Multipole expansion

10268: 8238: 2299: 9535: 7626: 1829: 5348: 10263:{\displaystyle {\begin{aligned}C_{1}^{0}&=\sum _{i=1}^{N}eZ_{i}\;z_{i}\\C_{1}^{1}&=\sum _{i=1}^{N}eZ_{i}\;x_{i}\\S_{1}^{1}&=\sum _{i=1}^{N}eZ_{i}\;y_{i}\\C_{2}^{0}&={\frac {1}{2}}\sum _{i=1}^{N}eZ_{i}\;(3z_{i}^{2}-r_{i}^{2})\\C_{2}^{1}&={\sqrt {3}}\sum _{i=1}^{N}eZ_{i}\;z_{i}x_{i}\\C_{2}^{2}&={\frac {1}{3}}{\sqrt {3}}\sum _{i=1}^{N}eZ_{i}\;(x_{i}^{2}-y_{i}^{2})\\S_{2}^{1}&={\sqrt {3}}\sum _{i=1}^{N}eZ_{i}\;z_{i}y_{i}\\S_{2}^{2}&={\frac {2}{3}}{\sqrt {3}}\sum _{i=1}^{N}eZ_{i}\;x_{i}y_{i}\end{aligned}}} 8233:{\displaystyle {\begin{aligned}U_{AB}={}&{\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell _{A}=0}^{\infty }\sum _{\ell _{B}=0}^{\infty }(-1)^{\ell _{B}}{\binom {2\ell _{A}+2\ell _{B}}{2\ell _{A}}}^{1/2}\\&\times \sum _{m_{A}=-\ell _{A}}^{\ell _{A}}\sum _{m_{B}=-\ell _{B}}^{\ell _{B}}(-1)^{m_{A}+m_{B}}I_{\ell _{A}+\ell _{B}}^{-m_{A}-m_{B}}(\mathbf {R} _{AB})\;Q_{\ell _{A}}^{m_{A}}Q_{\ell _{B}}^{m_{B}}\;\langle \ell _{A},m_{A};\ell _{B},m_{B}\mid \ell _{A}+\ell _{B},m_{A}+m_{B}\rangle .\end{aligned}}} 2539: 4923: 2294:{\displaystyle {\begin{aligned}v(\mathbf {r} -\mathbf {R} )&=v(\mathbf {-R} )+\sum _{\alpha =x,y,z}r_{\alpha }v_{\alpha }(\mathbf {-R} )+{\frac {1}{2}}\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}r_{\alpha }r_{\beta }v_{\alpha \beta }(\mathbf {-R} )+\cdots +\cdots \\&=v(\mathbf {R} )-\sum _{\alpha =x,y,z}r_{\alpha }v_{\alpha }(\mathbf {R} )+{\frac {1}{2}}\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}r_{\alpha }r_{\beta }v_{\alpha \beta }(\mathbf {R} )-\cdots +\cdots \end{aligned}}} 3901: 7495: 2304: 4461: 3583: 1549: 7072: 6025: 5343:{\displaystyle {\begin{aligned}V(\mathbf {R} )&={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {R} )Q_{\ell }^{m}\\&={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\left^{1/2}\;{\frac {1}{R^{\ell +1}}}\sum _{m=-\ell }^{\ell }(-1)^{m}Y_{\ell }^{-m}({\hat {R}})Q_{\ell }^{m},\qquad R>r_{\mathrm {max} }\end{aligned}}} 6995: 2974: 3574: 4161: 3318: 9357: 1269: 8580: 2534:{\displaystyle v_{\alpha }(\mathbf {R} )\equiv \left({\frac {\partial v(\mathbf {r} -\mathbf {R} )}{\partial r_{\alpha }}}\right)_{\mathbf {r} =\mathbf {0} }\quad {\text{and}}\quad v_{\alpha \beta }(\mathbf {R} )\equiv \left({\frac {\partial ^{2}v(\mathbf {r} -\mathbf {R} )}{\partial r_{\alpha }\partial r_{\beta }}}\right)_{\mathbf {r} =\mathbf {0} }.} 5764: 6693: 2707: 3896:{\displaystyle {\begin{aligned}4\pi \varepsilon _{0}V(\mathbf {R} )&\equiv \sum _{i=1}^{N}q_{i}v(\mathbf {r} _{i}-\mathbf {R} )\\&={\frac {q_{\mathrm {tot} }}{R}}+{\frac {1}{R^{3}}}\sum _{\alpha =x,y,z}P_{\alpha }R_{\alpha }+{\frac {1}{2R^{5}}}\sum _{\alpha ,\beta =x,y,z}Q_{\alpha \beta }R_{\alpha }R_{\beta }+\cdots \end{aligned}}} 3323: 5759: 7490:{\displaystyle R_{L}^{M}(\mathbf {r} _{Ai}-\mathbf {r} _{Bj})=\sum _{\ell _{A}=0}^{L}(-1)^{L-\ell _{A}}{\binom {2L}{2\ell _{A}}}^{1/2}\times \sum _{m_{A}=-\ell _{A}}^{\ell _{A}}R_{\ell _{A}}^{m_{A}}(\mathbf {r} _{Ai})R_{L-\ell _{A}}^{M-m_{A}}(\mathbf {r} _{Bj})\;\langle \ell _{A},m_{A};L-\ell _{A},M-m_{A}\mid LM\rangle ,} 6579: 3117: 9099: 1610:. The basic idea is to decompose the particles into groups; particles within a group interact normally (i.e., by the full potential), whereas the energies and forces between groups of particles are calculated from their multipole moments. The efficiency of the fast multipole method is generally similar to that of 5533: 8258: 8622:

atoms) have one or more non-vanishing permanent multipole moments. Different definitions can be found in the literature, but the following definition in spherical form has the advantage that it is contained in one general equation. Because it is in complex form it has as the further advantage that it

217:

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 are located far from the origin and the potential is observed close to the origin. In the first

2698: 4456:{\displaystyle V(\mathbf {R} )\equiv \sum _{i=1}^{N}{\frac {q_{i}}{4\pi \varepsilon _{0}|\mathbf {r} _{i}-\mathbf {R} |}}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {R} )\sum _{i=1}^{N}q_{i}R_{\ell }^{m}(\mathbf {r} _{i}),} 10301:

over a one-particle charge distribution. Remember that in the case of a one-particle quantum mechanical system the expectation value is nothing but an integral over the charge distribution (modulus of wavefunction squared), so that the definition of this article is a quantum mechanical

1590:

multipoles of the electronic orbitals. The multipole moments of the nuclei report on the distribution of charges within the nucleus and, thus, on the shape of the nucleus. Truncation of the multipole expansion to its first non-zero term is often useful for theoretical calculations.

1752:. The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required. Its disadvantage is that the derivations are fairly cumbersome (in fact a large part of it is the implicit rederivation of the Legendre expansion of 1544:{\displaystyle V(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }C_{\ell }^{m}(r)\,Y_{\ell }^{m}(\theta ,\varphi )=\sum _{j=1}^{\infty }\,\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }{\frac {D_{\ell ,j}^{m}}{r^{j}}}\,Y_{\ell }^{m}(\theta ,\varphi ).} 406: 805: 4895: 5549: 8977:

hold for the expectation value of the multipole operator, or in other words, that the expectation value may vanish because of symmetry. A well-known example of this is the fact that molecules with an inversion center do not carry a dipole (the expectation values of

6413: 6020:{\displaystyle V_{\ell =1}(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}R^{3}}}(R_{x}P_{x}+R_{y}P_{y}+R_{z}P_{z})={\frac {\mathbf {R} \cdot \mathbf {P} }{4\pi \varepsilon _{0}R^{3}}}={\frac {{\hat {\mathbf {R} }}\cdot \mathbf {P} }{4\pi \varepsilon _{0}R^{2}}}.} 10465:

In practice, many fields can be well approximated with a finite number of multipole moments (although an infinite number may be required to reconstruct a field exactly). A typical application is to approximate the field of a localized charge distribution by its

