Need Help with Some Advanced Integration By Parts Methods

Question

Note: I am asking this question for someone to check my work for me. The problem started out with me finding z! which is equal to the $\mathbf P \mathbf i$ $\mathbf f \mathbf u \mathbf n \mathbf c \mathbf t \mathbf i \mathbf o \mathbf n$ $(z)$
$\mathbf P \mathbf i$ $\mathbf f \mathbf u \mathbf n \mathbf c \mathbf t \mathbf i \mathbf o \mathbf n$ $(z)$ = $\int_0^\infty$ $t^z$ $e^{-t}$ $dt$, Note: I got this online from the Wikipedia on Factorials explaining the extension of factorial to non-integer values of argument. I tried answering it to see if I can do it. Can somebody check my answer and give me a detailed way of actually showing the integration by parts way correctly if I am wrong?

Answer 1

$\mathbf P \mathbf i$ $\mathbf f \mathbf u \mathbf n \mathbf c \mathbf t \mathbf i \mathbf o \mathbf n$ $(z)$ = $\int_0^\infty$ $t^z$$e^-t$ $dt$
By using integration by parts, I got

$-t^z$ $e^{-t}$ $\mid_0^\infty$ $+$ $z$ $\int_0^\infty$ $e^{-t}$ $t^{z-1}$ $dt$
$= z$ $\int_0^\infty$ $e^{-t}$ $t^{z-1}$ $dt$
=$\mathbf P \mathbf i$ $\mathbf f \mathbf u \mathbf n \mathbf c \mathbf t \mathbf i \mathbf o \mathbf n$ $(z-1)$
Note: I let $u= t^z$ and $v= e^{-t}$
Hence the left-hand side cancelling in the first row because it goes to $0$ once I computed it.
The last step someone told me that my second line would be equal to $\mathbf P \mathbf i$ $\mathbf f \mathbf u \mathbf n \mathbf c \mathbf t \mathbf i \mathbf o \mathbf n$ $(z-1)$. That part was not not clear to me