Laplace transform

Laplace transform

In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily soluble algebraic equation from an ordinary differential equation. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory.

In mathematics, it is used for solving differential and integral equations. In physics, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In this analysis, the Laplace transform is often interpreted as a transformation from the "time-domain", in which inputs and outputs are functions of time, to the "frequency-domain", where the same inputs and outputs are functions of complex angular frequency, or radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Denoted displaystylemathcal{L} left{f(t) ight}, it is a linear operator on a function "f"("t") ("original") with a real argument "t" ("t" ≥ 0) that transforms it to a function "F"("s") ("image") with a complex argument "s". This transformation is essentially bijective for the majority of practical uses; the respective pairs of "f(t)" and "F(s)" are matched in tables. The Laplace transform has the useful property that many relationships and operations over the originals "f"("t") correspond to simpler relationships and operations over the images "F"("s") [Korn and Korn, Section 8.1] .

History

The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory.

From 1744, Leonhard Euler investigated integrals of the form:

: z = int X(x) e^{ax}, dx ext{ and } z = int X(x) x^A , dx,

— as solutions of differential equations but did not pursue the matter very far. [Euler (1744), (1753) and (1769)] Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form:

: int X(x) e^{- a x } a^x, dx

— which some modern historians have interpreted within modern Laplace transform theory. [Lagrange (1773)] [Grattan-Guinness (1997) "p."260]

These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations. [Grattan-Guinness (1997) "p."261] However, in 1785, Laplace took the critical step forward when, rather than just look for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form:

: int x^s phi (s), dx,

— akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power. [Grattan-Guinness (1997) "p."261-262]

Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space as the solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space. [Grattan-Guinness (1997) "p." 262-266]

Formal definition

The Laplace transform of a function "f"("t"), defined for all real numbers "t" ≥ 0, is the function "F"("s"), defined by:

:F(s) = mathcal{L} left{f(t) ight}=int_{0^-}^infty e^{-st} f(t) ,dt.

The lower limit of 0 is short notation to mean

:lim_{varepsilon o 0+}int_{-varepsilon}^infty

and assures the inclusion of the entire Dirac delta function δ("t") at 0 if there is such an impulse in "f"("t") at 0.

The parameter "s" is in general complex:

:s = sigma + i omega ,

This integral transform has a number of properties that make it useful for analyzing linear dynamic systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, by "s". (This is similar to the way that logarithms change an operation of multiplication of numbers to addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts back to the time domain.

Bilateral Laplace transform

When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is normally intended. The Laplace transform can be alternatively defined as the "bilateral Laplace transform" or two-sided Laplace transform by extending the limits of integration to be the entire real axis. If that is done the common unilateral transform simply becomes a special case of the bilateral transform where the definition of the function being transformed is multiplied by the Heaviside step function.

The bilateral Laplace transform is defined as follows:

: F(s) = mathcal{L}left{f(t) ight} =int_{-infty}^{+infty} e^{-st} f(t),dt.

Inverse Laplace transform

The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral, the Fourier-Mellin integral, and Mellin's inverse formula):

: f(t) = mathcal{L}^{-1} {F(s)} = frac{1}{2 pi i} int_{ gamma - i cdot infty}^{ gamma + i cdot infty} e^{st} F(s),ds,

where gamma is a real number so that the contour path of integration is in the "region of convergence" of "F"("s") normally requiring gamma > Re("s"p) for every singularity "s"p of "F"("s") and "i"2 = −1. If all singularities are in the left half-plane, that is Re("s"p) < 0 for every "s"p, then gamma can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform.

An alternative formula for the inverse Laplace transform is given by Post's inversion formula.

Region of convergence

The Laplace transform "F"("s") typically exists for all complex numbers such that Re{"s"} > "a", where "a" is a real constant which depends on the growth behavior of "f"("t"), whereas the two-sided transform is defined in a range"a" < Re{"s"} < "b". The subset of values of "s" for which the Laplace transform exists is called the "region of convergence" (ROC) or the "domain of convergence". In the two-sided case, it is sometimes called the "strip of convergence."

The integral defining the Laplace transform of a function may fail to exist for various reasons. For example, when the function has infinite discontinuities in the interval of integration, or when it increases so rapidly that exp(-pt) cannot damp it sufficiently for convergence on the interval to take place. There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken, other than to say the defining integral converges. It is however easy to give theorems on cases where it may or may not be taken.

