LAPLACE TRANSFORM: Gamma Function

Let tex2html_wrap_inline28 , with tex2html_wrap_inline30 . Set tex2html_wrap_inline32 . It is easy to see that F(s) is defined for s > 0. Consider the new variable r = st. Then we have

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which implies

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where the Gamma function tex2html_wrap_inline44 is defined by

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The domain of the Gamma function is x > 1. Using integration by part, one can easily prove the fundamental formula

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which implies (knowing that tex2html_wrap_inline52 )

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Consequently, we have

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If you are interested to learn more about the Gamma function, click HERE.

[Gamma Function] [Beta Function]
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