Summation of Powers over Product of Differences/Proof 2

Proof
By Cauchy's Residue Theorem:


 * $\ds \sum_{j \mathop = 1}^n \begin {pmatrix} {\dfrac { {x_j}^r} {\ds \prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {x_j - x_k} } } \end {pmatrix} = \dfrac 1 {2 \pi i} \int \limits_{\size z \mathop = R} \dfrac {z^r \rd z} {\paren {z - z_1} \cdots \paren {z - z_n} }$

where $R > \size {z_1}, \ldots, \size {z_n}$.

The Laurent series of the integrand converges uniformly on $\size z = R$:

On integrating term my term, everything vanishes except the coefficient of $z^{-1}$.

Thus:
 * $\ds \sum_{\substack {j_1 \mathop + \mathop \cdots \mathop + j_n \mathop = r \mathop - n \mathop + 1 \\ j_1, \mathop \ldots j_n \mathop \ge 0} } {x_1}^{j_1} {x_2}^{j_2} \cdots {x_n}^{j_n}$