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Theorems Involv ing the Laplace Transform

Definition The Laplace Transform of a function is an integral operator defined by

It is naturally assumed that he function has certain properties on 0, that permit the integral toconverge for at least some values of. We denote the resulting function of the variablesby F(s). That is, which is defined for values ofs for which the integral converges.Functions of Exponential Order

A functionfis of exponential order at on the Interval 0, if there exists a constant 0 andconstants and such that || for all .Piecewise Continuous Functions

f is piecewise (or sectionally) continuous on an open interval , if there are at most a finite number ofpoints

with

such that 1) f

is continuous on each subinterval

, and

2) f has one-sided limits at the endpoints of all subintervals. That is, for all , lim andlim both exist as well as the limits at the endpointslim and lim .Theorem 1 Iff is piecewise continuous and of exponential order on0,, then exists.Corollary 1 If exists, thenlim 0.Theorem 2 The Laplace Transform is a linear operator: If and are piecewise continuous and ofexponential order on0,, then

for any constants and.In what follows, assume that and exist. Then:Theorem 3 in words, the transform of is the transform of evaluated at :

|Theorem 4 Alternatively, Theorem 5 Ifn is a positive integer,

1 Theorem 6 Iff is periodic with periodT,

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The Convolut ion Integral

Iffandg are piecewise continuous on0, we the define theconvolution offandg, denoted by ,by the integral

Theorem 7 The convolution of two functions is commutative: Theorem 8 . That is,

Theorem 9

1 Theorem 10 If has a transform andlim exists and is finite, then

The Inverse Laplace Transform

A functionf(t) is the Inverse Laplace Transform of a functionsF(s) if The inverse transform ofF(s) is denoted by :

The inverse transform has the following integral form:

12 lim

This requires knowledge ofcontour integration, a subject that is taken up in aComplex Analysis course.

Corresponding to each of formula for the transform, there is a inverse transform formula:

Theorem 2 The Inverse Laplace Transform is a linear operator. If and are functions which posesinverse transforms and, respectively, then

Theorem 3 Alternatively, we can write this formula as

Theorem 4

Alternatively, |Although the transform of a product is not the product of the transforms, the convolution has thatmultiplicative property. Therefore,

Theorem 8

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Theorem 10

or