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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 Laplacian field ： ウィキペディア英語版 Harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function ''f'' : ''U'' → R (where ''U'' is an open subset of R''n'') which satisfies Laplace's equation, i.e. :$\backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; +\; \backslash frac\; =\; 0$ everywhere on ''U''. This is usually written as :$\backslash nabla^2\; f\; =\; 0$ or :$\backslash textstyle\; \backslash Delta\; f\; =\; 0$ == Examples == Examples of harmonic functions of two variables are: * The real or imaginary part of any holomorphic function * The function $\backslash ,\backslash !\; f(x,y)=e^\; \backslash sin\; y$; this is a special case of the example above, as $f(x,y)=\backslash operatorname(e^)$, and $e^$ is a holomorphic function. * The function ::$\backslash ,\backslash !\; f(x\_1,x\_2)=\backslash ln\; (x\_1^2+x\_2^2)$ : defined on $\backslash mathbb^2\; \backslash setminus\; \backslash lbrace\; 0\; \backslash rbrace$ (e.g. the electric potential due to a line charge, and the gravity potential due to a long cylindrical mass) Examples of harmonic functions of three variables are given in the table below with $r^2=x^2+y^2+z^2$: : |Unit point charge at origin |- |align=center|$\backslash frac$ |x-directed dipole at origin |- |align=center|$-\backslash ln(r^2-z^2)\backslash ,$ |Line of unit charge density on entire z-axis |- |align=center|$-\backslash ln(r+z)\backslash ,$ |Line of unit charge density on negative z-axis |- |align=center|$\backslash frac\backslash ,$ |Line of x-directed dipoles on entire z axis |- |align=center|$\backslash frac\backslash ,$ |Line of x-directed dipoles on negative z axis |} Harmonic functions that arise in physics are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions). On regions without boundaries, adding the real or imaginary part of any entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution goes to 0 as you go to infinity. In this case, uniqueness follows by Liouville's theorem. The singular points of the harmonic functions above are expressed as "charges" and "charge densities" using the terminology of electrostatics, and so the corresponding harmonic function will be proportional to the electrostatic potential due to these charge distributions. Each function above will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are the images of the original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will yield another harmonic function. Finally, examples of harmonic functions of ''n'' variables are: * The constant, linear and affine functions on all of R''n'' (for example, the electric potential between the plates of a capacitor, and the gravity potential of a slab) * The function $\backslash ,\backslash !\; f(x\_1,\backslash dots,x\_n)=(^2+\backslash cdots+^2)^$ on $\backslash mathbb^n\; \backslash setminus\; \backslash lbrace\; 0\; \backslash rbrace$ for ''n'' > 2. 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「Harmonic function」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.