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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. List of Laplace transforms - Wikipedia

    en.wikipedia.org/wiki/List_of_Laplace_transforms

    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).

  4. 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:

  5. 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.

  6. Convolution - Wikipedia

    en.wikipedia.org/wiki/Convolution

    A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints.

  7. Laplace's equation - Wikipedia

    en.wikipedia.org/wiki/Laplace's_equation

    In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.

  8. Green's function for the three-variable Laplace equation

    en.wikipedia.org/wiki/Green's_function_for_the...

    Similarly, the Green's function for the three-variable Laplace equation can be given as a Fourier integral cosine transform of the difference of vertical heights whose kernel is given in terms of the order-zero modified Bessel function of the second kind as | ′ | = (+ ′ ′ ⁡ (′)) ⁡ [(′)].

  9. Inverse Laplace transform - Wikipedia

    en.wikipedia.org/wiki/Inverse_Laplace_transform

    Post's inversion formula for Laplace transforms, named after Emil Post, [3] is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform. The statement of the formula is as follows: Let f ( t ) {\displaystyle f(t)} be a continuous function on the interval [ 0 , ∞ ) {\displaystyle [0,\infty )} of exponential ...