Given ...
(1)
... and the Euler's constant...
(2)
... demonstrate from definition (2) that is...
(3)
Kind regards
Problem: Prove that if then find
Proof:
Lemma: Let then
Proof: It can be readily seen that . So that . Furthermore . Thus the functional equation and the boundary condition are satisfied
Now using this we can get somehwere. If we look closely at we can see that . Now using the common fact that yields . Now using a litle snake-oil we can see that where . So then or equivalently . So that
The 'lemma' ...
(1)
... where...
(2)
... must be demonstrated and that can be done with the 'little nice formula' discussed in a previous thread...
(3)
If we compute the integral (3) with applying systematically the integration by parts we obtain...
(4)
Now if we take into account the identity...
(5)
... from (4) we derive the 'infinite product'...
(6)
Kind regards