Multi-Dimensional Analytic Functions for Laplace Equations and Generalized Cauchy–Riemann Equations

A new concept of projective solution is introduced for the multi-dimensional Laplace equations. We project the field point onto a characteristic vector to obtain a projective variable, which can be used to reduce the Laplace equations to a second-order ordinary differential equation with only a leading term multiplied by the squared norm of the characteristic vector. The projective solutions involve characteristic vectors as parameters, which must be complex numbers to satisfy a null equation. Since the projective variable is a complex variable, we can construct the analytic function based on the conventional complex analytic function theory. Both the analytic function and the Cauchy–Riemann equations are generalized for the multi-dimensional Laplace equations. A powerful numerical technique to solve the 3D Laplace equation with high accuracy is available by further developing the Trefftz-type bases. Numerical experiments confirm the accuracy and efficiency of the projective solutions method (PSM).