Laplace transform intuition

[member 428835]

hello pf!

i am wondering if anyone here knows of a geometric, intuitive explanation for the laplace transform? if so, please direct me to the source of if you could, explain to me your understanding?

thanks!

 

[homeomorphic]

The Laplace transform always bothered me. It took me a very long time to understand it deeply.

Here's a start.

http://ocw.mit.edu/courses/mathemat...ure-19-introduction-to-the-laplace-transform/

Also, it helps to understand Fourier transforms and Fourier series. That gives you more physical/geometric intuition. Once you understand that stuff, you can interpret Laplace transforms in terms of them. Places to get intuition about Fourier series/transforms would include signal processing books, or if you want a more mathematical approach that's still pretty intuitive, you could look at Discourse on Fourier Series by Lanczos.

 

[member 428835]

thanks homeomorphic!

 

[micromass]

See this: https://www.physicsforums.com/showthread.php?t=461628 The videos in that thread basically show that the Laplace transform is the continuous extension of power series. Much in the same way that Fourier transforms are the continuous analogue of Fourier series.

 

[blue_raver22]

The Laplace transform is a mathematical tool used to analyze systems in the time domain. It essentially converts a function of time into a function of a complex variable, s. This transformation allows us to solve differential equations and analyze the behavior of systems in the frequency domain.

One way to understand the Laplace transform intuitively is to think of it as a way to "zoom in" on the behavior of a system. Just like how a microscope allows us to see details that are not visible to the naked eye, the Laplace transform allows us to see the behavior of a system at a more detailed level.

Another way to think about it is in terms of frequency response. The Laplace transform essentially breaks down a function into its individual frequency components. This allows us to analyze how the system responds to different frequencies, which can be useful in understanding the stability and behavior of a system.

Overall, the Laplace transform is a powerful tool that allows us to gain a deeper understanding of systems and their behavior. Its geometric and intuitive interpretation can help us visualize and analyze complex systems in a more accessible way.