Beta and Gamma functions relation proof. I found a proof of the Beta-Gamma functions relation, but I don't understand what happened between 3rd and 2nd last line. There was s^z and in the next line there is s^(z-1). Where did it come from? Or is the proof incorrect?
Can anyone explain it, please?
See the proof here
$$\int_0^\infty s^z \ e^{-s}$$
becomes
$$\int_0^\infty s^{z-1} \ e^{-s}$$

I believe you are asking about the line
$$
\begin{align}
&\phantom{=\ }\int_0^\infty s^ze^{-s}\int_0^\infty(us)^{w-1}e^{-su}\,\mathrm{d}u\,\mathrm{d}s\\
&=\int_0^\infty s^{z-1}e^{-s}\,\Gamma(w)\,\mathrm{d}s
\end{align}
$$
That is because, using the substitution $u\mapsto\frac ts$, we get
$$
\begin{align}
\int_0^\infty(us)^{w-1}e^{-su}\,\mathrm{d}u
&=\int_0^\infty t^{w-1}e^{-t}\frac1s\,\mathrm{d}t\\
&=\frac1s\,\Gamma(w)
\end{align}
$$
