converges for . But is the following solution only valid for ? Is it valid at all?
The issue is that does not converge for .
let
The fact is that the definition of the Gamma function...
(1)
... is valid only if , that is necessary condition for convergence of the integral in (1). For the Gamma function can be computed using the relation...
(2)
Kind regards
What Random Variable is asking for is probably: is the formula...
(1)
... valid for ?...
All right!... a simple way is try to compute the (1) setting [for example...] in some way and verify. If (1) is valid it should be...
(2)
Now we define...
(3)
... and then compute...
(4)
The integral in (4) can be attacked with the usual complex analysis and we obtain...
(5)
... and from (5)...
(6)
... so that is...
(7)
... and the (1) is verified for . Following the same procedure it is [probably...] possible to demonstrate that the same is for ...
Kind regards