4726: 6990:{\displaystyle {\frac {1}{|\mathbf {r} _{j}-\mathbf {r} _{i}|}}={\frac {1}{|\mathbf {R} _{AB}-(\mathbf {r} _{Ai}-\mathbf {r} _{Bj})|}}=\sum _{L=0}^{\infty }\sum _{M=-L}^{L}\,(-1)^{M}I_{L}^{-M}(\mathbf {R} _{AB})\;R_{L}^{M}(\mathbf {r} _{Ai}-\mathbf {r} _{Bj}),} 2969:{\displaystyle \sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}r_{\alpha }r_{\beta }v_{\alpha \beta }(\mathbf {R} )={\frac {1}{3}}\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}\left(3r_{\alpha }r_{\beta }-\delta _{\alpha \beta }r^{2}\right)v_{\alpha \beta }(\mathbf {R} ),} 3988: 1179: 5384: 4107: 3569:{\displaystyle q_{\mathrm {tot} }\equiv \sum _{i=1}^{N}q_{i},\quad P_{\alpha }\equiv \sum _{i=1}^{N}q_{i}r_{i\alpha },\quad {\text{and}}\quad Q_{\alpha \beta }\equiv \sum _{i=1}^{N}q_{i}(3r_{i\alpha }r_{i\beta }-\delta _{\alpha \beta }r_{i}^{2}),} 206:, terms of higher order are conventionally named by adding "-pole" to the number of poles—e.g., 32-pole (rarely dotriacontapole or triacontadipole) and 64-pole (rarely tetrahexacontapole or hexacontatetrapole). A multipole moment usually involves 6349: 6684: 2563: 3108: 8806: 6159: 3313:{\displaystyle v(\mathbf {R} )={\frac {1}{R}},\quad v_{\alpha }(\mathbf {R} )=-{\frac {R_{\alpha }}{R^{3}}},\quad {\hbox{and}}\quad v_{\alpha \beta }(\mathbf {R} )={\frac {3R_{\alpha }R_{\beta }-\delta _{\alpha \beta }R^{2}}{R^{5}}}.} 9352:{\displaystyle M_{1}^{1}=-{\tfrac {1}{\sqrt {2}}}\sum _{i=1}^{N}eZ_{i}\langle \Psi |x_{i}+iy_{i}|\Psi \rangle \quad {\hbox{and}}\quad M_{1}^{-1}={\tfrac {1}{\sqrt {2}}}\sum _{i=1}^{N}eZ_{i}\langle \Psi |x_{i}-iy_{i}|\Psi \rangle .} 8959: 7596: 9460: 9017:. For a molecule without symmetry, no selection rules are operative and such a molecule will have non-vanishing multipoles of any order (it will carry a dipole and simultaneously a quadrupole, octupole, hexadecapole, etc.). 3905:

This expansion of the potential of a discrete charge distribution is very similar to the one in real solid harmonics given below. The main difference is that the present one is in terms of linearly dependent quantities, for

8575:{\displaystyle I_{\ell _{A}+\ell _{B}}^{-(m_{A}+m_{B})}(\mathbf {R} _{AB})\equiv \left^{1/2}\;{\frac {Y_{\ell _{A}+\ell _{B}}^{-(m_{A}+m_{B})}\left({\widehat {\mathbf {R} }}_{AB}\right)}{R_{AB}^{\ell _{A}+\ell _{B}+1}}},} 280: 593: 4771: 4604: 1017: 7631: 3909: 1022: 9540: 4928: 3588: 1834: 9094: 8866:(also known as Schmidt's semi-normalization). If the molecule has total normalized wave function Ψ (depending on the coordinates of electrons and nuclei), then the multipole moment of order 8856: 4021: 453: 588: 6229: 4507: 4591: 6584: 1770:

in the 1780s). Also it is difficult to give a closed expression for a general term of the multipole expansion—usually only the first few terms are given followed by an ellipsis.

5754:{\displaystyle \mathbf {R} =(R_{x},R_{y},R_{z}),\quad \mathbf {P} =(P_{x},P_{y},P_{z})\quad {\hbox{with}}\quad P_{\alpha }\equiv \sum _{i=1}^{N}q_{i}r_{i\alpha },\quad \alpha =x,y,z.} 1202:

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,

80:, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be 6574:{\displaystyle \mathbf {R} _{AB}+\mathbf {r} _{Bj}+\mathbf {r} _{ji}+\mathbf {r} _{iA}=0\quad \iff \quad \mathbf {r} _{ij}=\mathbf {R} _{AB}-\mathbf {r} _{Ai}+\mathbf {r} _{Bj}.} 3036: 275: 8705: 6040: 940:. 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 144: 115: 74: 9530: 9498: 8893: 485: 7504: 4548: 4766: 9008: 7059: 7027: 517: 9361: 876: 856: 5528:{\displaystyle V_{\ell =0}(\mathbf {R} )={\frac {q_{\mathrm {tot} }}{4\pi \varepsilon _{0}R}}\quad {\hbox{with}}\quad q_{\mathrm {tot} }\equiv \sum _{i=1}^{N}q_{i}.} 8884: 927: 832: 10649:

Ikeda, Hiroaki; Suzuki, Michi-To; Arita, Ryotaro; Takimoto, Tetsuya; Shibauchi, Takasada; Matsuda, Yuji (3 June 2012). "Emergent rank-5 nematic order in URu2Si2".

5370:

summation is invariant under a unitary transformation of both factors simultaneously and since transformation of complex spherical harmonics to real form is by a

179:, 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 1264: 1244: 1220: 900: 165: 6037:

term, we have to introduce shorthand notations for the five real components of the quadrupole moment and the real spherical harmonics. Notations of the type

5366:

It is of interest to consider the first few terms in real form, which are the only terms commonly found in undergraduate textbooks. Since the summand of the

10431:

to point sources that are brought infinitely close to each other. These can be thought of as arranged in various geometrical shapes, or, in the sense of

2693:{\displaystyle \left(\nabla ^{2}v(\mathbf {r} -\mathbf {R} )\right)_{\mathbf {r} =\mathbf {0} }=\sum _{\alpha =x,y,z}v_{\alpha \alpha }(\mathbf {R} )=0,} 943: 10282: 7599: 6161:

can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation.

401:{\displaystyle f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }\,C_{\ell }^{m}\,Y_{\ell }^{m}(\theta ,\varphi )} 9027: 800:{\displaystyle f(\theta ,\varphi )=C+C_{i}n^{i}+C_{ij}n^{i}n^{j}+C_{ijk}n^{i}n^{j}n^{k}+C_{ijk\ell }n^{i}n^{j}n^{k}n^{\ell }+\cdots } 4768:. (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as 4890:{\displaystyle I_{\ell }^{m}(\mathbf {R} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}{\frac {Y_{\ell }^{m}({\hat {R}})}{R^{\ell +1}}}} 10446:. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of 5352:

This expansion is completely general in that it gives a closed form for all terms, not just for the first few. It shows that the

8887: 1603: 4721:{\displaystyle Q_{\ell }^{m}\equiv \sum _{i=1}^{N}q_{i}R_{\ell }^{m}(\mathbf {r} _{i}),\quad \ -\ell \leq m\leq \ell .} 10583: 10558: 8811: 411: 522: 7617:

and covering of the summation ranges in a somewhat different order (which is only allowed for an infinite range of

1187:

functions are by far the most common application of multipole expansions, they may also be generalized to describe

4728:

Note that a multipole moment is solely determined by the charge distribution (the positions and magnitudes of the

4466: 10442:

Multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated

10289:

by Jackson, except for the normalization. Moreover, in the classical definition of Jackson the equivalent of the

4553: 9468:

one can transform the complex multipole operators to real ones. The real multipole operators are of cosine type

10775: 9021: 7498: 3983:{\displaystyle \sum _{\alpha }v_{\alpha \alpha }=0\quad {\hbox{and}}\quad \sum _{\alpha }Q_{\alpha \alpha }=0.} 10497: 10363: 6687: 4155: 3111: 1174:{\displaystyle C=C^{\ast };\ C_{i}=C_{i}^{\ast };\ C_{ij}=C_{ij}^{\ast };\ C_{ijk}=C_{ijk}^{\ast };\ \ldots } 879: 10309:

The definition in this article agrees with, among others, the one of Fano and Racah and Brink and Satchler.

8863: 10350: 5353: 1749: 245: 36: 4102:{\displaystyle V(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}R^{3}}}(\mathbf {P} \cdot \mathbf {R} ),} 3995:

If the charge distribution consists of two charges of opposite sign which are an infinitesimal distance

202:

moment, the fourth (third-order) term is called the octupole moment, and so on. Given the limitation of

10432: 8966: 120: 91: 50: 10832: 10822: 10601:"High-resolution infrared spectroscopy of solid hydrogen: The tetrahexacontapole-induced transitions" 10487: 10389: 10286: 6344:{\displaystyle U_{AB}=\sum _{i\in A}\sum _{j\in B}{\frac {q_{i}q_{j}}{4\pi \varepsilon _{0}r_{ij}}}.} 10827: 9503: 9471: 8962: 8616: 7066: 2700:

and the expansion can be rewritten in terms of the components of a traceless Cartesian second rank

458: 6679:{\displaystyle |\mathbf {R} _{AB}|>|\mathbf {r} _{Bj}-\mathbf {r} _{Ai}|{\text{ for all }}i,j.} 1617: 1614:, but is superior if the particles are clustered, i.e. the system has large density fluctuations. 213:

In principle, a multipole expansion provides an exact description of the potential, and generally

186:

The multipole expansion is expressed as a sum of terms with progressively finer angular features (

183:. Such a combination gives an expansion describing a function throughout three-dimensional space. 10526: 10459: 10393: 10337: 4520: 1606:, a general technique for efficient computation of energies and forces in systems of interacting 4742: 10467: 10436: 8981: 7032: 7000: 490: 191: 28: 861: 10492: 10443: 4110: 1767: 1724: 1595: 841: 172: 10718: 3580:

of the total potential, which is the sum of the Coulomb potentials of the separate charges:

