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?
$$\int_0^\infty s^z \ e^{-s}$$ 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 $$ \begin{align} \int_0^\infty(us)^{w-1}e^{-su}\,\mathrm{d}u &=\int_0^\infty t^{w-1}e^{-t}\frac1s\,\mathrm{d}t\\ &=\frac1s\,\Gamma(w) \end{align} $$
