Laplace transform

In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by:

{1 over 1 - e^{-Ts}} int_0^T e^{-st} f(t),dt

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Laplace transformTime function

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1

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delta(t), unit impulse

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rac{1}{s}

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u(t), unit step

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rac{1}{(s+a)^n}

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rac{t^{n-1}}{(n-1)!}e^{-at}

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rac{a}{s(s+a)}

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1-e^{-at}

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rac{1}{(s+a)(s+b)}

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rac{1}{b-a}left(e^{-at}-e^{-bt} ight)

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rac{s+c}{(s+a)^2+b^2}

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e^{-at}left(cos{(bt)}+left(rac{c-a}{b} ight)sin{(bt)} ight)

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rac{ssinarphi+acosarphi}{s^2+a^2}

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sin{(at+arphi)}

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External links

• Online Computation of the transform or inverse transform, wims.unice.fr
• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.

Bibliography

• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
• William McC. Siebert, Circuits, Signals, and Systems, MIT Press, Cambridge, Massachusetts, 1986. ISBN 0-262-19229-2