The pole problem: A persistent challenge in spherical harmonic applications

Since the inception of spherical harmonic theory, the pole problem has been an inescapable companion in its many applications. A prevailing misconception holds that local corrections outside the standard framework can resolve it. However, our analysis reveals that the mapping from the spherical harmonic polynomial space to its tangent space disrupts completeness at the poles, demonstrating that the pole problem is fundamentally global in nature. Quantitative calculations further confirm that no representation on the tangent space can be constructed while preserving the completeness of spherical harmonic expansions. Despite this, spherical harmonics remain widely—and paradoxically—employed in fields such as planetary gravimetry and geomagnetic field modeling, where they are used to derive tensor representations of gravitational and magnetic fields. This amounts to an unjustifiable persistence: a practice sustained for 240 years in direct violation of representation theory’s completeness requirements. The time has come to abandon this deeply entrenched but mathematically flawed approach.