3103:{\displaystyle v(\mathbf {r} -\mathbf {R} )\equiv {\frac {1}{|\mathbf {r} -\mathbf {R} |}}.} 10733: 10668: 10615: 10511: 10502: 10447: 10420: 9465: 8869: 8801:{\displaystyle Q_{\ell }^{m}\equiv \sum _{i=1}^{N}eZ_{i}\;R_{\ell }^{m}(\mathbf {r} _{i}),} 8246:

of the interaction energy of two non-overlapping charge distributions which are a distance

6154:{\displaystyle Q_{z^{2}}\equiv \sum _{i=1}^{N}q_{i}\;{\frac {1}{2}}(3z_{i}^{2}-r_{i}^{2}),} 5371: 1575: 1184: 905: 810: 24: 10600: 5374:, we can simply substitute real irregular solid harmonics and real multipole moments. The 8: 8970: 1745: 1594:

Multipole expansions are also useful in numerical simulations, and form the basis of the

1559: 239: 176: 10737: 10672: 10619: 590:

represent the dipole; and so on. Equivalently, the series is also frequently written as

10764: 10684: 10658: 10547: 10475: 10404: 10385: 8604: 4736: 4514: 1266:, from a source in a small region near the origin, the coefficients may be written as: 1249: 1229: 1205: 1196: 885: 210:(or inverse powers) of the distance to the origin, as well as some angular dependence. 150: 8954:{\displaystyle M_{\ell }^{m}\equiv \langle \Psi \mid Q_{\ell }^{m}\mid \Psi \rangle .} 10807:, 2nd edition, Clarendon Press, Oxford, UK (1968). p. 64. See also footnote on p. 90. 10771: 10688: 10631: 10579: 10554: 10507: 10326: 10294: 10278: 7591:{\displaystyle R_{\ell }^{m}(-\mathbf {r} )=(-1)^{\ell }R_{\ell }^{m}(\mathbf {r} ).} 214: 10741: 10676: 10623: 10424: 10322: 8965:, then this is reflected in the wave function: Ψ transforms according to a certain 5536: 2557: 1719:. Two ways of making this expansion can be found in the literature: The first is a 1599: 1192: 187: 10521: 10516: 10428: 9455:{\displaystyle M_{1}^{0}=\sum _{i=1}^{N}eZ_{i}\langle \Psi |z_{i}|\Psi \rangle .} 8859: 7062: 2988: 1611: 44: 10451: 8974: 6212: 4510: 1618:

Multipole expansion of a potential outside an electrostatic charge distribution

1583: 1571: 1567: 1223: 937: 207: 203: 85: 10745: 10627: 10816: 8644: 1798: 1720: 77: 10277:

The definition of the complex molecular multipole moment given above is the

3004:. Removing the trace is common, because it takes the rotationally invariant 10635: 1648:. We assume the charges to be clustered around the origin, so that for all 6164: 10474:

terms. Problems solved once for a given order of multipole moment may be

10416: 10317:

There are many types of multipole moments, since there are many types of

9096:(the total charge of the molecule). The (complex) dipole components are: 6351:

This energy can be expanded in a power series in the inverse distance of

933: 835: 81: 3320:

Define a monopole, dipole, and (traceless) quadrupole by, respectively,

218:(more common) case, the coefficients of the series expansion are called 10455: 199: 10680: 10576:

Optically polarized atoms : understanding light-atom interactions

1195:

in electromagnetism, or the metric perturbation in the description of

10318: 6215:(positive point charges). The total electrostatic interaction energy 10462:

to extract the corresponding solutions for the radial dependencies.

10330: 10298: 9020:

The lowest explicit forms of the regular solid harmonics (with the

8623:

is easier to manipulate in calculations than its real counterpart.

6208: 6204: 1607: 10663: 10599:

Okumura, Mitchio; Chan, Man-Chor; Oka, Takeshi (2 January 1989).

1191:

of arbitrary rank. This finds use in multipole expansions of the

487:

are constant coefficients which depend on the function. The term

40: 10427:

for the decomposition of a function, based on the response of a

10333:

of the charge distribution. The most common expansions include:

4018:, it is easily shown that the dominant term in the expansion is 10471: 2701: 1188: 195: 180: 1019:

In the multi-vector expansion, each coefficient must be real:

929:

is a set of three numbers representing the dipole; and so on.

32: 1574:

of charge and current distributions, and the propagation of

1012:{\displaystyle C_{\ell }^{-m}=(-1)^{m}C_{\ell }^{m\ast }\,.} 10478:

to create a final approximate solution for a given source.

10285:, which follows the definition of the standard textbook on 6027:

This term is identical to the one found in Cartesian form.

1563: 1558:

Multipole expansions are widely used in problems involving

10574:

Auzinsh, Marcis; Budker, Dmitry; Rochester, Simon (2010).

1246:. For example, to describe the electromagnetic potential, 10454:, which leads to spherical harmonics and related sets of 8973:("Ψ has symmetry type λ"). This has the consequence that 171:

Multipole expansions are used frequently in the study of

10648: 4593:

is a regular solid harmonic (a spherical harmonic times

194:

moment, the second (the first-order) term is called the

6207:, and recall that a molecule by definition consists of 6165:

Interaction of two non-overlapping charge distributions

10573: 9251: 9222: 9125: 6375:

In order to derive this multipole expansion, we write

5659: 5463: 3944: 3208: 1622:

Consider a discrete charge distribution consisting of

9538: 9506: 9474: 9364: 9102: 9030: 8984: 8896: 8872: 8814: 8708: 8261: 7629: 7507: 7075: 7035: 7003: 6696: 6587: 6581:

We assume that the two distributions do not overlap:

