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The static susceptibility of a free electron gas in D dimensions at T=0 is obtained by techniques of dimensional regularization. Our solutions for the susceptibility χ(k,D) are given in terms of the hypergeometric function. For any integer dimensions analytic expressions are possible. The high- and low-k series solutions are shown to be related by an analytic continuation if D is an odd integer, but not related if D is an even integer. The singularity at 2kF is a branch point, whereupon the series solutions are absolutely convergent, yielding χ(k=2kF,D)=(D−1)−1. The relationship of χkD has the appearance of a PVT diagram.
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Sharma et al. (1986) studied this question.
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