Cylindrical harmonics

In mathematics, the cylindrical harmonics are a set of linearly independent solutions to Laplace's differential equation, ∇ 2 V = 0 {displaystyle abla ^{2}V=0} , expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function Vn(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). In mathematics, the cylindrical harmonics are a set of linearly independent solutions to Laplace's differential equation, ∇ 2 V = 0 {displaystyle abla ^{2}V=0} , expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function Vn(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 V n ( k ) {displaystyle V_{n}(k)} of this basis consists of the product of three functions: where ( ρ , φ , z ) {displaystyle ( ho ,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. Since all of the surfaces of constant ρ, φ and z  are conicoid, Laplace's equation is separable in cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written: and Laplace's equation, divided by V, is written: The Z  part of the equation is a function of z alone, and must therefore be equal to a constant: where k  is, in general, a complex number. For a particular k, the Z(z) function has two linearly independent solutions. If k is real they are: