I'm going over my notes and trying to write the Beta function in terms of gamma functions. However, I just can't get $(1.73)$ from $(1.72)$.
Even if I substitute $t/(1-t)$ with $u$, I can't remove $t$. Can anyone shed some light on this? Thanks.
2026-03-25 16:39:24.1774456764
Writing beta function in terms of gamma functions (by substitution)
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Hint: If $u = \frac{t}{1-t}$, then $u(1-t)=t$, that is, $$\begin{align*} u-ut &=t \\ \Rightarrow u &= ut+t \\ \Rightarrow u &=t(1+u).\end{align*}$$ So $t= \frac{u}{1+u}$. Try substituting this now.