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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).

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  • Cylindrical harmonics (en)
  • Harmonique cylindrique (fr)
  • 柱諧函數 (zh)
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  • 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). (en)
  • En mathématiques, les harmoniques cylindriques sont un ensemble de solutions linéairement indépendantes de l'équation différentielle de Laplace exprimées en coordonnées cylindriques ρ (rayon), φ (azimut) et z (cote). Chaque fonction Vn(k) est le produit de trois termes, chacun ne dépendant que d'une coordonnée. Le terme dépendant de ρ s'exprime avec les fonctions de Bessel (qui sont parfois également appelées harmoniques cylindriques). (fr)
  • 在數學中，柱諧函數是指在柱坐標中，拉普拉斯方程， ，的一系列的解。每一個柱諧函數 都是三個函數的積： 其中 是柱坐標下的坐標（分別為半徑、極角和高度），而 n 和 k 則是兩個常數，用以區分不同的柱諧函數。所有的柱諧函數一起，組成一組正交完備的基底，任何一個拉普拉斯方程的解都可以寫成這些函數的線性組合。 有時候，柱諧函數也用來指代貝塞爾函數（柱諧函數最重要的組成部分）。 (zh)
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  • 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). (en)
  • En mathématiques, les harmoniques cylindriques sont un ensemble de solutions linéairement indépendantes de l'équation différentielle de Laplace exprimées en coordonnées cylindriques ρ (rayon), φ (azimut) et z (cote). Chaque fonction Vn(k) est le produit de trois termes, chacun ne dépendant que d'une coordonnée. Le terme dépendant de ρ s'exprime avec les fonctions de Bessel (qui sont parfois également appelées harmoniques cylindriques). (fr)
  • 在數學中，柱諧函數是指在柱坐標中，拉普拉斯方程， ，的一系列的解。每一個柱諧函數 都是三個函數的積： 其中 是柱坐標下的坐標（分別為半徑、極角和高度），而 n 和 k 則是兩個常數，用以區分不同的柱諧函數。所有的柱諧函數一起，組成一組正交完備的基底，任何一個拉普拉斯方程的解都可以寫成這些函數的線性組合。 有時候，柱諧函數也用來指代貝塞爾函數（柱諧函數最重要的組成部分）。 (zh)
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