Properties and theorems

Given the functions "f"("t") and "g"("t"), and their respective Laplace transforms "F"("s") and "G"("s"):: f(t) = mathcal{L}^{-1} { F(s) } : g(t) = mathcal{L}^{-1} { G(s) }

the following table is a list of properties of unilateral Laplace transform:

Starting with the Laplace transform

:X(s) = frac{s+eta}{(s+alpha)^2+omega^2},

we find the inverse transform by first adding and subtracting the same constant α to the numerator:

:X(s) = frac{s+alpha } { (s+alpha)^2+omega^2} + frac{eta - alpha }{(s+alpha)^2+omega^2}.

By the shift-in-frequency property, we have

: x(t) = e^{-alpha t} mathcal{L}^{-1} left{ {s over s^2 + omega^2} + { eta - alpha over s^2 + omega^2 } ight}

::: = e^{-alpha t} mathcal{L}^{-1} left{ {s over s^2 + omega^2} + left( { eta - alpha over omega } ight) left( { omega over s^2 + omega^2 } ight) ight}

::: = e^{-alpha t} left [ mathcal{L}^{-1} left{ {s over s^2 + omega^2} ight} + left( { eta - alpha over omega } ight) mathcal{L}^{-1} left{ { omega over s^2 + omega^2 } ight} ight] .

Finally, using the Laplace transforms for sine and cosine (see the table, above), we have

:x(t) = e^{-alpha t} left [ cos{(omega t)}u(t)+left(frac{eta-alpha}{omega} ight)sin{(omega t)}u(t) ight] .

:x(t) = e^{-alpha t} left [ cos{(omega t)}+left(frac{eta-alpha}{omega} ight)sin{(omega t)} ight] u(t).

Example #6: Phase delay

To simplify this answer, we must recall the trigonometric identity that

:a sin omega t + b cos omega t = sqrt{a^2+b^2} cdot sin left(omega t + arctan (b/a) ight)

and apply it to our value for x("t"):

:

We can apply similar logic to find that

:mathcal{L}^{-1} left{ frac{scosphi - omega sinphi}{s^2+omega^2} ight} = cos{(omega t+phi)}.

See also

* Pierre-Simon Laplace
* Fourier transform
* Analog signal processing
* Laplace transform applied to differential equations
* Moment-generating_function

References

Bibliography

Modern

* G.A. Korn and T.M. Korn, "Mathematical Handbook for Scientists and Engineers", McGraw-Hill Companies; 2nd edition (June 1967). ISBN 0-0703-5370-0
* 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
* Davies, Brian, "Integral transforms and their applications", Third edition, Springer, New York, 2002. ISBN 0-387-95314-0
* Wolfgang Arendt, Charles J.K. Batty, Matthias Hieber, and Frank Neubrander. "Vector-Valued Laplace Transforms and Cauchy Problems", Birkhäuser Basel, 2002. ISBN-10:3764365498

Historical

* cite journal | author=Deakin, M. A. B. | year=1981 | title=The development of the Laplace transform | journal=Archive for the History of the Exact Sciences | volume=25 | pages=343–390 | doi=10.1007/BF01395660
* cite journal | author=— | year=1982 | title=The development of the Laplace transform | journal=Archive for the History of the Exact Sciences | volume=26 | pages=351–381
*Euler, L. (1744) "De constructione aequationum", "Opera omnia" 1st series, 22:150-161
*&mdash; (1753) "Methodus aequationes differentiales", "Opera omnia" 1st series, 22:181-213
*&mdash; (1769) "Institutiones calculi integralis" 2, Chs.3-5, in "Opera omnia" 1st series, 12
*Grattan-Guinness, I (1997) "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. "Pierre Simon Laplace 1749-1827: A Life in Exact Science", Princeton: Princeton University Press, ISBN 0-691-01185-0
*Lagrange, J. L. (1773) "Mémoire sur l'utilité de la méthode", "Œuvres de Lagrange", 2:171-234

External links

* [http://wims.unice.fr/wims/wims.cgi?lang=en&+module=tool%2Fanalysis%2Ffourierlaplace.en Online Computation] of the transform or inverse transform, wims.unice.fr
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
*
* [http://math.fullerton.edu/mathews/c2003/LaplaceTransformMod.html Laplace Transform Module by John H. Mathews]
* [http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/ Good explanations of the initial and final value theorems]
* [http://www.intmath.com/Laplace/1a_lap_unitstepfns.php Laplace and Heaviside] at Interactive maths.
* [http://www.vibrationdata.com/Laplace.htm Laplace Transform Table and Examples] at Vibrationdata.
* [http://www.syscompdesign.com/tech.htm Laplace Transform Cookbook] at Syscomp Electronic Design.
* [http://www.exampleproblems.com/wiki/index.php/PDE:Laplace_Transforms Examples] of solving boundary value problems (PDEs) with Laplace Transforms


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