Laplace transform

An integral transform that represents a function through its weighted integrals against an exponential kernel. It often converts differential equations into algebraic equations.

Bilateral Laplace transform: An integral transform that integrates a function over the entire real line using an exponential kernel. Unlike the usual one-sided transform, it includes values before time zero.
Ordinary differential equation: An equation relating a function of one variable to its derivatives. Laplace methods turn many linear ordinary differential equations into solvable algebraic relations.
Improper integral: An integral defined through a limit when its interval is unbounded or its integrand is unbounded. The transform's infinite integration interval makes convergence an essential part of its definition.
Pierre-Simon Laplace: A French mathematician and astronomer whose work shaped probability, celestial mechanics, and mathematical physics. The transform bears Laplace's name and grew from methods associated with his work.
Z-transform: A transform that represents a discrete-time sequence as a function of a complex variable. It plays a comparable role for discrete systems, where Laplace methods address continuous time.
Region of convergence: The set of complex values for which a transform integral converges. It determines where a Laplace transform exists and helps identify the original signal.
Initial value problem: A differential equation together with conditions specifying a solution at an initial point. Initial values enter naturally into derivative-transform formulas.
Complex number: A number with real and imaginary parts, often represented as a point in the complex plane. The transform variable is generally complex, and its real part controls exponential weighting.
Oliver Heaviside: An English electrical engineer and mathematician who developed operational methods for solving circuit problems. Heaviside's operational calculus anticipated practical uses of transform methods in engineering.
Mellin transform: An integral transform using powers of its variable as its kernel. Its scale-oriented kernel differs from the exponential weighting used by the Laplace transform.

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