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  2. 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).

  3. Laplace transform - Wikipedia

    en.wikipedia.org/wiki/Laplace_transform

    The following table provides Laplace transforms for many common functions of a single variable. [31] [32] For definitions and explanations, see the Explanatory Notes at the end of the table. Because the Laplace transform is a linear operator, The Laplace transform of a sum is the sum of Laplace transforms of each term.

  4. Perron's formula - Wikipedia

    en.wikipedia.org/wiki/Perron's_formula

    In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform.

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

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

    Examples of these can be seen to exist in rotational cylindrical coordinates as an integral Laplace transform in the difference of vertical heights whose kernel is given in terms of the order-zero Bessel function of the first kind as | ′ | = (+ ′ ′ ⁡ (′)) (> <), where > (<) are the greater (lesser) variables and ′.

  6. 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:

  7. Initial value theorem - Wikipedia

    en.wikipedia.org/wiki/Initial_value_theorem

    Of course we don't really need DCT here, one can give a very simple proof using only elementary calculus: Start by choosing A {\displaystyle A} so that ∫ A ∞ e − t d t < ϵ {\displaystyle \int _{A}^{\infty }e^{-t}\,dt<\epsilon } , and then note that lim s → ∞ f ( t s ) = α {\displaystyle \lim _{s\to \infty }f\left({\frac {t}{s ...

  8. Two-sided Laplace transform - Wikipedia

    en.wikipedia.org/wiki/Two-sided_Laplace_transform

    Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If f ( t ) is a real- or complex-valued function of the real variable t defined for all real numbers, then the two-sided Laplace transform is defined by the integral

  9. 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 .