Lemma 7.18 . [02XF]

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Lemma 7.18.

Let $\Delta^{r}$ be the standard simplex of $\mathbb{R}^{r}$ and $\beta=(\beta_{0},\dots,\beta_{r-1})\in\mathbb{N}^{r}$ . Let $f\in\mathscr{C}^{|\beta|+r}(\mathbb{R})$ where $|\beta|=\beta_{0}+\dots+\beta_{r-1}$ . For $(w_{1},\dots,w_{r})\in\Delta^{r}$ write $w_{0}=1-w_{1}-\dots-w_{r}$ . Then

\int_{\Delta^{r}}\bigg(\prod_{i=0}^{r-1}\frac{w_{i}^{\beta_{i}}}{\beta_{i}!}\bigg)f^{(|\beta|+r)}(w_{r})\,\text{\rm d}w_{1}\wedge\cdots\wedge\,\text{\rm d}w_{r}=f(1)-\sum_{j=0}^{|\beta|+r-1}\frac{f^{(j)}(0)}{j!}.