Answer by Michael Engelhardt for Motivation and physical interpretation of...
The Laplace transform is the fundamental operation encoding the canonical ensemble in statistical mechanics. It converts the density of states $d(\epsilon )$ (a non-statistical concept) into the...
View ArticleAnswer by Alexandre Eremenko for Motivation and physical interpretation of...
Besides the important physical motivation pointed by Carlo Beenaker, there is another one, purely mathematical. Laplace transform is a generalization of a power series (and Dirichlet series).In...
View ArticleAnswer by Carlo Beenakker for Motivation and physical interpretation of the...
The physical motivation for the Laplace transform is causality.Consider the linear input-output relation$$f_{\text{output}}(t)=\int_{-\infty}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$Causality...
View ArticleMotivation and physical interpretation of the Laplace transform
Concerning the one-sided Laplace transform,$$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$what is a motivation to come up with that formula? I am particularly interested in "physical"...
View ArticleAnswer by Giuseppe Negro for Motivation and physical interpretation of the...
From R. N. Bracewell, "The Fourier Transform and Applications", McGraw Hill 3rd ed., pp.381:"Advantages of the Laplace transform over the Fourier transform for handling electrical transient and other...
View ArticleAnswer by Michael Hardy for Motivation and physical interpretation of the...
Interest is continuously compounded at rate $r,$ so that if you deposit $\\\$1$ now it will be worth $\\\$e^{rx}$ at time $x.$ How much do you need to deposit now in order to withdraw at rate...
View ArticleAnswer by Liviu Nicolaescu for Motivation and physical interpretation of the...
I will not discuss the uses of the Laplace transform.Myself I think of the analogy with sequences. A sequence $(a_n)_{n\geq 0}$ is determined by its generating function (convergence issues...
View ArticleAnswer by Tom Copeland for Motivation and physical interpretation of the...
Re "I have not yet encountered any good explanation of how the Laplace transform formula arises."I}Before its use in probability theory by Laplace (characteristic functions), as a weight function for...
View ArticleAnswer by Wahome for Motivation and physical interpretation of the Laplace...
The following simple motivation for the formula was given in my undergrad class, which I hope I'm not misremembering:Typically the way the Laplace transform arises in applications is when solving...
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