LAPLACE TRANSFORM

Basic Definitions and Results

Let f(t) be a function defined on tex2html_wrap_inline134 . The Laplace transform of f(t) is a new function defined as

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The domain of tex2html_wrap_inline138 is the set of tex2html_wrap_inline140, such that the improper integral converges.

(1)
We will say that the function f(t) has an exponential order at infinity if, and only if, there exist tex2html_wrap_inline144 and M such that

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(2)
Existence of Laplace transform
Let f(t) be a function piecewise continuous on [0,A] (for every A>0) and have an exponential order at infinity with tex2html_wrap_inline154 . Then, the Laplace transform tex2html_wrap_inline138 is defined for tex2html_wrap_inline158 , that is tex2html_wrap_inline160 .
(3)
Uniqueness of Laplace transform
Let f(t), and g(t), be two functions piecewise continuous with an exponential order at infinity. Assume that

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then f(t)=g(t) for tex2html_wrap_inline168 , for every B > 0, except maybe for a finite set of points.

(4)
If tex2html_wrap_inline172 , then

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(5)
Suppose that f(t), and its derivatives tex2html_wrap_inline176 , for tex2html_wrap_inline178 , are piecewise continuous and have an exponential order at infinity. Then we have

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This is a very important formula because of its use in differential equations.

(6)
Let f(t) be a function piecewise continuous on [0,A] (for every A>0) and have an exponential order at infinity. Then we have

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where tex2html_wrap_inline186 is the derivative of order n of the function F.

(7)
Let f(t) be a function piecewise continuous on [0,A] (for every A>0) and have an exponential order at infinity. Suppose that the limit tex2html_wrap_inline198, is finite. Then we have

displaymath114

(8)
Heaviside function
The function

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is called the Heaviside function at c. It plays a major role when discontinuous functions are involved. We have

displaymath116

When c=0, we write tex2html_wrap_inline204 . The notation tex2html_wrap_inline206, is also used to denote the Heaviside function.

(9)
Let f(t) be a function which has a Laplace transform. Then

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and

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Hence,

displaymath119

Example: Find

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Solution: Since

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we get

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Hence,

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In particular, we have

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The next example deals with the Gamma Function.

[Differential Equations] [First Order D.E.] [Second Order D.E.]
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Author: Mohamed Amine Khamsi

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