the laplace transform is an

8/13/2019 The Laplace Transform is An

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The Laplace transform is anintegral transformperhaps second only to theFourier transformin its utility insolving physical problems. The Laplace transform is particularly useful in solving linearordinarydifferential equationssuch as those arising in the analysis of electronic circuits.

The (unilateral) Laplace transform (not to be confused with theLie derivative,also commonly

denoted ) is defined by

(1)

where is defined for (Abramowitz and Stegun 1972). The unilateral Laplace transform is almostalways what is meant by "the" Laplace transform, although a bilateral Laplace transformis sometimesalso defined as

(2)

(Oppenheim et al. 1997). The unilateral Laplace transform is implemented

inMathematicaasLaplaceTransform[f[t], t, s].

The inverse Laplace transform is known as theBromwich integral,sometimes known as the Fourier-Mellin integral (see also the relatedDuhamel's convolution principle).

A table of several important one-sided Laplace transforms is given below.

conditions

1
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In the above table, is the zeroth-orderBessel function of the first kind, is thedelta function,

and is theHeaviside step function.

The Laplace transform has many important properties. The Laplace transform existence theorem states

that, if ispiecewise continuouson every finite interval in satisfying

(3)

for all , then exists for all . The Laplace transform is alsounique,in the sense

that, given two functions and with the same transform so that

(4)

thenLerch's theoremguarantees that the integral

(5)

vanishes for all for anull functiondefined by

(6)

The Laplace transform islinearsince

(7)

(8)

(9)

The Laplace transform of aconvolutionis given by

(10)
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Now considerdifferentiation.Let be continuously differentiable times in . If, then

(11)

This can be proved byintegration by parts,

(12)

(13)

(14)

(15)

Continuing for higher-order derivatives then gives

(16)

This property can be used to transform differential equations into algebraic equations, a procedure knownas theHeaviside calculus,which can then be inverse transformed to obtain the solution. For example,applying the Laplace transform to the equation

(17)

gives

(18)

(19)

which can be rearranged to

(20)

If this equation can be inverse Laplace transformed, then the original differential equation is solved.

The Laplace transform satisfied a number of useful properties. Considerexponentiation.

If for (i.e., is the Laplace transform of ), then

for . This follows from

(21)

(22)

(23)
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The Laplace transform also has nice properties when applied to integralsof functions. If ispiecewise

continuousand , then
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the laplace transform is an

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