How could we express the above integral in a similar form if the upper limit of the integral was some constant T rather than oo?

beefypy 2022-10-14 Answered
Try to deal with a problem, that might actually be related to Laplace transform. Here is brief overview. Let P ( t ) = p m t m + + p 1 t + p 0 . Then we know that,it is possible to express the following integral in the form.
0 e z t P ( t ) = a = 0 m p a a ! z a + 1 ,
where the equality comes from the property of Laplace transform.
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Answers (1)

Jean Deleon
Answered 2022-10-15 Author has 14 answers
You can use the lower incomplete gamma function
(1) γ ( s , x ) = 0 x t s 1 e t   d t
In your case
(2) 0 T t k e z t   d t = z k 1 0 T ( z t ) k e z t   d ( z t ) = ( 1 ) z k 1 γ ( k + 1 , z T )
So that
(3) 0 T P ( t ) e z t   d t = k = 0 m p k 0 T t k e z t   d t = ( 2 ) k = 0 m p k γ ( k + 1 , z T ) z k + 1
If you take T you have γ ( k + 1 , z T ) Γ ( k + 1 ) = k ! which leads back to your last expression
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