Lec 19 | MIT 18.03 Differential Equations, Spring 2006
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone asks where the Laplace transform comes from or how to understand it intuitively, especially as a continuous version of power series and generating functions.
Recommended by 3 people across 5 different months, 2014 to 2022.
What it is
A 48-minute MIT OpenCourseWare lecture by Arthur Mattuck from the 18.03 Differential Equations course. It introduces the Laplace transform and its basic formulas, motivating it as a continuous analogue of a power series.
What people said
Strong rec Kolja · 2018-05 · Mathematics Stack Exchange · on “Laplace Transform: How to Interpret the Output”
I highly reccomend watching this analysis lecture from Arthur Mattuck [this video] . It explains the Laplace transfrom in a crystal clear way.
Strong rec Kolja · 2019-01 · Mathematics Stack Exchange · on “Visualizing the Laplace Transform (for beginners)”
I very highly recommend to watch this lecture given by Arthur Mattuck - [this video] It explains the Laplace transform in a beautiful manner, and shows the intuition behind it.
Recommends user91498 · 2014-07 · Mathematics Stack Exchange · on “Origin of Laplace Transform”
Is the Laplace transform the continuous version of the infinite power series? $$ \sum_{n=0}^\infty a_nx^n$$ becomes $$\int_0^\infty f(t)e^{-st}dt$$ I learned this by watching this video lecture: [this video] .
Recommends Math1000 · 2019-11 · Mathematics Stack Exchange · on “In what sense does the Laplace Transform give components of a signal in something called the 's-domain'?”
This may not answer your exact question, but this video should lend some insight into where the Laplace transform "comes from" and what it is actually doing: [this video]
Recommends Math1000 · 2020-02 · Mathematics Stack Exchange · on “Reference request: why integral transforms?”
This video gives a good intuition as to how the Laplace transform can be seen as a continuous version of generating functions: [this video]
Cites Math1000 · 2015-01 · Mathematics Stack Exchange · on “Connection between the Laplace transform and generating functions”
Using Laplace transforms to solve linear ODEs is analogous to using generating functions to solve linear recurrences. Here is a lecture from an MIT OCW course where the professor talks about this: [this video]
Cites vecohah · 2019-09 · Mathematics Stack Exchange · on “How is the following derived?”
To give some context, I am viewing the following lecture which derives the Laplace Transform. Lec 19 | MIT 18.03 Differential Equations, Spring 2006 [this video] At the 6:40 mark, the lecturer seems to skip over how the continuous analogue of the following discrete power series is derived. $$\sum_{0}^{\infty} a(n)x^n$$ Afterwards, the lecturer presents the following integral. $$\int_{0}^{\infty} a(t)x^tdt$$ I want to understand how the continuous analogue was derived as I am having a hard time of it myself. (Disclaimer: I am an engineering student, not a mathematics student.)
Cites 能够可能 · 2022-07 · Mathematics Stack Exchange · on “Power series and integrals”
@Paul Sinclair Course address: [this video] written as Riemann sum, it seems to be the sum of continuous functions. But the extra "dt" is incomprehensible, and I don't need "dt" at all.