Laplace’s transform turns calculus into algebra in s-space
Named for Pierre-Simon Laplace, the Laplace transform converts a function of a real variable, usually time, into a function of a complex variable s in the so-called s-domain. Differentiation and integration become multiplication and division, much as logarithms turn multiplication into addition, which makes linear differential equations far easier to solve.
It is defined by integrating f(t) multiplied by e to the power −st from zero to infinity. Engineers use it to replace differential and integral equations with polynomial ones and to turn convolution into multiplication. Applied to a simple harmonic oscillator obeying Hooke’s law, it yields an algebraic equation in X(s) that already contains the initial position and velocity; once that is solved, lookup tables help invert the answer back to time.
Laplace used a similar transform in probability theory and wrote extensively on generating functions in 1814. Euler had studied related integrals from 1744 as solutions of differential equations, introducing the gamma function along the way, and Lagrange followed his lead. Laplace took notice in 1782, and in 1785 made the key step of transforming a whole difference equation rather than seeking a solution in integral form. In 1809 he used his transform to find diffusion solutions that spread indefinitely, where Fourier’s periodic series could not reach.
Cauchy developed an operational calculus for it in 1821, and Oliver Heaviside popularized, or perhaps rediscovered, the method near 1900. Bernhard Riemann used the transform in his 1859 paper on counting primes, deriving the functional equation of the zeta function and an inversion theorem. Hjalmar Mellin studied it rigorously around the turn of the 20th century. In 1929 Vannevar Bush and Norbert Wiener’s Operational Circuit Analysis contained an early forerunner of today’s transform tables, and in 1934 Wiener and Raymond Paley treated Fourier transforms in the complex domain.
Widespread engineering use came during and soon after World War II, displacing Heaviside’s calculus, with Gustav Doetsch championing its advantages. Unlike its Fourier cousin, a Laplace transform is often analytic, with power-series coefficients that record the moments of the original function, and contour integration can simplify the work. It is essentially the same as the Mellin transform.
Source: Laplace transform
