An Elementary Derivation of the Fundamental Solution of the Wave Equation in All Dimensions

摘要: We give an elementary, self-contained derivation of the fundamental solution for the wave equation in every integer dimension. The argument uses only single-variable calculus: spherical coordinates in Fourier space, exponential regularization, and the complex power function. Huygens' principle---sharp wavefronts in odd dimensions, wakes in even ones---is seen to arise from the single-valuedness or branching of $z^{(n-1)/2}$, because the exponent $(n-1)/2$ is an integer when $n$ is odd and a half-integer when $n$ is even (Figure~\ref{fig:branch}). No Bessel functions, special-function identities, the theory of tempered distributions, or radial transform formulas are needed; the only limiting processes used are the classical approximate identities for the Dirac delta, and the passage to the limit is justified by explicit elementary estimates.