In mathematics, the cylindrical harmonics are a set of linearly independent functions that are 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: $${\displaystyle V_{n}(k;\rho ,\varphi ,z)=P_{n}(k,\rho )\Phi _{n}(\varphi )Z(k,z)\,}$$ where ${\displaystyle (\rho ,\varphi ,z)}$ are the cylindrical coordinates, and n and k constants that differentiate the members of the set. 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 surfaces with 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 can be expressed as: $${\displaystyle V=P(\rho )\,\Phi (\varphi )\,Z(z)}$$ and Laplace's equation, divided by V, is written: $${\displaystyle {\frac {\ddot {P}}{P}}+{\frac {1}{\rho }}\,{\frac {\dot {P}}{P}}+{\frac {1}{\rho ^{2}}}\,{\frac {\ddot {\Phi }}{\Phi }}+{\frac {\ddot {Z}}{Z}}=0}$$

The Z  part of the equation is a function of z alone, and must therefore be equal to a constant: $${\displaystyle {\frac {\ddot {Z}}{Z}}=k^{2}}$$ 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: $${\displaystyle Z(k,z)=\cosh(kz)\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,\sinh(kz)\,}$$ or by their behavior at infinity: $${\displaystyle Z(k,z)=e^{kz}\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,e^{-kz}\,}$$

If k is imaginary: $${\displaystyle Z(k,z)=\cos(|k|z)\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,\sin(|k|z)\,}$$ or: $${\displaystyle Z(k,z)=e^{i|k|z}\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,e^{-i|k|z}\,}$$

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 ${\displaystyle k^{2}}$ for ${\displaystyle {\ddot {Z}}/Z}$ , Laplace's equation may now be written: $${\displaystyle {\frac {\ddot {P}}{P}}+{\frac {1}{\rho }}\,{\frac {\dot {P}}{P}}+{\frac {1}{\rho ^{2}}}{\frac {\ddot {\Phi }}{\Phi }}+k^{2}=0}$$

Multiplying by ${\displaystyle \rho ^{2}}$, we may now separate the P  and Φ functions and introduce another constant (n) to obtain: $${\displaystyle {\frac {\ddot {\Phi }}{\Phi }}=-n^{2}}$$ $${\displaystyle \rho ^{2}{\frac {\ddot {P}}{P}}+\rho {\frac {\dot {P}}{P}}+k^{2}\rho ^{2}=n^{2}}$$