• In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable...
    76 KB (9,635 words) - 14:49, 12 July 2025
  • In mathematics, the inverse Laplace transform of a function F {\displaystyle F} is a real function f {\displaystyle f} that is piecewise-continuous,...
    8 KB (1,012 words) - 13:27, 30 June 2025
  • following is a list of Laplace transforms for many common functions of a single variable. The Laplace transform is an integral transform that takes a function...
    10 KB (574 words) - 16:09, 28 April 2025
  • Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms...
    18 KB (1,995 words) - 01:40, 28 February 2025
  • Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform....
    9 KB (1,354 words) - 06:50, 5 January 2025
  • Thumbnail for Pierre-Simon Laplace
    probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of...
    107 KB (13,313 words) - 19:32, 7 June 2025
  • representation. It can be considered a discrete-time equivalent of the Laplace transform (the s-domain or s-plane). This similarity is explored in the theory...
    43 KB (5,636 words) - 03:09, 8 July 2025
  • Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is...
    33 KB (4,679 words) - 18:09, 17 June 2025
  • Thumbnail for Meijer G-function
    integral transforms like the Hankel transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels...
    49 KB (10,023 words) - 12:56, 16 June 2025
  • Thumbnail for Heaviside step function
    distributions. The Laplace transform of the Heaviside step function is a meromorphic function. Using the unilateral Laplace transform we have: H ^ ( s )...
    14 KB (2,157 words) - 11:06, 13 June 2025
  • Thumbnail for Linear time-invariant system
    system is the Laplace transform or Z-transform of the system's impulse response, respectively. As a result of the properties of these transforms, the output...
    37 KB (5,902 words) - 18:02, 1 June 2025
  • Borel measure on the real line is of this kind. One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral...
    10 KB (1,336 words) - 21:43, 12 March 2025
  • Thumbnail for Fourier transform
    Hankel transform Hartley transform Laplace transform Least-squares spectral analysis Linear canonical transform List of Fourier-related transforms Mellin...
    177 KB (21,313 words) - 19:14, 8 July 2025
  • 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...
    3 KB (657 words) - 16:23, 10 May 2025
  • like the Fourier transform. The major difference is that the Laplace transform has a region of convergence for which the transform is valid. This implies...
    10 KB (1,621 words) - 11:19, 23 June 2025
  • f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplace transform F ( s ) {\displaystyle F(s)} , then a final value theorem establishes...
    18 KB (3,260 words) - 04:46, 6 January 2025
  • Thumbnail for Ramp function
    delta (in this formula, its derivative appears). The single-sided Laplace transform of R(x) is given as follows, L { R ( x ) } ( s ) = ∫ 0 ∞ e − s x R...
    7 KB (1,005 words) - 03:45, 8 August 2024
  • dividing the Laplace transform of the output, Y ( s ) = L { y ( t ) } {\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}} , by the Laplace transform of the...
    15 KB (2,498 words) - 06:36, 5 May 2025
  • Thumbnail for Gamma distribution
    _{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}} The Laplace transform of the gamma PDF, which is the moment-generating function of the gamma...
    66 KB (9,100 words) - 06:11, 7 July 2025
  • }^{+\infty }{g(\omega )\ e^{i\ \omega \ t}\ \mathrm {d} \ \omega }} The Laplace transform is another integral operator and is involved in simplifying the process...
    13 KB (1,857 words) - 21:52, 8 May 2024
  • frequency domain. Employing the inverse transform, i.e., the inverse procedure of the original Laplace transform, one obtains a time-domain solution. In...
    13 KB (1,278 words) - 17:01, 18 November 2024
  • In mathematics, the Laplace–Carson transform, named after Pierre Simon Laplace and John Renshaw Carson, is an integral transform with significant applications...
    2 KB (226 words) - 10:32, 7 June 2024
  • filter is u(t). Apply z-transform and Laplace transform on these two inputs to obtain the converted output signal. Perform z-transform on step input Z [ u...
    15 KB (2,867 words) - 01:36, 2 July 2025
  • poles and zeros of the Laplace transform in the complex plane. (In discrete time, one can similarly consider the Z-transform of the impulse response...
    32 KB (4,696 words) - 20:25, 28 February 2025
  • } The Laplace transform is the fractional Laplace transform when θ = 90 ∘ . {\displaystyle \theta =90^{\circ }.} The inverse Laplace transform corresponds...
    19 KB (2,884 words) - 04:13, 24 February 2025
  • by ξ {\displaystyle \xi } , then η {\displaystyle \eta } has the Laplace transform L η ( f ) = exp ⁡ ( − ∫ 1 − exp ⁡ ( − f ( x ) ) ξ ( d x ) ) {\displaystyle...
    3 KB (398 words) - 16:47, 25 January 2022
  • Thumbnail for Convolution
    f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u   f ( u )   d u {\displaystyle...
    67 KB (8,819 words) - 22:44, 19 June 2025
  • Thumbnail for Maple (software)
    viewpoint=[path=M]); Laplace transform f := (1+A*t+B*t^2)*exp(c*t); ( 1 + A t + B t 2 ) e c t {\displaystyle \left(1+A\,t+B\,t^{2}\right)e^{ct}} inttrans:-laplace(f, t...
    25 KB (2,644 words) - 21:36, 20 February 2025
  • Mellin inversion theorem (category Laplace transforms)
    which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function. If φ ( s )...
    5 KB (797 words) - 15:49, 18 July 2024
  • Thumbnail for Dirichlet integral
    improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign, contour...
    15 KB (2,929 words) - 21:29, 17 June 2025