6416: 6232: 6043: 5767: 5552: 5387: 4926: 4774: 4745: 4607: 4556: 4523: 4469: 4164: 4024: 3912: 3586: 3326: 3120: 3039: 2710: 2566: 2307: 1832: 1773: 1272: 1252: 1232: 1208: 1025: 946: 908: 888: 864: 844: 813: 596: 525: 493: 461: 414: 283: 248: 153: 123: 94: 53: 8702:. The (complex) electrostatic multipole operator is 233: 190:). The first (the zeroth-order) term is called the 10763: 10546: 10410: 10306:-particle generalization of Jackson's definition. 10262: 9524: 9492: 9454: 9351: 9088: 9002: 8953: 8878: 8850: 8800: 8574: 8232: 7590: 7489: 7053: 7021: 6989: 6678: 6573: 6343: 6153: 6019: 5753: 5527: 5342: 4889: 4760: 4720: 4585: 4542: 4501: 4455: 4101: 3982: 3895: 3568: 3312: 3102: 2968: 2692: 2533: 2293: 1543: 1258: 1238: 1214: 1173: 1011: 921: 894: 870: 850: 826: 799: 582: 511: 479: 447: 400: 269: 159: 138: 109: 68: 10719:"Multipole Expansions of Gravitational Radiation" 7818: 7765: 7497:where the quantity between pointed brackets is a 7227: 7197: 3576:and we obtain finally the first few terms of the 932:In the above expansions, the coefficients may be 238:Most commonly, the series is written as a sum of 10814: 10321:and many ways of approximating a potential by a 9089:{\displaystyle M_{0}^{0}=\sum _{i=1}^{N}eZ_{i},} 8851:{\displaystyle R_{\ell }^{m}(\mathbf {r} _{i})} 448:{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} 10598: 8630:particles (electrons and nuclei) with charges 1578:. A classic example is the calculation of the 583:{\displaystyle C_{1}^{-1},C_{1}^{0},C_{1}^{1}} 1690:, due to the charge distribution, at a point 226:whereas, in the second case, they are called 9446: 9417: 9343: 9298: 9217: 9172: 8945: 8915: 8220: 8116: 7481: 7408: 6169:Consider two sets of point charges, one set 4918:outside the charge distribution is given by 4502:{\displaystyle I_{\ell }^{-m}(\mathbf {R} )} 10757: 10755: 4586:{\displaystyle R_{\ell }^{m}(\mathbf {r} )} 10235: 10137: 10030: 9932: 9822: 9741: 9670: 9599: 8761: 8582:this expansion is manifestly in powers of 8422: 8115: 8056: 7407: 6929: 6497: 6493: 6095: 5190: 455:are the standard spherical harmonics, and 10662: 8643:-value of −1, while for nuclei it is the 7067:translation of the regular solid harmonic 6870: 1586:from their interaction energies with the 1507: 1447: 1425: 1370: 1321: 1005: 367: 351: 326: 198:moment, the third (the second-order) the 126: 97: 56: 10794:, Academic Press, New York (1959). p. 31 10752: 10761: 10544: 1696:outside the charge distribution, i.e., 10815: 10716: 10538: 10272: 6686:Under this condition we may apply the 4601:of the charge distribution as follows 4138:outside the charge distribution, i.e. 2560:, then by the above expansion we have 10549:Angular Momentum in Quantum Mechanics 10458:functions. One uses the technique of 8626:We consider a molecule consisting of 1766:, which was done once and for all by 1679:has some finite value. The potential 838:in the direction given by the angles 10701: 8610: 35:—usually the two angles used in the 3018:Consider now the following form of 270:{\displaystyle f(\theta ,\varphi )} 13: 10396:of point sources. An example of a 9443: 9420: 9340: 9301: 9214: 9175: 8942: 8918: 7769: 7730: 7702: 7201: 6841: 6402:, which is a vector pointing from 5482: 5479: 5476: 5432: 5429: 5426: 5330: 5327: 5324: 5136: 4993: 4309: 3712: 3709: 3706: 3339: 3336: 3333: 2574: 2492: 2479: 2446: 2367: 2340: 1774:Expansion in Cartesian coordinates 1744:, while the second is in terms of 1442: 1420: 1316: 321: 242:. Thus, we might write a function 14: 10844: 10578:. Oxford: New York. p. 100. 6359:. This expansion is known as the 6226:between the two distributions is 4116: 88:-valued and is defined either on 10803:D. M. Brink and G. R. Satchler, 10770:(2d ed.). New York: Wiley. 9532:. A few of the lowest ones are: 8886:of the molecule is given by the 8835: 8782: 8651:has spherical polar coordinates 8615:All atoms and molecules (except 8499: 8331: 8040: 7578: 7530: 7391: 7329: 7114: 7096: 6968: 6950: 6913: 6798: 6780: 6759: 6725: 6710: 6641: 6623: 6595: 6555: 6537: 6519: 6501: 6473: 6455: 6437: 6419: 5979: 5965: 5920: 5912: 5788: 5608: 5554: 5408: 5063: 4938: 4794: 4677: 4576: 4492: 4437: 4379: 4253: 4239: 4172: 4089: 4081: 4032: 3681: 3667: 3614: 3232: 3166: 3128: 3085: 3077: 3055: 3047: 2956: 2804: 2674: 2620: 2612: 2598: 2590: 2522: 2514: 2470: 2462: 2427: 2397: 2389: 2358: 2350: 2322: 2268: 2155: 2093: 2057: 2054: 1941: 1938: 1876: 1873: 1852: 1844: 234:Expansion in spherical harmonics 139:{\displaystyle \mathbb {R} ^{n}} 110:{\displaystyle \mathbb {R} ^{3}} 69:{\displaystyle \mathbb {R} ^{3}} 10411:General mathematical properties 9228: 9220: 6498: 6492: 5726: 5665: 5657: 5606: 5469: 5461: 5311: 4693: 3950: 3942: 3449: 3443: 3382: 3214: 3206: 3151: 3010:out of the second rank tensor. 2409: 2403: 1712:, can be expanded in powers of 1553: 10797: 10784: 10710: 10695: 10642: 10592: 10567: 10553:. Princeton University Press. 10067: 10031: 9862: 9823: 9439: 9424: 9336: 9305: 9210: 9179: 8845: 8830: 8792: 8777: 8485: 8459: 8344: 8326: 8321: 8295: 8053: 8035: 7948: 7938: 7745: 7735: 7582: 7574: 7550: 7540: 7534: 7523: 7404: 7386: 7342: 7324: 7171: 7161: 7127: 7091: 6981: 6945: 6926: 6908: 6881: 6871: 6815: 6811: 6775: 6753: 6736: 6704: 6655: 6617: 6609: 6589: 6494: 6145: 6106: 5969: 5902: 5833: 5792: 5784: 5654: 5615: 5600: 5561: 5412: 5404: 5356:appear as coefficients in the 5290: 5284: 5275: 5248: 5238: 5067: 5059: 5032: 5022: 4942: 4934: 4865: 4859: 4850: 4798: 4790: 4752: 4687: 4672: 4580: 4572: 4496: 4488: 4447: 4432: 4383: 4375: 4348: 4338: 4258: 4233: 4176: 4168: 4093: 4077: 4036: 4028: 3685: 3662: 3618: 3610: 3560: 3497: 3236: 3228: 3170: 3162: 3132: 3124: 3090: 3072: 3059: 3043: 2960: 2952: 2808: 2800: 2678: 2670: 2602: 2586: 2474: 2458: 2431: 2423: 2362: 2346: 2326: 2318: 2272: 2264: 2159: 2151: 2097: 2089: 2061: 2050: 1945: 1934: 1880: 1869: 1856: 1840: 1535: 1523: 1398: 1386: 1367: 1361: 1294: 1276: 978: 968: 834:represent the components of a 612: 600: 442: 430: 395: 383: 299: 287: 264: 252: 43:angles) for three-dimensional 1: 10717:Thorne, Kip S. (April 1980). 10706:. John Wiley & Sons, Inc. 10532: 10364:Cylindrical multipole moments 9525:{\displaystyle S_{\ell }^{m}} 9493:{\displaystyle C_{\ell }^{m}} 6211:(negative point charges) and 480:{\displaystyle C_{\ell }^{m}} 10762:Jackson, John David (1975). 10407:of an infinite line charge. 8961:If the molecule has certain 8888:expectation (expected) value 5363:expansion of the potential. 