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  2. 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 ...

  3. Z-transform - Wikipedia

    en.wikipedia.org/wiki/Z-transform

    The properties of Z-transforms (listed in § Properties) ... The inverse Laplace transform is a mathematical abstraction known as an impulse-sampled function.

  4. Laplace transform - Wikipedia

    en.wikipedia.org/wiki/Laplace_transform

    Finally, using the linearity property and the known transform for exponential decay (see Item #3 in the Table of Laplace Transforms, above), we can take the inverse Laplace transform of H(s) to obtain = {()} = (), which is the impulse response of the system.

  5. 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).

  6. Mellin inversion theorem - Wikipedia

    en.wikipedia.org/wiki/Mellin_inversion_theorem

    In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.

  7. 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

  8. Convolution theorem - Wikipedia

    en.wikipedia.org/wiki/Convolution_theorem

    In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).

  9. 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