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Monday, 25 March 2013

LAPLACE TRANSFORM BASIC


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
displaymath108
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 thatdisplaymath109
(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 istex2html_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 thatdisplaymath110
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 , thendisplaymath111
(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 havedisplaymath112
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 havedisplaymath113
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 havedisplaymath114
(8)
Heaviside function
The functiondisplaymath115
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. Thendisplaymath117,
and
displaymath118
Hence,
displaymath119
Example: Find
displaymath210.
Solution: Since
displaymath212,
we get
displaymath214
Hence,
displaymath216
In particular, we have
displaymath218
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