In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, , 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). Definition Each function of this basis consists of the product of three functions: where 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: or by their behavior at infinity: If k is imaginary: or: It can be seen that the Z(k,z) functions are the kernels of the Fourier transform or Laplace transform of the Z(z) function and so k may be a discrete variable for periodic boundary conditions, or it may be a continuous variable for non-periodic boundary conditions. Substituting for , Laplace's equation may now be written: Multiplying by , we may now separate the P and Φ functions and introduce another constant (n) to obtain: Since is periodic, we may take n to be a non-negative integer and accordingly, the the constants are subscripted. Real solutions for are or, equivalently: The differential equation for is a form of Bessel's equation. If k is zero, but n is not, the solutions are: If both k and n are zero, the solutions are: If k is a real number we may write a real solution as: where and are ordinary Bessel functions. If k is an imaginary number, we may write a real solution as: where and are modified Bessel functions. The cylindrical harmonics for (k,n) are now the