Explicit Green's Functions for the Robin Problem of the Laplace Equation on Balls in All Dimensions

Abstract: This paper establishes a unified elementary approach to explicitGreen's functions for the Laplace equation with Robin boundary
conditions on disks and $n$-dimensional balls for all $n\geq 3$. Expanding the fundamental solution into Fourier or Gegenbauer
series and matching coefficients yields the harmonic part explicitly.
Moreover, the Dirichlet and Neumann Green's functions emerge as limiting
cases ($\alpha\to\infty$ and $\alpha\to 0^+$), unifying all three
classical boundary value problems within a single elementary framework.