf and Bishop amzn.to/331JQAm
Fundamentals of Differential Equations - Nagle, Saff, and Snider amzn.to/3qeGD9C

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$Introduction to the Laplace Transform This fast paced video gives an advanced introduction to the Laplace transform for delta functions and distributions. This necessitates the definition of a Schwartz space for the Laplace transform, and it defines the Laplace transform of a distribution through a limit of its action on a particular sequence of functions. //Additional Reading Zemanian - An Introduction to Generalized Functions and the Generalized Laplace and Legendre Transformations (JSTOR) https://www.jstor.org/stable/2027808?seq=1#metadata_info_tab_contents (SIAM) https://epubs.siam.org/doi/abs/10.1137/1010001?journalCode=siread //Exercises - Show that regular functionals defined through functions of exponential type with support in [M,infinity) for some M in R are continuous over L. - Find an explicit function phi in L such that phi is 1 for t less than 1 and 0 for t greater than 2. - Find the Laplace transform of the distribution that maps u in L to its derivative at zero from the definition of Laplace transforms. - Show that the function g(t) = e^{-st} is not in L for any s. //Watch Next The Schwartz Space (Full Lecture) https://youtu.be/StzEO8B1pFY Undergraduate Intro to the Laplace Transform https://youtu.be/mJSqt7nchRQ Introduction to Control Theory https://youtu.be/0v4WFmOm764 What if I were to tell you that the Fourier Transform is not an integral? https://youtu.be/YaeGIgtE6R4 Fixed Point Iteration - Loki gets slapped 16 times for SCIENCE https://youtu.be/_I790SqNDjs //Music Provided by Epidemic Sound Mt Fuji - Oozy Opposite - Jones Meadow Euphoria - Catiso Leaving at Midnight - waykap Pomme - Chantarelle Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referral/644nao //Books Advanced Mathematical Analysis - Beals https://amzn.to/3fn2jdP From Vector Spaces to Function Spaces - Yutaka Yamamoto https://amzn.to/3sU2CVj Feedback Control Theory - Doyle, Francis, Tannenbaum https://amzn.to/3HFvuov Pick Interpolation and Hilbert Function Spaces - Agler and McCarthy https://amzn.to/3EOJU3K Nonlinear Systems - Khalil https://amzn.to/32Zyk8E Modern Control Systems - Dorf and Bishop https://amzn.to/331JQAm Fundamentals of Differential Equations - Nagle, Saff, and Snider https://amzn.to/3qeGD9C //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://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!$

$Moreras Theorem in Practice - Laplace Transform Here we examine an application of Moreras Theorem in complex analysis to show the analyticity of the Laplace transform. What sort of function is the transform of a signal in the frequency domain? To be perfectly honest, when I first learned it, I didnt think that Moreras theorem would have been very useful. It says that if you have a function that is continuous and its integrals along each triangle in the domain of the function is zero, then the function is analytic. I thought how in the world would you know those two conditions without already knowing that a function was analytic. Well, thats egg on my face, because, Moreras theorem helps a lot. In fact, whenever you have a transform like the Laplace tranform, it really comes in to save the day, and this all comes down to the basis of the Laplace transform being e^{-st}, or entire. The smoothness and analyticity of the Laplace transform is leveraged throughout systems theory and control. In your ordinary differential equations class, every example that is presented is not only analytic (or meromorphic) but is also rational. This is really handy when you have to invert a Laplace transform using tables, where a partial fraction decomposition breaks down a Laplace transform of a solution to a differential equation into a collection of pieces that you can match up easily. However, analytic functions are much more than just rational functions, and so the question arises as to what functions (or signals) give rise to an analytic Laplace transform. The resolution to this question is the content of the Pailey Wiener theorem (or well, one of them), where an L2 signal may be extracted as the inverse of a function that is analytic in the right half of the complex plane. What about the other way around? If we have an L2 signal, can we show that its Laplace transform is analytic in the right half plane? The answer, perhaps unsurprisingly, is yes. And the proof of this comes down to a touch of the Dominated Convergence Theorem and Moreras theorem. If you are curious to see Moreras Theorem in action, then check out this video, which is Part 1 of two laying out the Pailey Wiener Theorem. //Additional Reading Zemanian - An Introduction to Generalized Functions and the Generalized Laplace and Legendre Transformations (JSTOR) https://www.jstor.org/stable/2027808?seq=1#metadata_info_tab_contents (SIAM) https://epubs.siam.org/doi/abs/10.1137/1010001?journalCode=siread //Exercises //Watch Next (The Other Pailey Wiener Theorem) https://youtu.be/jcY5Tq1JsVc Undergraduate Intro to the Laplace Transform https://youtu.be/mJSqt7nchRQ A Mature Look at the Laplace transform https://youtu.be/F-xS27UfF30 Introduction to Control Theory https://youtu.be/0v4WFmOm764 //Music Provided by Epidemic Sound Mt. Fuji - Ooyy Vegas - Onda Norte bam. bam. - Yomoti Tiny Things - Giants Nest Late Nights - Daxten Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referral/644nao //Books Functions of One Complex Variable - Conway https://amzn.to/3uhUCOH Real and Comlex Analysis - Rudin https://amzn.to/3riipvZ Real Analysis - Folland https://amzn.to/3ogVuiG Fundamentals of Differential Equations - Nagle, Saff, and Snider https://amzn.to/3qeGD9C Advanced Mathematical Analysis - Beals https://amzn.to/3fn2jdP From Vector Spaces to Function Spaces - Yutaka Yamamoto https://amzn.to/3sU2CVj Feedback Control Theory - Doyle, Francis, Tannenbaum https://amzn.to/3HFvuov Pick Interpolation and Hilbert Function Spaces - Agler and McCarthy https://amzn.to/3EOJU3K Nonlinear Systems - Khalil https://amzn.to/32Zyk8E Modern Control Systems - Dorf and Bishop https://amzn.to/331JQAm //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://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!$

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