# Laplace equation in N dimensional space

Is it possible to obtain a general expression for the solution of laplace equation that is valid in an euclidean space with an arbitrary dimension ?

I don't think there is a 'general expression for the solution' to laplace's equation in any set of spaces. Solutions of laplace's equation are any combination of harmonic functions.

vanhees71
Gold Member
2021 Award
Solutions of Laplces's equation are also called harmonic. One special case are the spherical harmonics. In 3D space they are simply known as spherical harmonics and are closely related to the Legendre Polynomials, which are the solution with "magnetic quantum number", $m=0$. The other functions are called the associated Legendre functions. With the factor $\exp(\mathrm{i} \varphi)$ and normalized this gives the spherical harmonics, $\mathrm{Y}_{lm}(\vartheta,\varphi)$ which are a complete set on the Hilbert space, $\mathrm{L}^2(\mathrm{S}_1)$, i.e., all functions that are defined on the unit sphere in $\mathbb{R}^3$ that are square integrable. For the spherical harmonics one has

$$\int_{\mathrm{S}_1} \mathrm{d} \Omega \mathrm{Y}_{lm}^*(\vartheta,\varphi)\mathrm{Y}_{l'm'}(\vartheta,\varphi)=\int_0^{\pi} \mathrm{d} \vartheta \int_0^{2 \pi} \mathrm{d} \varphi \sin \vartheta \mathrm{Y}_{lm}^*(\vartheta,\varphi)\mathrm{Y}_{l'm'}(\vartheta,\varphi)=\delta_{ll'} \delta_{mm'}.$$

Any harmonic function on $\mathbb{R}^3$ then can be expanded in the sense of convergence with respect to the corresponding Hilbert-space norm by

$$f(\vec{x})=\sum_{l=0}^{\infty} \sum_{m=-l}^{l} \left [f_{1lm}(r) r^l + f_{2lm} \frac{1}{r^{l+1}} \right ]\mathrm{Y}_{lm}.$$

In arbitrary dimensions the analogues of the spherical harmonics are known as Gegenbauer functions.

Laplace equation can be solved in 3 dimensional space by separation of variables and also in 4 , 5 and 6 dimensional spaces with cartesian coordinates but If I want to find the solution in N dimensional space is it possible to write a computer algorithm to find the solution for N dimensions

Ben Niehoff
Gold Member
You don't need spherical harmonics or Gegenbauer polynomials. The Green function for the Laplace equation in 3 dimensions is

$$G(\vec r, \vec r') = \frac{1}{\lvert \vec r - \vec r' \rvert}$$

and in n dimensions, it is easy to show that the Green function is

$$G(\vec r, \vec r') = \frac{1}{\lvert \vec r - \vec r' \rvert^{n-2}}$$

To show this, write the Laplace equation in spherical coordinates, and assume your solution is a function of r alone. Then the Green function follows by translational invariance.