Corollary 7.19 . [02XH]

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Corollary 7.19.

Let $\alpha\in\mathbb{N}^{r+1}$ . For $(w_{1},\dots,w_{r})\in\Delta^{r}$ , write $w_{0}=1-w_{1}-\dots-w_{r}$ . Then

\int_{\Delta^{r}}w_{0}^{\alpha_{0}}w_{1}^{\alpha_{1}}\dots w_{r}^{\alpha_{r}}\,\text{\rm d}w_{1}\wedge\dots\wedge\,\text{\rm d}w_{r}=\frac{\alpha_{0}!\dots\alpha_{r}!}{(|\alpha|+r)!}

and, for $i=0,\dots,r$ ,

\int_{\Delta^{r}}w_{0}^{\alpha_{0}}w_{1}^{\alpha_{1}}\dots w_{r}^{\alpha_{r}}\log(w_{i})\,\text{\rm d}w_{1}\wedge\dots\wedge\,\text{\rm d}w_{r}=-\frac{\alpha_{0}!\dots\alpha_{r}!}{(|\alpha|+r)!}\sum_{j=\alpha_{i}+1}^{|\alpha|+r}\frac{1}{j}.