7: 10481: 10448:irreducible representations 10351:Spherical multipole moments 10312: 10281:of the definition given in 6203:. Think for example of two 5354:spherical multipole moments 4739:depends on the unit vector 4543:{\displaystyle R^{\ell +1}} 1750:spherical polar coordinates 37:spherical coordinate system 10: 10849: 10792:Irreducible Tensorial Sets 8967:irreducible representation 8675:and Cartesian coordinates 7499:Clebsch–Gordan coefficient 7069:gives a finite expansion, 7061:are irregular and regular 4761:{\displaystyle {\hat {R}}} 4599:spherical multipole moment 3013: 228:interior multipole moments 220:exterior multipole moments 10766:Classical electrodynamics 10746:10.1103/RevModPhys.52.299 10726:Reviews of Modern Physics 10628:10.1103/PhysRevLett.62.32 10287:classical electrodynamics 9003:{\displaystyle Q_{1}^{m}} 7598:Use of the definition of 7054:{\displaystyle R_{L}^{M}} 7022:{\displaystyle I_{L}^{M}} 6199:clustered around a point 6182:clustered around a point 4154:, can be expanded by the 2301:with Taylor coefficients 519:represents the monopole; 512:{\displaystyle C_{0}^{0}} 10297:expectation value is an 871:{\displaystyle \varphi } 10608:Physical Review Letters 10545:Edmonds, A. R. (1960). 10527:Dynamic toroidal dipole 10460:separation of variables 10437:directional derivatives 10394:gravitational potential 10384:potentials include the 10338:Axial multipole moments 4111:dipolar potential field 1222:—most frequently, as a 851:{\displaystyle \theta } 10790:U. Fano and G. Racah, 10444:differential equations 10264: 10221: 10123: 10016: 9918: 9808: 9727: 9656: 9585: 9526: 9494: 9464:Note that by a simple 9456: 9403: 9353: 9284: 9158: 9090: 9069: 9004: 8955: 8880: 8852: 8802: 8747: 8576: 8234: 7937: 7892: 7734: 7706: 7592: 7491: 7294: 7160: 7055: 7023: 6991: 6869: 6845: 6690:in the following form 6680: 6575: 6345: 6155: 6084: 6030:In order to write the 6021: 5755: 5699: 5529: 5511: 5372:unitary transformation 5344: 5237: 5140: 5021: 4997: 4891: 4762: 4722: 4646: 4587: 4544: 4503: 4457: 4406: 4337: 4313: 4202: 4103: 3984: 3897: 3648: 3570: 3486: 3416: 3368: 3314: 3104: 2970: 2694: 2535: 2295: 1637:with position vectors 1545: 1471: 1446: 1424: 1345: 1320: 1260: 1240: 1216: 1175: 1013: 923: 896: 872: 852: 828: 801: 584: 513: 481: 449: 402: 350: 325: 271: 204:Greek numeral prefixes 161: 140: 111: 70: 10702:Thompson, William J. 10512:particle accelerators 10493:Fast multipole method 10488:Barnes–Hut simulation 10415:Multipole moments in 10265: 10201: 10103: 9996: 9898: 9788: 9707: 9636: 9565: 9527: 9495: 9457: 9383: 9354: 9264: 9138: 9091: 9049: 9022:Condon-Shortley phase 9005: 8956: 8881: 8879:{\displaystyle \ell } 8864:Racah's normalization 8853: 8803: 8727: 8577: 8235: 7893: 7848: 7707: 7679: 7593: 7492: 7250: 7133: 7056: 7024: 6992: 6846: 6825: 6681: 6576: 6346: 6156: 6064: 6022: 5756: 5679: 5530: 5491: 5345: 5214: 5120: 4998: 4977: 4892: 4763: 4723: 4626: 4588: 4545: 4504: 4458: 4386: 4314: 4293: 4182: 4104: 3985: 3898: 3628: 3571: 3466: 3396: 3348: 3315: 3105: 2971: 2695: 2536: 2296: 1797:for convenience. The 1725:Cartesian coordinates 1596:fast multipole method 1582:multipole moments of 1576:electromagnetic waves 1546: 1448: 1426: 1404: 1322: 1300: 1261: 1241: 1217: 1176: 1014: 924: 922:{\displaystyle C_{i}} 897: 873: 853: 829: 827:{\displaystyle n^{i}} 802: 585: 514: 482: 450: 403: 327: 305: 272: 162: 141: 112: 71: 10503:Legendre polynomials 10421:mathematical physics 9536: 9504: 9472: 9362: 9100: 9028: 8982: 8963:point group symmetry 8894: 8870: 8812: 8706: 8639:. (Electrons have a 8259: 7627: 7600:spherical multipoles 7505: 7073: 7065:, respectively. The 7033: 7001: 6694: 6585: 6414: 6230: 6041: 5765: 5550: 5385: 4924: 4772: 4743: 4605: 4554: 4521: 4517:function divided by 4513:(defined below as a 4467: 4162: 4022: 3910: 3584: 3324: 3118: 3037: 2708: 2564: 2305: 1830: 1560:gravitational fields 1270: 1250: 1230: 1206: 1183:While expansions of 1023: 944: 906: 886: 862: 842: 811: 594: 523: 491: 459: 412: 281: 246: 177:gravitational fields 151: 121: 92: 51: 10738:1980RvMP...52..299T 10673:2012NatPh...8..528I 10620:1989PhRvL..62...32O 10433:distribution theory 10325:, depending on the 10273:Note on conventions 10176: 10088: 10066: 10048: 9971: 9883: 9861: 9843: 9770: 9699: 9628: 9557: 9521: 9489: 9379: 9246: 9117: 9045: 8999: 8938: 8911: 8829: 8776: 8723: 8566: 8489: 8325: 8244:multipole expansion 8114: 8085: 8034: 7573: 7522: 7385: 7323: 7090: 7050: 7018: 6944: 6907: 6661: for all 6361:multipole expansion 6144: 6126: 5307: 5274: 5084: 5058: 4899:multipole expansion 4849: 4789: 4671: 4622: 4571: 4487: 4431: 4374: 3578:multipole expansion 3559: 1746:spherical harmonics 1522: 1494: 1385: 1360: 1197:gravitational waves 1161: 1115: 1075: 1004: 964: 579: 561: 543: 508: 476: 429: 382: 366: 240:spherical harmonics 117:, or less often on 25:mathematical series 21:multipole expansion 16:Mathematical series 10508:Quadrupole magnets 10450:of the rotational 10405:electric potential 10390:magnetic potential 10386:electric potential 10295:quantum mechanical 10260: 10258: 10162: 10074: 10052: 10034: 9957: 9869: 9847: 9829: 9756: 9685: 9614: 9543: 9522: 9507: 9490: 9475: 9466:linear combination 9452: 9365: 9349: 9262: 9229: 9226: 9136: 9103: 9086: 9031: 9000: 8985: 8951: 8924: 8897: 8876: 8848: 8815: 8798: 8762: 8709: 8605:spherical harmonic 8572: 8523: 8426: 8262: 8230: 8228: 8086: 8057: 7977: 7588: 7559: 7508: 7501:. Further we used 7487: 7345: 7295: 7076: 7051: 7036: 7019: 7004: 6987: 6930: 6890: 6676: 6571: 6341: 6280: 6264: 6151: 6130: 6112: 6017: 5751: 5663: 5546:term we introduce 5525: 5467: 5340: 5338: 5293: 5257: 5070: 5041: 4887: 4835: 4775: 4758: 4737:spherical harmonic 4718: 4657: 4608: 4583: 4557: 4540: 4515:spherical harmonic 4499: 4470: 4453: 4417: 4357: 4099: 3980: 3960: 3948: 3922: 3893: 3891: 3849: 3770: 3566: 3545: 3310: 3212: 3100: 2966: 2879: 2851: 2766: 2738: 2690: 2656: 2531: 2291: 2289: 2230: 2202: 2130: 2016: 1988: 1913: 1826:can be written as 1816:around the origin 1541: 1508: 1474: 1371: 1346: 1256: 1236: 1212: 1171: 1141: 1098: 1061: 1009: 987: 947: 919: 892: 878:, and indices are 868: 848: 824: 797: 580: 565: 547: 526: 509: 494: 477: 462: 445: 415: 398: 368: 352: 267: 157: 136: 107: 66: 10681:10.1038/nphys2330 10498:Laplace expansion 10476:linearly combined 10403:potential is the 10279:complex conjugate 10199: 10192: 10101: 9994: 9987: 9896: 9786: 9261: 9260: 9225: 9135: 9134: 8611:Molecular moments 8567: 8506: 8402: 7816: 7677: 7225: 6820: 6741: 6688:Laplace expansion 6662: 6336: 6265: 6249: 6104: 6012: 5972: 5953: 5831: 