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Last visit:  Fri 26 Jul 2019, 09:53
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Let $\mathbf{x}=(x_1,\ldots,x_d) \in \mathbbm{R}^d$, $a>0$, $d \geqslant 2$, $n \in \mathbbm{N}$, then: \begin{multline} M_{ijkl}[n] \doteqdot \int_{\mathbbm{R}^d} \dd \mathbf{x} \, |\mathbf{x}|^n \mathrm{e}^{-a x^2} x_i x_j x_k x_l \ = \pi^{d/2} \frac{3}{4} \frac{(d+n) (d+n+2)}{d(d+2)} \frac{\Gamma \left[(d+n)/2 \right]}{\Gamma\left( d/2 \right)} \frac{1}{a^{(d+n+4)/2}} \ \times \left\{ \delta_{ijkl} + \frac{1}{3} \Big[ \delta_{ij}\delta_{kl}(1-\delta_{ik}) + \delta_{ik} \delta_{jl}(1-\delta_{ij}) + \delta_{il} \delta_{jk}(1-\delta_{ij}) \Big] \right\}. \label{mijkl} \end{multline}

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