Cylindrical harmonics

In mathematics, the cylindrical harmonics are a set of linearly independent solutions to Laplace's differential equation, ${\displaystyle \nabla ^{2}V=0}$, expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function V_n(k) is the product of three terms, each depending on one coordinate alone. The ρ-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics).

Each function ${\displaystyle V_{n}(k)}$ of this basis consists of the product of three functions:

where ${\displaystyle (\rho ,\varphi ,z)}$ are the cylindrical coordinates, and n and k are constants which distinguish the members of the set from each other. As a result of the superposition principle applied to Laplace's equation, very general solutions to Laplace's equation can be obtained by linear combinations of these functions.

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