I have been trying to test my knowledge of the gamma function by calculating 1! and 4!. I got the right result for 1! but I cannot get 4! analytically.
To be more specific, I think I am not integrating the gamma function correctly for 4!
I have $4! = \int_0^\infty e^{-t}t^{z-1}dt$ with $z=4$. How can you do it?
Do I have to integrate by parts?
Thanks
Using integration by parts, we see that the gamma function satisfies the functional equation $\Gamma(z+1)=z\Gamma(z)$ and so $\Gamma(n+1)=n!$. So you can do it for $\Gamma(5)$.
ATTENTION $4!=\Gamma(5)$.
$$\small \Gamma(z+1)=\int_0^\infty\operatorname{e}^{-t}t^{z}\operatorname{d}t=\underbrace{\left[-\operatorname{e}^{-t}t^{z}\right]_0^\infty}_0-\int_0^\infty\left(-\operatorname{e}^{-t}\right)\left(zt^{z-1}\right)\operatorname{d}t=z\underbrace{\int_0^\infty\operatorname{e}^{-t}t^{z-1}\operatorname{d}t}_{\Gamma(z)}=z\Gamma(z) $$