A list of Laplace transforms is provided in the table below; one lists some of the common properties, and the other lists some common examples. The tables are followed by a subsection outlining the Physics and Engineering areas in which the Laplace transforms are intensely utilized at present. A list of references is also provided in relation to possible non-commutative or higher dimensional extensions of the classical Laplace transforms (LTs).
Original | Transformed | comment | derivation |
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linearity | |
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convolution property | here |
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integration with respect to a parametre | here |
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diffentiation with respect to a parameter | |
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here |
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here |
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here |
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here |
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here |
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here |
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here |
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conditions | explanation | derivation |
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trivial | |
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here | |
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here | |
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here | |
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here | |
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See sinc function | here |
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gamma function
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here |
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See error function | here |
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See error function | here |
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here | |
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Bessel function
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here |
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See error function | here |
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Euler'sconstant ![]() |
here |
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Dirac delta function | ||
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Dirac delta with delay |
Although possibly `less popular' with physicists than the Fourier transform, the Laplace transform has applications both in Astrophysics, Engineering and Mathematical Biophysics.
In Astrophysics the Laplace transform is employed to succesfully `sharpen up' images of distant planets obtained by satellite mounted-telescopes of various kinds without having the disadvantage of FT that may lose fine detail through exponential multiplication “smoothing” of partially fuzzy images.
On the other hand, in Engineering applications the Laplace transform is often employed to calculate the transfer function of an engineered system such as an electrical network or electronic circuit.
In Mathematical Biophysics (and also in Optimal Control theories) both the Laplace and Fourier transforms are employed to model living systems and their components, and also to optimize such models.
[More to be added...]
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