5662: 5466: 5459: 5287: 5212: 5170: 5118: 4975: 4885: 4862: 4830: 4829: 4755: 4696: 4597:). We define the 4291: 4263: 4156:Laplace expansion 4075: 3951: 3947: 3913: 3816: 3814: 3743: 3741: 3721: 3447: 3305: 3211: 3201: 3146: 3095: 2852: 2824: 2822: 2739: 2711: 2629: 2506: 2407: 2381: 2203: 2175: 2173: 2103: 1989: 1961: 1959: 1886: 1505: 1259:{\displaystyle V} 1239:{\displaystyle r} 1215:{\displaystyle r} 1167: 1121: 1081: 1047: 902:is the monopole; 895:{\displaystyle C} 882:. Here, the term 880:implicitly summed 224:multipole moments 160:{\displaystyle n} 10840: 10833:Moment (physics) 10823:Potential theory 10808: 10805:Angular Momentum 10801: 10795: 10788: 10782: 10781: 10769: 10759: 10750: 10749: 10723: 10714: 10708: 10707: 10704:Angular Momentum 10699: 10693: 10692: 10666: 10646: 10640: 10639: 10605: 10596: 10590: 10589: 10571: 10565: 10564: 10552: 10542: 10425:orthogonal basis 10402: 10383: 10372: 10359: 10346: 10323:series expansion 10269: 10267: 10266: 10261: 10259: 10255: 10254: 10245: 10244: 10234: 10233: 10220: 10215: 10200: 10195: 10193: 10185: 10175: 10170: 10157: 10156: 10147: 10146: 10136: 10135: 10122: 10117: 10102: 10097: 10087: 10082: 10065: 10060: 10047: 10042: 10029: 10028: 10015: 10010: 9995: 9990: 9988: 9980: 9970: 9965: 9952: 9951: 9942: 9941: 9931: 9930: 9917: 9912: 9897: 9892: 9882: 9877: 9860: 9855: 9842: 9837: 9821: 9820: 9807: 9802: 9787: 9779: 9769: 9764: 9751: 9750: 9740: 9739: 9726: 9721: 9698: 9693: 9680: 9679: 9669: 9668: 9655: 9650: 9627: 9622: 9609: 9608: 9598: 9597: 9584: 9579: 9556: 9551: 9531: 9529: 9528: 9523: 9520: 9515: 9499: 9497: 9496: 9491: 9488: 9483: 9461: 9459: 9458: 9453: 9442: 9437: 9436: 9427: 9416: 9415: 9402: 9397: 9378: 9373: 9358: 9356: 9355: 9350: 9339: 9334: 9333: 9318: 9317: 9308: 9297: 9296: 9283: 9278: 9263: 9256: 9252: 9245: 9237: 9227: 9223: 9213: 9208: 9207: 9192: 9191: 9182: 9171: 9170: 9157: 9152: 9137: 9130: 9126: 9116: 9111: 9095: 9093: 9092: 9087: 9082: 9081: 9068: 9063: 9044: 9039: 9016: 9009: 9007: 9006: 9001: 8998: 8993: 8960: 8958: 8957: 8952: 8937: 8932: 8910: 8905: 8885: 8883: 8882: 8877: 8857: 8855: 8854: 8849: 8844: 8843: 8838: 8828: 8823: 8807: 8805: 8804: 8799: 8791: 8790: 8785: 8775: 8770: 8760: 8759: 8746: 8741: 8722: 8717: 8603:is a normalized 8602: 8591: 8581: 8579: 8578: 8573: 8568: 8565: 8558: 8557: 8545: 8544: 8534: 8522: 8521: 8517: 8516: 8508: 8507: 8502: 8497: 8488: 8484: 8483: 8471: 8470: 8454: 8453: 8452: 8440: 8439: 8424: 8421: 8420: 8416: 8407: 8403: 8401: 8394: 8393: 8378: 8377: 8364: 8356: 8343: 8342: 8334: 8324: 8320: 8319: 8307: 8306: 8290: 8289: 8288: 8276: 8275: 8239: 8237: 8236: 8231: 8229: 8219: 8218: 8206: 8205: 8193: 8192: 8180: 8179: 8167: 8166: 8154: 8153: 8141: 8140: 8128: 8127: 8113: 8112: 8111: 8101: 8100: 8099: 8084: 8083: 8082: 8072: 8071: 8070: 8052: 8051: 8043: 8033: 8032: 8031: 8019: 8018: 8005: 8004: 8003: 7991: 7990: 7976: 7975: 7974: 7973: 7961: 7960: 7936: 7935: 7934: 7924: 7923: 7922: 7907: 7906: 7891: 7890: 7889: 7879: 7878: 7877: 7862: 7861: 7841: 7837: 7836: 7832: 7823: 7822: 7821: 7815: 7814: 7813: 7800: 7799: 7798: 7783: 7782: 7768: 7760: 7759: 7758: 7757: 7733: 7728: 7721: 7720: 7705: 7700: 7693: 7692: 7678: 7676: 7675: 7674: 7655: 7651: 7646: 7645: 7621:) gives finally 7620: 7616: 7615: 7614: 7597: 7595: 7594: 7589: 7581: 7572: 7567: 7558: 7557: 7533: 7521: 7516: 7496: 7494: 7493: 7488: 7471: 7470: 7452: 7451: 7433: 7432: 7420: 7419: 7403: 7402: 7394: 7384: 7383: 7382: 7366: 7365: 7364: 7341: 7340: 7332: 7322: 7321: 7320: 7310: 7309: 7308: 7293: 7292: 7291: 7281: 7280: 7279: 7264: 7263: 7246: 7245: 7241: 7232: 7231: 7230: 7224: 7223: 7222: 7209: 7200: 7192: 7191: 7190: 7189: 7159: 7154: 7147: 7146: 7126: 7125: 7117: 7108: 7107: 7099: 7089: 7084: 7060: 7058: 7057: 7052: 7049: 7044: 7028: 7026: 7025: 7020: 7017: 7012: 6996: 6994: 6993: 6988: 6980: 6979: 6971: 6962: 6961: 6953: 6943: 6938: 6925: 6924: 6916: 6906: 6898: 6889: 6888: 6868: 6863: 6844: 6839: 6821: 6819: 6818: 6810: 6809: 6801: 6792: 6791: 6783: 6771: 6770: 6762: 6756: 6747: 6742: 6740: 6739: 6734: 6733: 6728: 6719: 6718: 6713: 6707: 6698: 6685: 6683: 6682: 6677: 6663: 6660: 6658: 6653: 6652: 6644: 6635: 6634: 6626: 6620: 6612: 6607: 6606: 6598: 6592: 6580: 6578: 6577: 6572: 6567: 6566: 6558: 6549: 6548: 6540: 6531: 6530: 6522: 6513: 6512: 6504: 6485: 6484: 6476: 6467: 6466: 6458: 6449: 6448: 6440: 6431: 6430: 6422: 6409: 6405: 6401: 6358: 6354: 6350: 6348: 6347: 6342: 6337: 6335: 6334: 6333: 6321: 6320: 6304: 6303: 6302: 6293: 6292: 6282: 6279: 6263: 6245: 6244: 6225: 6202: 6198: 6185: 6181: 6160: 6158: 6157: 6152: 6143: 6138: 6125: 6120: 6105: 6097: 6094: 6093: 6083: 6078: 6060: 6059: 6058: 6057: 6036: 6026: 6024: 6023: 6018: 6013: 6011: 6010: 6009: 6000: 5999: 5983: 5982: 5974: 5973: 5968: 5963: 5959: 5954: 5952: 5951: 5950: 5941: 5940: 5924: 5923: 5915: 5909: 5901: 5900: 5891: 5890: 5878: 5877: 5868: 5867: 5855: 5854: 5845: 5844: 5832: 5830: 5829: 5828: 5819: 5818: 5799: 5791: 5783: 5782: 5760: 5758: 5757: 5752: 5722: 5721: 5709: 5708: 5698: 5693: 5675: 5674: 5664: 5660: 5653: 5652: 5640: 5639: 5627: 5626: 5611: 5599: 5598: 5586: 5585: 5573: 5572: 5557: 5545: 5535:This is in fact 5534: 5532: 5531: 5526: 5521: 5520: 5510: 5505: 5487: 5486: 5485: 5468: 5464: 5460: 5458: 5454: 5453: 5437: 5436: 5435: 5419: 5411: 5403: 5402: 5380: 5362: 5349: 5347: 5346: 5341: 5339: 5335: 5334: 5333: 5306: 5301: 5289: 5288: 5280: 5273: 5265: 5256: 5255: 5236: 5231: 5213: 5211: 5210: 5192: 5189: 5188: 5184: 5175: 5171: 5169: 5155: 5147: 5139: 5134: 5119: 5117: 5116: 5115: 5096: 5088: 5083: 5078: 5066: 5057: 5049: 5040: 5039: 5020: 5015: 4996: 4991: 4976: 4974: 4973: 4972: 4953: 4941: 4917: 4911: 4896: 4894: 4893: 4888: 4886: 4884: 4883: 4868: 4864: 4863: 4855: 4848: 4843: 4833: 4831: 4828: 4814: 4806: 4805: 4797: 4788: 4783: 4767: 4765: 4764: 4759: 4757: 4756: 4748: 4727: 4725: 4724: 4719: 4694: 4686: 4685: 4680: 4670: 4665: 4656: 4655: 4645: 4640: 4621: 4616: 4596: 4592: 4590: 4589: 4584: 4579: 4570: 4565: 4549: 4547: 4546: 4541: 4539: 4538: 4509:is an irregular 4508: 4506: 4505: 4500: 4495: 4486: 4478: 4462: 4460: 4459: 4454: 4446: 4445: 4440: 4430: 4425: 4416: 4415: 4405: 4400: 4382: 4373: 4365: 4356: 4355: 4336: 4331: 4312: 4307: 4292: 4290: 4289: 4288: 4269: 4264: 4262: 4261: 4256: 4248: 4247: 4242: 4236: 4231: 4230: 4214: 4213: 4204: 4201: 4196: 4175: 4153: 4145: 4137: 4131: 4108: 4106: 4105: 4100: 4092: 4084: 4076: 4074: 4073: 4072: 4063: 4062: 4043: 4035: 4017: 3998: 3989: 3987: 3986: 3981: 3973: 3972: 3959: 3949: 3945: 