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A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex in general. Multipole expansions are very frequently used in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angular features. For example, the initial term — called the zero-th, or monopole, moment — is a constant, independent of angle. The following term — the first, or dipole, moment — varies once from positive to negative around the sphere. Higher-order terms (like the quadrupole and octupole) vary more quickly with angles. A multipole moment usually involves powers (or inverse powers) of the distance to the origin, as well as some angular dependence. In principle, a multipole expansion provides an exact description of the potential and generally converges under two conditions: (1) if the sources (e.g., charges) are localized close to the origin and the point at which the potential is observed is far from the origin; or (2) the reverse, i.e., if the sources (e.g., charges) are located far from the origin and the potential is observed close to the origin. In the first (more common) case, the coefficients of the series expansion are called ''exterior multipole moments'' or simply ''multipole moments'' whereas, in the second case, they are called ''interior multipole moments''. The first (the zeroth-order) term in the expansion is called the monopole moment, the second (the first-order) term is denoted as the dipole moment, and the third (the second-order), fourth (the third-order), etc. terms are denoted as quadrupole, octupole, etc. moments. ==Expansion in spherical harmonics== Most commonly, the series is written as a sum of spherical harmonics. Thus, we might write a function as the sum : Here, are the standard spherical harmonics, and are constant coefficients which depend on the function. The term represents the monopole; represent the dipole; and so on. Equivalently, the series is also frequently written as : Here, the represent the components of a unit vector in the direction given by the angles and , and indices are implicitly summed. Here, the term is the monopole; is a set of three numbers representing the dipole; and so on. In the above expansions, the coefficients may be real or complex. If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties. In the spherical harmonic expansion, we must have : In the multi-vector expansion, each coefficient must be real: : While expansions of scalar functions are by far the most common application of multipole expansions, they may also be generalized to describe tensors of arbitrary rank. This finds use in multipole expansions of the vector potential in electromagnetism, or the metric perturbation in the description of gravitational waves. For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, —most frequently, as a Laurent series in powers of . For example, to describe the electromagnetic potential, , from a source in a small region near the origin, the coefficients may be written as: : 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「multipole expansion」の詳細全文を読む スポンサード リンク
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