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In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ l ə ˈ p l ɑː s /), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane).
The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t). The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0).
The Reverend Henry Harte, a fellow at Trinity College, Dublin translated the entire first volume of Mécanique Céleste, with Book 1 published in 1822 and Book 2 published separately in 1827. [10] Similarly to Bowditch (see below), Harte felt that Laplace's exposition was too brief, making his work difficult to understand:
Abraham, R.; Marsden, J. E. (2008). Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics with an Introduction to the Qualitative Theory of Dynamical Systems (2nd ed.).
The Laplace transform is a frequency-domain approach for continuous time signals irrespective of whether the system is stable or unstable. The Laplace transform of a function f ( t ) , defined for all real numbers t ≥ 0 , is the function F ( s ) , which is a unilateral transform defined by
In mathematics, transform theory is the study of transforms, which relate a function in one domain to another function in a second domain. The essence of transform theory is that by a suitable choice of basis for a vector space a problem may be simplified—or diagonalized as in spectral theory .
Let (,) be a function and a complex variable. The Laplace–Carson transform is defined as: [1] (,) = (,)The inverse Laplace–Carson transform is: (,) = + (,)where is a real-valued constant, refers to the imaginary axis, which indicates the integral is carried out along a straight line parallel to the imaginary axis lying to the right of all the singularities of the following expression:
It consolidated into one unified and harmonious system, the scattered developments of contributors such as Alexis Clairaut, Jean le Rond d'Alembert, Pierre-Simon Laplace, Leonhard Euler, and Johann and Jacob Bernoulli in the historical transition from geometrical methods, as presented in Newton's Principia, to the methods of mathematical analysis.