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  2. Laplace transform - Wikipedia

    en.wikipedia.org/wiki/Laplace_transform

    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).

  3. Classical control theory - Wikipedia

    en.wikipedia.org/wiki/Classical_control_theory

    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

  4. List of Laplace transforms - Wikipedia

    en.wikipedia.org/wiki/List_of_Laplace_transforms

    The following is a list of Laplace transforms for many common functions of a single variable. [1] The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency ).

  5. Riemann–Lebesgue lemma - Wikipedia

    en.wikipedia.org/wiki/Riemann–Lebesgue_lemma

    In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L 1 function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis .

  6. Two-sided Laplace transform - Wikipedia

    en.wikipedia.org/wiki/Two-sided_Laplace_transform

    In pure mathematics the argument t can be any variable, and Laplace transforms are used to study how differential operators transform the function. In science and engineering applications, the argument t often represents time (in seconds), and the function f ( t ) often represents a signal or waveform that varies with time.

  7. Multidimensional transform - Wikipedia

    en.wikipedia.org/wiki/Multidimensional_transform

    The multidimensional Laplace transform is useful for the solution of boundary value problems. Boundary value problems in two or more variables characterized by partial differential equations can be solved by a direct use of the Laplace transform. [3] The Laplace transform for an M-dimensional case is defined [3] as

  8. Laplace transform applied to differential equations - Wikipedia

    en.wikipedia.org/wiki/Laplace_transform_applied...

    In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. First consider the following property of the Laplace transform:

  9. Integro-differential equation - Wikipedia

    en.wikipedia.org/wiki/Integro-differential_equation

    Consider the following second-order problem, ′ + + = () =, where = {,, <is the Heaviside step function.The Laplace transform is defined by, = {()} = ().Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,