3935: 3934: 3921: 3902: 3900: 3899: 3894: 3892: 3882: 3881: 3872: 3871: 3862: 3861: 3848: 3815: 3813: 3812: 3811: 3795: 3790: 3789: 3780: 3779: 3769: 3742: 3740: 3739: 3727: 3722: 3717: 3716: 3715: 3699: 3691: 3684: 3676: 3675: 3670: 3658: 3657: 3647: 3642: 3617: 3606: 3605: 3575: 3573: 3572: 3567: 3558: 3553: 3544: 3543: 3528: 3527: 3515: 3514: 3496: 3495: 3485: 3480: 3462: 3461: 3448: 3445: 3439: 3438: 3426: 3425: 3415: 3410: 3392: 3391: 3378: 3377: 3367: 3362: 3344: 3343: 3342: 3319: 3317: 3316: 3311: 3306: 3304: 3303: 3294: 3293: 3292: 3283: 3282: 3267: 3266: 3257: 3256: 3243: 3235: 3227: 3226: 3213: 3209: 3202: 3200: 3199: 3190: 3189: 3180: 3169: 3161: 3160: 3147: 3139: 3131: 3114:it follows that 3109: 3107: 3106: 3101: 3096: 3094: 3093: 3088: 3080: 3075: 3066: 3058: 3050: 3032: 3009: 3003: 3001: 2986: 2975: 2973: 2972: 2967: 2959: 2951: 2950: 2938: 2934: 2933: 2932: 2923: 2922: 2907: 2906: 2897: 2896: 2878: 2850: 2823: 2815: 2807: 2799: 2798: 2786: 2785: 2776: 2775: 2765: 2737: 2699: 2697: 2696: 2691: 2677: 2669: 2668: 2655: 2625: 2624: 2623: 2615: 2609: 2605: 2601: 2593: 2582: 2581: 2558:Laplace equation 2555: 2540: 2538: 2537: 2532: 2527: 2526: 2525: 2517: 2511: 2507: 2505: 2504: 2503: 2491: 2490: 2477: 2473: 2465: 2454: 2453: 2443: 2430: 2422: 2421: 2408: 2405: 2402: 2401: 2400: 2392: 2386: 2382: 2380: 2379: 2378: 2365: 2361: 2353: 2338: 2325: 2317: 2316: 2300: 2298: 2297: 2292: 2290: 2271: 2263: 2262: 2250: 2249: 2240: 2239: 2229: 2201: 2174: 2166: 2158: 2150: 2149: 2140: 2139: 2129: 2096: 2079: 2060: 2049: 2048: 2036: 2035: 2026: 2025: 2015: 1987: 1960: 1952: 1944: 1933: 1932: 1923: 1922: 1912: 1879: 1855: 1847: 1825: 1815: 1799:Taylor expansion 1796: 1765: 1763: 1748:which depend on 1743: 1737: 1731: 1718: 1711: 1703: 1695: 1689: 1678: 1669: 1647: 1636: 1625: 1550: 1548: 1547: 1542: 1521: 1516: 1506: 1504: 1503: 1493: 1488: 1473: 1470: 1465: 1445: 1440: 1423: 1418: 1384: 1379: 1359: 1354: 1344: 1339: 1319: 1314: 1265: 1263: 1262: 1257: 1245: 1243: 1242: 1237: 1221: 1219: 1218: 1213: 1193:vector potential 1180: 1178: 1177: 1172: 1165: 1160: 1155: 1137: 1136: 1119: 1114: 1109: 1094: 1093: 1079: 1074: 1069: 1057: 1056: 1045: 1041: 1040: 1018: 1016: 1015: 1010: 1003: 995: 986: 985: 963: 955: 928: 926: 925: 920: 918: 917: 901: 899: 898: 893: 877: 875: 874: 869: 857: 855: 854: 849: 833: 831: 830: 825: 823: 822: 806: 804: 803: 798: 790: 789: 780: 779: 770: 769: 760: 759: 750: 749: 728: 727: 718: 717: 708: 707: 698: 697: 679: 678: 669: 668: 659: 658: 643: 642: 633: 632: 589: 587: 586: 581: 578: 573: 560: 555: 542: 534: 518: 516: 515: 510: 507: 502: 486: 484: 483: 478: 475: 470: 454: 452: 451: 446: 428: 423: 407: 405: 404: 399: 381: 376: 365: 360: 349: 344: 324: 319: 276: 274: 273: 268: 168: 166: 164: 163: 158: 145: 143: 142: 137: 135: 134: 129: 116: 114: 113: 108: 106: 105: 100: 75: 73: 72: 67: 65: 64: 59: 31:that depends on 10848: 10847: 10843: 10842: 10841: 10839: 10838: 10837: 10828:Vector calculus 10813: 10812: 10811: 10802: 10798: 10789: 10785: 10778: 10760: 10753: 10721: 10715: 10711: 10700: 10696: 10647: 10643: 10603: 10597: 10593: 10586: 10572: 10568: 10561: 10543: 10539: 10535: 10522:Toroidal moment 10517:Solid harmonics 10484: 10413: 10397: 10378: 10367: 10354: 10341: 10315: 10275: 10257: 10256: 10250: 10246: 10240: 10236: 10229: 10225: 10216: 10205: 10194: 10184: 10177: 10171: 10166: 10159: 10158: 10152: 10148: 10142: 10138: 10131: 10127: 10118: 10107: 10096: 10089: 10083: 10078: 10071: 10070: 10061: 10056: 10043: 10038: 10024: 10020: 10011: 10000: 9989: 9979: 9972: 9966: 9961: 9954: 9953: 9947: 9943: 9937: 9933: 9926: 9922: 9913: 9902: 9891: 9884: 9878: 9873: 9866: 9865: 9856: 9851: 9838: 9833: 9816: 9812: 9803: 9792: 9778: 9771: 9765: 9760: 9753: 9752: 9746: 9742: 9735: 9731: 9722: 9711: 9700: 9694: 9689: 9682: 9681: 9675: 9671: 9664: 9660: 9651: 9640: 9629: 9623: 9618: 9611: 9610: 9604: 9600: 9593: 9589: 9580: 9569: 9558: 9552: 9547: 9539: 9537: 9534: 9533: 9516: 9511: 9505: 9502: 9501: 9484: 9479: 9473: 9470: 9469: 9438: 9432: 9428: 9423: 9411: 9407: 9398: 9387: 9374: 9369: 9363: 9360: 9359: 9335: 9329: 9325: 9313: 9309: 9304: 9292: 9288: 9279: 9268: 9250: 9238: 9233: 9221: 9209: 9203: 9199: 9187: 9183: 9178: 9166: 9162: 9153: 9142: 9124: 9112: 9107: 9101: 9098: 9097: 9077: 9073: 9064: 9053: 9040: 9035: 9029: 9026: 9025: 9011: 8994: 8989: 8983: 8980: 8979: 8975:selection rules 8933: 8928: 8906: 8901: 8895: 8892: 8891: 8871: 8868: 8867: 8839: 8834: 8833: 8824: 8819: 8813: 8810: 8809: 8786: 8781: 8780: 8771: 8766: 8755: 8751: 8742: 8731: 8718: 8713: 8707: 8704: 8703: 8701: 8692: 8683: 8674: 8668: 8659: 8638: 8613: 8601: 8593: 8592:. The function 8589: 8583: 8553: 8549: 8540: 8536: 8535: 8527: 8509: 8498: 8496: 8495: 8494: 8490: 8479: 8475: 8466: 8462: 8455: 8448: 8444: 8435: 8431: 8430: 8425: 8423: 8412: 8408: 8389: 8385: 8373: 8369: 8365: 8357: 8355: 8351: 8350: 8335: 8330: 8329: 8315: 8311: 8302: 8298: 8291: 8284: 8280: 8271: 8267: 8266: 8260: 8257: 8256: 8254: 8227: 8226: 8214: 8210: 8201: 8197: 8188: 8184: 8175: 8171: 8162: 8158: 8149: 8145: 8136: 8132: 8123: 8119: 8107: 8103: 8102: 8095: 8091: 8090: 8078: 8074: 8073: 8066: 8062: 8061: 8044: 8039: 8038: 8027: 8023: 8014: 8010: 8006: 7999: 7995: 7986: 7982: 7981: 7969: 7965: 7956: 7952: 7951: 7947: 7930: 7926: 7925: 7918: 7914: 7902: 7898: 7897: 7885: 7881: 7880: 7873: 7869: 7857: 7853: 7852: 7839: 7838: 7828: 7824: 7817: 7809: 7805: 7801: 7794: 7790: 7778: 7774: 7770: 7764: 7763: 7762: 7761: 7753: 7749: 7748: 7744: 7729: 7716: 7712: 7711: 7701: 7688: 7684: 7683: 7670: 7666: 7659: 7654: 7652: 7650: 7638: 7634: 7630: 7628: 7625: 7624: 7618: 7613: 7608: 7607: 7606: 7602: 7577: 7568: 7563: 7553: 7549: 7529: 7517: 7512: 7506: 7503: 7502: 7466: 7462: 7447: 7443: 7428: 7424: 7415: 7411: 7395: 7390: 7389: 7378: 7374: 7367: 7360: 7356: 7349: 7333: 7328: 7327: 7316: 7312: 7311: 7304: 7300: 7299: 7287: 7283: 7282: 7275: 7271: 7259: 7255: 7254: 7237: 7233: 7226: 7218: 7214: 7210: 7202: 7196: 7195: 7194: 7193: 7185: 7181: 7174: 7170: 7155: 7142: 7138: 7137: 7118: 7113: 7112: 7100: 7095: 7094: 7085: 7080: 7074: 7071: 7070: 7063:solid harmonics 7045: 7040: 7034: 7031: 7030: 7013: 7008: 7002: 6999: 6998: 6972: 6967: 6966: 6954: 6949: 6948: 6939: 6934: 6917: 6912: 6911: 6899: 6894: 6884: 6880: 6864: 6850: 6840: 6829: 6814: 6802: 6797: 6796: 6784: 6779: 6778: 6763: 6758: 6757: 6752: 6751: 6746: 6735: 6729: 6724: 6723: 6714: 6709: 6708: 6703: 6702: 6697: 6695: 6692: 6691: 6659: 6654: 6645: 6640: 6639: 6627: 6622: 6621: 6616: 6608: 6599: 