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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cylindrical harmonics ： ウィキペディア英語版 Cylindrical harmonics In mathematics, the cylindrical harmonics are a set of linearly independent solutions to Laplace's differential equation, $\backslash nabla^2\; V\; =\; 0$, 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). ==Definition== Each function $V\_n(k)$ of this basis consists of the product of three functions: :$V\_n(k;\backslash rho,\backslash varphi,z)=P\_n(k,\backslash rho)\backslash Phi\_n(\backslash varphi)Z(k,z)\backslash ,$ where $(\backslash rho,\backslash varphi,z)$ 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: :$V=P(\backslash rho)\backslash ,\backslash Phi(\backslash varphi)\backslash ,Z(z)$ and Laplace's equation, divided by ''V'', is written: :$$ \frac+\frac\,\frac+\frac\,\frac+\frac=0 The ''Z''  part of the equation is a function of ''z'' alone, and must therefore be equal to a constant: :$\backslash frac=k^2$ where ''k''  is, in general, a complex number. For a particular ''k'', the ''Z(z)'' function has two linearly independent solutions. If ''k'' is real they are: :$Z(k,z)=\backslash cosh(kz)\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash sinh(kz)\backslash ,$ or by their behavior at infinity: :$Z(k,z)=e^\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,e^\backslash ,$ If ''k'' is imaginary: :$Z(k,z)=\backslash cos(|k|z)\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash sin(|k|z)\backslash ,$ or: :$Z(k,z)=e^\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,e^\backslash ,$ It can be seen that the ''Z(k,z)'' functions are the kernels of the Fourier transform or Laplace transform of the ''Z(z)'' function and so ''k'' may be a discrete variable for periodic boundary conditions, or it may be a continuous variable for non-periodic boundary conditions. Substituting $k^2$ for $\backslash ddot/Z$ , Laplace's equation may now be written: :$$ \frac+\frac\,\frac+\frac\frac+k^2=0 Multiplying by $\backslash rho^2$, we may now separate the ''P''  and Φ functions and introduce another constant (''n'') to obtain: :$\backslash frac\; =-n^2$ :$\backslash rho^2\backslash frac+\backslash rho\backslash frac+k^2\backslash rho^2=n^2$ Since $\backslash varphi$ is periodic, we may take ''n'' to be a non-negative integer and accordingly, the $\backslash Phi(\backslash varphi)$ the constants are subscripted. Real solutions for $\backslash Phi(\backslash varphi)$ are :$\backslash Phi\_n=\backslash cos(n\backslash varphi)\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash sin(n\backslash varphi)$ or, equivalently: :$\backslash Phi\_n=e^\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,e^$ The differential equation for $\backslash rho$ is a form of Bessel's equation. If ''k'' is zero, but ''n'' is not, the solutions are: :$P\_n(0,\backslash rho)=\backslash rho^n\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash rho^\backslash ,$ If both k and n are zero, the solutions are: :$P\_0(0,\backslash rho)=\backslash ln\backslash rho\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,1\backslash ,$ If ''k'' is a real number we may write a real solution as: :$P\_n(k,\backslash rho)=J\_n(k\backslash rho)\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,Y\_n(k\backslash rho)\backslash ,$ where $J\_n(z)$ and $Y\_n(z)$ are ordinary Bessel functions. If ''k''  is an imaginary number, we may write a real solution as: :$P\_n(k,\backslash rho)=I\_n(|k|\backslash rho)\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash mathrm\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,K\_n(|k|\backslash rho)\backslash ,$ where $I\_n(z)$ and $K\_n(z)$ are modified Bessel functions. The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions: :$V(\backslash rho,\backslash varphi,z)=\backslash sum\_n\; \backslash int\; dk\backslash ,\backslash ,\; A\_n(k)\; P\_n(k,\backslash rho)\; \backslash Phi\_n(\backslash varphi)\; Z(k,z)\backslash ,$ where the $A\_n(k)$ are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary conditions of the problem. Note that the integral may be replaced by a sum for appropriate boundary conditions. The orthogonality of the $J\_n(x)$ is often very useful when finding a solution to a particular problem. The $\backslash Phi\_n(\backslash varphi)$ and $Z(k,z)$ functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions. When $P\_n(k\backslash rho)$ is simply $J\_n(k\backslash rho)$ , the orthogonality of $J\_n$, along with the orthogonality relationships of $\backslash Phi\_n(\backslash varphi)$ and $Z(k,z)$ allow the constants to be determined. If $(x)\_k$ is the sequence of the positive zeros of $J\_n$ then: :$\backslash int\_0^1\; J\_n(x\_k\backslash rho)J\_n(x\_k\text{'}\backslash rho)\backslash rho\backslash ,d\backslash rho\; =\; \backslash fracJ\_(x\_k)^2\backslash delta\_$ In solving problems, the space may be divided into any number of pieces, as long as the values of the potential and its derivative match across a boundary which contains no sources. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cylindrical harmonics」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.