Summation of Powers over Product of Differences/Proof 2

Proof
By the Residue Theorem:


 * $\displaystyle \sum_{j \mathop = 1}^n \begin{pmatrix} {\dfrac { {x_j}^r} {\displaystyle \prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \left({x_j - x_k}\right)} } \end{pmatrix} = \dfrac 1 {2 \pi i} \int \limits_{\left\lvert{z}\right\rvert \mathop = R} \dfrac {z^r \rd z} {\left({z - z_1}\right) \cdots \left({z - z_n}\right)}$

where $R > \left\lvert{z_1}\right\rvert, \ldots, \left\lvert{z_n}\right\rvert$.

The Laurent series of the integrand converges uniformly on $\left\lvert{z}\right\rvert = R$:

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

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