Laplace Transformation

Prerequisites: Calculus.

• Intuitive
• Concrete
• Abstract

Intuitive

The Laplace transform takes a signal that starts at some initial time and turns it into a function of a rate. For each value of the rate we multiply the signal by an exponential that decays at that rate and add up the result over all times. A large rate means that only the very beginning of the signal counts; a small rate means that the long-time behaviour counts too. The quote below gives a familiar example: discounting a future cash flow at a fixed interest rate and adding up the discounted payments is evaluating a Laplace transform at one value of the rate.

The main reason physicists and engineers use it is that it turns differentiation into multiplication by the rate variable. A linear differential equation with constant coefficients, such as the equation for a damped oscillator or an electric circuit, becomes an algebraic equation for the transformed function. The initial values, such as the starting position and velocity, enter automatically. One solves the algebra, and then transforms back.

It is closely related to the Fourier transform. The Fourier transform decomposes a signal into oscillations that last forever. The Laplace transform uses oscillations multiplied by exponentials, which can grow or decay, and so it also works for signals that grow in time and for problems that start at a definite moment.

Concrete

" Ever study finance, John? It made a lot more sense to me once I understood that the net present value of a future cash flow is simply one particular value of the Laplace transform of that cash flow, the value corresponding to the selected discount rate. "https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/

Abstract

For a locally integrable function $f$ on $[0,\infty)$ the Laplace transform is $$F(s)=\mathcal{L}\{f\}(s)=\int_0^\infty f(t)\,e^{-st}\,dt,\qquad s\in\mathbb{C}.$$ If the integral converges absolutely for some $s_0$, it converges absolutely in the half-plane $\mathrm{Re}\,s\ge\mathrm{Re}\,s_0$, so there is an abscissa of absolute convergence $\sigma_a$, and $F$ is holomorphic for $\mathrm{Re}\,s>\sigma_a$. Writing $s=\sigma+i\omega$, $$F(\sigma+i\omega)=\int_{-\infty}^{\infty}\Theta(t)f(t)e^{-\sigma t}\,e^{-i\omega t}\,dt,$$ so the Laplace transform is the Fourier transform of the damped, causal function $\Theta(t)f(t)e^{-\sigma t}$. Fourier inversion then gives the Bromwich integral $$f(t)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}F(s)\,e^{st}\,ds,\qquad \gamma>\sigma_a,$$ valid at points where $f$ is continuous (under the usual regularity conditions). Closing the contour to the left turns the inversion into a sum over the poles of $F$, which is how the exponentials and damped oscillations in the solution of a linear ODE arise.

The algebraic rules follow from integration by parts and Fubini's theorem: $$\mathcal{L}\{f'\}(s)=sF(s)-f(0),\qquad \mathcal{L}\Big\{\int_0^t f(t-\tau)g(\tau)\,d\tau\Big\}=F(s)G(s),$$ so causal convolution becomes multiplication. For a linear system with response $y=h*x$, the function $H(s)=\mathcal{L}\{h\}$ is the transfer function, and for a rational $H$ stability means that its poles lie in $\mathrm{Re}\,s<0$.

The transform extends to positive measures, $F(s)=\int_{[0,\infty)}e^{-st}\,d\mu(t)$. Bernstein's theorem states that a function on $(0,\infty)$ is of this form if and only if it is completely monotone, $(-1)^nF^{(n)}\ge0$ for all $n$. In statistical mechanics the canonical partition function is the Laplace transform of the density of states in the energy, with the inverse temperature in the role of $s$: $$Z(\beta)=\int_0^\infty g(E)\,e^{-\beta E}\,dE.$$ For exponents that scale with a large parameter $N$, Laplace's method gives $$\lim_{N\to\infty}\frac1N\ln\int e^{N(px-f(x))}\,dx=\sup_x\big(px-f(x)\big),$$ the Legendre transform of $f$ (for continuous $f$ growing fast enough that the integral converges). This is the precise sense in which the Legendre transform is the zero-temperature or large-$N$ limit of a Laplace transform, and it is how thermodynamic potentials related by Legendre transforms arise from partition functions related by Laplace transforms.

Contributing authors:

Authors: Jakob Schwichtenberg