The Laplace-Carson transform of a real-valued function is an integral transform defined by the formula
(1)
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In most cases, the function is defined only for certain functions which lie in a class of real-valued functions. Functions in satisfy three properties, namely:
1. is integrable in every interval of finite length,
2. for all ,
3. There exists a real number such that for all values .
In particular, implies that exists for all real numbers .
One may regard the Laplace-Carson Transform as a variation of the regular Laplace transform specifically devised by Carson to make the transform of the Heaviside step function --a function whose Laplace transform is given by --equal to 1 for all values . Indeed, from the definition of alone, one can easily deduce this property of as well as a collection of other straightforward elementary properties of the transform itself. For example, if is a function whose Laplace-Carson transform is denoted and if is used as shorthand for applying the Laplace-Carson transform to and arriving at , the following identities hold:
(2)
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(3)
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and
(4)
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Moreover, one can show that for arbitrary real numbers and ,
(5)
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(6)
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and
(7)
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The identities in () and () are known as the lag and displacement theorems, respectively.
Given functions whose Laplace-Carson transforms are , respectively, one can show the convolution/multiplication theorem:
(8)
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Finally, one can show that
(9)
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and
(10)
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In addition to the above, one can prove a number of more colorful results about the Laplace-Carson identity using various other methods; many such results require a bit more sophistication (Rubinstein and Rubinstein 1999).