Laplace’s spherical harmonics “form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions” [1]. I have three related questions about this statement: (1) I can prove their orthonormality, but how do you prove that they form a complete set? (2) What does completeness mean for […]

I’m creating meshes for spherical harmonics, and I need a normal at a given point. Whenever I’m at the poles, $\cos{\theta} = \pm 1$, and I do not know how to find the derivative there. All the formulas I have found to describe the derivative have an $1 – x^2$ in the denominator, and I […]

Adding to the for dummies. The real spherical harmonics are orthonormal basis functions on the surface of a sphere. I’d like to fully understand that sentence and what it means. Still grappling with Orthonormal basis functions (I believe this is like Fourier Transform’s basis functions are sines and cosines, and sin is orthogonal to cos, […]

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