Laplace transform

In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually ${\displaystyle t}$, in the time domain) to a function of a complex variable ${\displaystyle s}$ (in the complex-valued frequency domain, also known as s-domain, or s-plane). The functions are often denoted by ${\displaystyle x(t)}$ for the time-domain representation, and ${\displaystyle X(s)}$ for the frequency-domain.

The transform is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations^[1] and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic polynomial equations, and by simplifying convolution into multiplication.^[2][3] For example, through the Laplace transform, the equation of the simple harmonic oscillator (Hooke's law) $${\displaystyle x''(t)+kx(t)=0}$$ is converted into the algebraic equation $${\displaystyle s^{2}X(s)-sx(0)-x'(0)+kX(s)=0,}$$ which incorporates the initial conditions ${\displaystyle x(0)}$ and ${\displaystyle x'(0)}$, and can be solved for the unknown function ${\displaystyle X(s).}$ Once solved, the inverse Laplace transform can be used to revert it back to the original domain. This is often aided by referencing tables such as that given below.

The Laplace transform is defined (for suitable functions ${\displaystyle f}$) by the integral $${\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}$$ here s is a complex number.

The Laplace transform is related to many other transforms, most notably the Fourier transform and the Mellin transform.

Formally, the Laplace transform can be converted into a Fourier transform by the substituting ${\displaystyle s=i\omega }$ where ${\displaystyle \omega }$ is real. However, unlike the Fourier transform, which decomposes a function into its frequency components, the Laplace transform of a function with suitable decay yields an analytic function. This analytic function has a convergent power series, the coefficients of which represent the moments of the original function. Moreover unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of complex analysis, and especially contour integrals, can be used for simplifying calculations.