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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:

int_{-infty}^{+infty} e^{-st} f(t),dt.

Alternative Definition

Moreover, both transforms are sometimes defined slightly differently, by

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: F(s)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

◄ Laplace transform: mathcal{L}left{f(t) ight}
Laplace transform: s int_{0^-}^{+infty} e^{-st} f(t),dt. ►

~ Table of Content ~

►Introduction
►mathcal{L} left{f(t) ight}
►int_{0^-}^infty e^{-st} f(t),dt.
► rac{1}{2 pi imath} int_{ gamma - imath infty}^{ gamma + imath infty} e^{st} F(s),ds,
►mathcal{L}left{f(t) ight}
►int_{-infty}^{+infty} e^{-st} f(t),dt.
►s int_{0^-}^{+infty} e^{-st} f(t),dt.
►s int_{-infty}^{+infty} e^{-st} f(t),dt.
►a mathcal{L}left{ f(t) ight} +
►s mathcal{L}{f} - f(0)
►s^2 mathcal{L}{f} - s f(0) - f'(0)
►s^n mathcal{L}{f} - s^{n - 1} f(0) - cdots - f^{(n - 1)}(0)
►-F'(s)
►(-1)^{n} F^{(n)}(s)
►mathcal{L}left{ 1 * f(t) ight} {1 over s} mathcal{L}{f}
►F(s - a)
►e^{at} f(t)
►e^{-as} F(s)
►f(t - a) u(t - a)
►mathcal{L}{ f } mathcal{L}{ g }
►{1 over 1 - e^{-Ts}} int_0^T e^{-st} f(t),dt

 

 

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