Coulomb electrostatics in arbitrary spatial dimensions

In three spatial dimensions, the electrostatic potential \(\phi(\mathbf{x})\) satisfies Poisson's equation:\[\left(\frac{\partial}{\partial \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \right) \phi(\mathbf{x}) = - \frac{\rho(\mathbf{x})}{\varepsilon_{0}}\]The solution to this partial differential equation can be written down in terms of the Green function for Laplace's operator:\[\phi(\mathbf{x}) = \phi_{0}(\mathbf{x}) + \int dy \left[ G(\mathbf{x}, \, \mathbf{y}) \rho(\mathbf{y})\right] \]where \(\phi_{0}\) is a harmonic function and \(G\) satisfies\[\left(\frac{\partial}{\partial \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \right) G(\mathbf{x}, \, \mathbf{y}) = -\frac{1}{\varepsilon_{0}}\delta(\mathbf{x} - \mathbf{y})\]In order to find the Green function, we write it as the Fourier transform of another function:\[G(\mathbf{x}, \, \mathbf{y}) = \int \int dbdc \left[ g(\mathbf{b}, \, \mathbf{c}) \exp{\left( -i \mathbf{x} \cdot \mathbf{b} +i\mathbf{y} \cdot \mathbf{c}\right)} \right]\]This leads to\[g(\mathbf{b}, \, \mathbf{c}) = \frac{1}{\varepsilon_{0}} \frac{\delta(\mathbf{b} - \mathbf{c})}{\mathbf{b}^{2}}= \frac{1}{2\varepsilon_{0}} \delta(\mathbf{b} - \mathbf{c}) \int_{0}^{\infty}d\tau \, \exp{\left(- \frac{\tau}{2}\mathbf{b}^{2} \right)} \]Then, the Green function yields\[G(\mathbf{x}, \, \mathbf{y}) = \frac{1}{2\varepsilon_{0}} \int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{3} \exp{\left(-\frac{1}{2\tau}(\mathbf{x} - \mathbf{y})^{2} \right)}\]This expression can be generalize to \(D\) spatial dimensions: \[G(\mathbf{x}, \, \mathbf{y}) = \frac{1}{2 \varepsilon_{D}}\int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{D} \exp{\left(- \frac{1}{2\tau} (\mathbf{x} - \mathbf{y})^{2} \right)}\]where we have introduced \(\varepsilon_{D}\) as the \(D\)-dimensional analog of \(\varepsilon_{0}\). The Coulomb potential corresponds to a localized charge density:\[\rho(\mathbf{x}) = e \delta(\mathbf{x} - \mathbf{x}_{e})\]In \(D\) spatial dimensions, the Coulomb potential is\[\phi_{C}(\mathbf{x}) = \frac{e}{2 \varepsilon_{D}}\int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{D} \exp{\left(- \frac{1}{2\tau} (\mathbf{x} - \mathbf{x}_{e})^{2} \right)} = \frac{e}{2 \varepsilon_{D}} \left( \frac{2}{(\mathbf{x} - \mathbf{x}_{e})^{2}} \right)^{\nu}\Gamma(\nu) \qquad \nu \equiv \frac{D - 2}{2}\]This expression reduces to the familiar form when \(D = 3\).

In three spatial dimensions, the Green function can be written as a Legendre series. For arbitrary spatial dimension \(D\) the Legendre series generalizes to a Gegenbauer series. With \(|\mathbf{x}| > |\mathbf{y}|\) this yields\[G(\mathbf{x}, \, \mathbf{y}) = \frac{\Gamma(\nu)}{2\varepsilon_{D}}\left(\frac{2}{|\mathbf{x}| |\mathbf{y}|}\right)^{\nu}\sum_{n = 0}^{\infty}\eta^{n+\nu}C_{n}^{\hphantom{n}\nu}(\xi)\]where\[\eta \equiv \frac{|\mathbf{y}|}{|\mathbf{x}|} \qquad \xi \equiv \frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}| |\mathbf{y}|}\]