26 votes

Motivation and physical interpretation of the Laplace transform

The physical motivation for the Laplace transform is causality. Consider the linear input-output relation $$f_{\text{output}}(t)=\int_{0}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$ Causality dictates ...
Carlo Beenakker's user avatar
23 votes

Motivation and physical interpretation of the Laplace transform

Besides the important physical motivation pointed out by Carlo Beenakker, there is another one, purely mathematical. Laplace transform is a generalization of a power series (and Dirichlet series). In $...
Alexandre Eremenko's user avatar
19 votes

Motivation and physical interpretation of the Laplace transform

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 $\\\$f(x)...
Michael Hardy's user avatar
15 votes

Motivation and physical interpretation of the Laplace transform

The Laplace transform is the fundamental operation encoding the canonical ensemble in statistical mechanics. It converts the density of states $d(\varepsilon )$ (a non-statistical concept) into the ...
Michael Engelhardt's user avatar
12 votes

Nicer expression for 2.1369288...?

With a few substitutions, we find that $$c=\frac{k^3}{k^2-k+1}\quad\text{where}\quad1+\frac k{(1-k)^2}=e^k.$$ The solution for $k$ requires a more advanced function than Lambert $W$.
TheSimpliFire's user avatar
9 votes
Accepted

Nicer expression for 2.1369288...?

Apply Lagrange reversion to @TheSimpliFire’s equation: $$\frac{k}{(k-1)^2}+1=e^k\iff k=1+\sqrt{\frac k{e^k-1}}\\\implies k=1+\sum_{n=1}^\infty\frac1{n!}\left.\frac{d^{n-1}}{dx^{n-1}}\left(\frac x{e^x-...
Tyma Gaidash's user avatar
6 votes

Motivation and physical interpretation of the Laplace transform

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 aside) ...
Liviu Nicolaescu's user avatar
4 votes

The Fourier transform of the Liouville function?

Your first formula for $\lambda(x)$ is equivalent to $$\lambda(x)=\underset{N\to \infty}{\text{lim}}\left(\sum\limits_{z=-N}^N \cos(\pi(\Omega(z)+x-z))\, \text{sinc}(\pi(x-z))\right)\tag{1}.$$ ...
Steven Clark's user avatar
  • 1,061
4 votes

Motivation and physical interpretation of the Laplace transform

Re "I have not yet encountered any good explanation of how the Laplace transform formula arises." I think it's useful to look at early origins of the LPT, relations to other transforms, and ...
Tom Copeland's user avatar
  • 9,847
4 votes

Motivation and physical interpretation of the Laplace transform

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 ...
Giuseppe Negro's user avatar
3 votes
Accepted

Bound on $L^1$ norm of solution of two-point boundary value problem

First of all, there is clearly no bound if $0$ is an eigenvalue of $Lu=(pu')'+qu$, $u(0)=u(1)=0$. (This will not happen if $p(x)<0, q(x)\ge 0$ because then the operator is positive, but if $p,q$ ...
Christian Remling's user avatar
3 votes

Fréchet-valued symbols

Check if this follows with appropriate adjustments from the arguments for Thm.44.1 and Exr.44.6 in Trèves, François, Topological vector spaces, distributions and kernels, Pure and Applied Mathematics ...
Igor Khavkine's user avatar
3 votes
Accepted

Existence of solution to nonlinear first order PDE with C^1 bounds

I don't think in the level of generality you are looking at you can say anything useful. Let me give two examples with very contrasting behaviors. Here I am, per your comment, allowing myself to think ...
Willie Wong's user avatar
  • 36.5k
1 vote

Motivation and physical interpretation of the Laplace transform

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 ...
Wahome's user avatar
  • 737

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