PlanetPhysics/Table of Laplace Transforms

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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).

\subsubsection*{Properties}

\hlineOriginal Transformed comment derivation
\hline linearity
convolution property here
integration with respect to a parametre here
diffentiation with respect to a parameter
here
here
here
here

here

here
here

\subsubsection*{Examples}

\hline conditions explanation derivation
\hline trivial
here
here
here
here
See sinc function

here

gamma function

here

See error function here
See error function here
here
Bessel function here
See error function here
Euler'sconstant here
Dirac delta function
Dirac delta with delay

\subsubsection*{Rational Functions}

\hline conditions explanation derivation
\hline1
here
here
exponential integral here
here
?
Laguerre polynomial

Applications of Laplace Transforms in Physics, Engineering and Mathematical Biophysics/Theoretical Biology

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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.

This contributed topic entry, in addition to the most used, or useful, Laplace transforms, may also contain information on how the convolution of Laplace transforms work, and also, possibly, higher dimensional generalizations of Laplace transforms, such as 2D Laplace transforms, non-commutative integral generalizations \`a la A. Connes of LTs, etc.

[More to be added...]

All Sources

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[1] [2] [3] [4]

References

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  1. A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, Lecture Notes in Math. , Springer-Verlag, Berlin, {\mathbf 725}: 19-14.
  2. A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids,J. Functional Anal . 148 : 314-367 (1997).
  3. A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) Free PDF file download.
  4. B. Aniszczyk. 1991. A rigid Borel space., Proceed. AMS. , 113 (4):1013-1015., available online.