6594: 6593: 6588: 6586: 6583: 6582: 6559: 6554: 6553: 6541: 6536: 6535: 6523: 6518: 6517: 6505: 6500: 6499: 6477: 6472: 6471: 6459: 6454: 6453: 6441: 6436: 6435: 6423: 6418: 6417: 6415: 6412: 6411: 6407: 6403: 6400: 6391: 6382: 6376: 6371: 6356: 6352: 6326: 6322: 6316: 6312: 6305: 6298: 6294: 6288: 6284: 6283: 6281: 6269: 6253: 6237: 6233: 6231: 6228: 6227: 6224: 6216: 6200: 6196: 6187: 6183: 6179: 6170: 6167: 6139: 6134: 6121: 6116: 6096: 6089: 6085: 6079: 6068: 6053: 6049: 6048: 6044: 6042: 6039: 6038: 6031: 6005: 6001: 5995: 5991: 5984: 5978: 5964: 5962: 5961: 5960: 5958: 5946: 5942: 5936: 5932: 5925: 5919: 5911: 5910: 5908: 5896: 5892: 5886: 5882: 5873: 5869: 5863: 5859: 5850: 5846: 5840: 5836: 5824: 5820: 5814: 5810: 5803: 5798: 5787: 5772: 5768: 5766: 5763: 5762: 5714: 5710: 5704: 5700: 5694: 5683: 5670: 5666: 5658: 5648: 5644: 5635: 5631: 5622: 5618: 5607: 5594: 5590: 5581: 5577: 5568: 5564: 5553: 5551: 5548: 5547: 5540: 5539:again. For the 5516: 5512: 5506: 5495: 5475: 5474: 5470: 5462: 5449: 5445: 5438: 5425: 5424: 5420: 5418: 5407: 5392: 5388: 5386: 5383: 5382: 5375: 5357: 5337: 5336: 5323: 5322: 5318: 5302: 5297: 5279: 5278: 5266: 5261: 5251: 5247: 5232: 5218: 5200: 5196: 5191: 5180: 5176: 5156: 5148: 5146: 5142: 5141: 5135: 5124: 5111: 5107: 5100: 5095: 5086: 5085: 5079: 5074: 5062: 5050: 5045: 5035: 5031: 5016: 5002: 4992: 4981: 4968: 4964: 4957: 4952: 4945: 4937: 4927: 4925: 4922: 4921: 4913: 4902: 4873: 4869: 4854: 4853: 4844: 4839: 4834: 4832: 4815: 4807: 4804: 4793: 4784: 4779: 4773: 4770: 4769: 4747: 4746: 4744: 4741: 4740: 4681: 4676: 4675: 4666: 4661: 4651: 4647: 4641: 4630: 4617: 4612: 4606: 4603: 4602: 4594: 4575: 4566: 4561: 4555: 4552: 4551: 4528: 4524: 4522: 4519: 4518: 4491: 4479: 4474: 4468: 4465: 4464: 4441: 4436: 4435: 4426: 4421: 4411: 4407: 4401: 4390: 4378: 4366: 4361: 4351: 4347: 4332: 4318: 4308: 4297: 4284: 4280: 4273: 4268: 4257: 4252: 4243: 4238: 4237: 4232: 4226: 4222: 4215: 4209: 4205: 4203: 4197: 4186: 4171: 4163: 4160: 4159: 4152: 4141: 4139: 4133: 4122: 4119: 4088: 4080: 4068: 4064: 4058: 4054: 4047: 4042: 4031: 4023: 4020: 4019: 4000: 3999:apart, so that 3996: 3965: 3961: 3955: 3943: 3927: 3923: 3917: 3911: 3908: 3907: 3890: 3889: 3877: 3873: 3867: 3863: 3854: 3850: 3820: 3807: 3803: 3799: 3794: 3785: 3781: 3775: 3771: 3747: 3735: 3731: 3726: 3705: 3704: 3700: 3698: 3689: 3688: 3680: 3671: 3666: 3665: 3653: 3649: 3643: 3632: 3621: 3613: 3601: 3597: 3587: 3585: 3582: 3581: 3554: 3549: 3536: 3532: 3520: 3516: 3507: 3503: 3491: 3487: 3481: 3470: 3454: 3450: 3444: 3431: 3427: 3421: 3417: 3411: 3400: 3387: 3383: 3373: 3369: 3363: 3352: 3332: 3331: 3327: 3325: 3322: 3321: 3299: 3295: 3288: 3284: 3275: 3271: 3262: 3258: 3252: 3248: 3244: 3242: 3231: 3219: 3215: 3207: 3195: 3191: 3185: 3181: 3179: 3165: 3156: 3152: 3138: 3127: 3119: 3116: 3115: 3112:differentiation 3110:Then by direct 3089: 3084: 3076: 3071: 3070: 3065: 3054: 3046: 3038: 3035: 3034: 3019: 3016: 3005: 2997: 2992: 2989:Kronecker delta 2985: 2977: 2955: 2943: 2939: 2928: 2924: 2915: 2911: 2902: 2898: 2892: 2888: 2884: 2880: 2856: 2828: 2814: 2803: 2791: 2787: 2781: 2777: 2771: 2767: 2743: 2715: 2709: 2706: 2705: 2673: 2661: 2657: 2633: 2619: 2611: 2610: 2597: 2589: 2577: 2573: 2572: 2568: 2567: 2565: 2562: 2561: 2542: 2521: 2513: 2512: 2499: 2495: 2486: 2482: 2478: 2469: 2461: 2449: 2445: 2444: 2442: 2438: 2437: 2426: 2414: 2410: 2404: 2396: 2388: 2387: 2374: 2370: 2366: 2357: 2349: 2339: 2337: 2333: 2332: 2321: 2312: 2308: 2306: 2303: 2302: 2288: 2287: 2267: 2255: 2251: 2245: 2241: 2235: 2231: 2207: 2179: 2165: 2154: 2145: 2141: 2135: 2131: 2107: 2092: 2077: 2076: 2053: 2041: 2037: 2031: 2027: 2021: 2017: 1993: 1965: 1951: 1937: 1928: 1924: 1918: 1914: 1890: 1872: 1859: 1851: 1843: 1833: 1831: 1828: 1827: 1817: 1802: 1779: 1776: 1755: 1753: 1739: 1733: 1727: 1713: 1710: 1699: 1697: 1691: 1680: 1677: 1671: 1668: 1661: 1653: 1646: 1638: 1635: 1627: 1623: 1620: 1612:Ewald summation 1572:magnetic fields 1556: 1517: 1512: 1499: 1495: 1489: 1478: 1472: 1466: 1452: 1441: 1430: 1419: 1408: 1380: 1375: 1355: 1350: 1340: 1326: 1315: 1304: 1271: 1268: 1267: 1251: 1248: 1247: 1231: 1228: 1227: 1207: 1204: 1203: 1156: 1145: 1126: 1122: 1110: 1102: 1086: 1082: 1070: 1065: 1052: 1048: 1036: 1032: 1024: 1021: 1020: 996: 991: 981: 977: 956: 951: 945: 942: 941: 913: 909: 907: 904: 903: 887: 884: 883: 863: 860: 859: 843: 840: 839: 818: 814: 812: 809: 808: 785: 781: 775: 771: 765: 761: 755: 751: 736: 732: 723: 719: 713: 709: 703: 699: 687: 683: 674: 670: 664: 660: 651: 647: 638: 634: 628: 624: 595: 592: 591: 574: 569: 556: 551: 535: 530: 524: 521: 520: 503: 498: 492: 489: 488: 471: 466: 460: 457: 456: 424: 419: 413: 410: 409: 377: 372: 361: 356: 345: 331: 320: 309: 282: 279: 278: 247: 244: 243: 236: 173:electromagnetic 152: 149: 148: 147: 146:for some other 130: 125: 124: 122: 119: 118: 101: 96: 95: 93: 90: 89: 76:. Similarly to 60: 55: 54: 52: 49: 48: 45:Euclidean space 39:(the polar and 27:representing a 17: 12: 11: 5: 10846: 10836: 10835: 10830: 10825: 10810: 10809: 10796: 10783: 10776: 10751: 10732:(2): 299–339. 10709: 10694: 10657:(7): 528–533. 10651:Nature Physics 10641: 10591: 10584: 10566: 10559: 10536: 10534: 10531: 10530: 10529: 10524: 10519: 10514: 10505: 10500: 10495: 10490: 10483: 10480: 10452:symmetry group 10412: 10409: 10375: 10374: 10361: 10360:potential; and 10348: 10314: 10311: 10274: 10271: 10253: 10249: 10243: 10239: 10232: 10228: 10224: 10219: 10214: 10211: 10208: 10204: 10198: 10191: 10188: 10183: 10180: 10178: 10174: 10169: 10165: 10161: 10160: 10155: 10151: 10145: 10141: 10134: 10130: 10126: 10121: 10116: 10113: 10110: 10106: 10100: 10095: 10092: 10090: 10086: 10081: 10077: 10073: 10072: 10069: 10064: 10059: 10055: 10051: 10046: 10041: 10037: 10033: 10027: 10023: 10019: 10014: 10009: 10006: 10003: 9999: 9993: 9986: 9983: 9978: 9975: 9973: 9969: 9964: 9960: 9956: 9955: 9950: 9946: 9940: 9936: 9929: 9925: 9921: 9916: 9911: 9908: 9905: 9901: 9895: 9890: 9887: 9885: 9881: 9876: 9872: 9868: 9867: 9864: 9859: 9854: 9850: 9846: 9841: 9836: 9832: 9828: 9825: 9819: 9815: 9811: 9806: 9801: 9798: 9795: 9791: 9785: 9782: 9777: 9774: 9772: 9768: 9763: 9759: 9755: 9754: 9749: 9745: 9738: 9734: 9730: 9725: 9720: 9717: 9714: 9710: 9706: 9703: 9701: 9697: 9692: 9688: 9684: 9683: 9678: 9674: 9667: 9663: 9659: 9654: 9649: 9646: 9643: 9639: 9635: 9632: 9630: 9626: 9621: 9617: 9613: 9612: 9607: 9603: 9596: 9592: 9588: 9583: 9578: 9575: 9572: 9568: 9564: 9561: 9559: 9555: 9550: 9546: 9542: 9541: 9519: 9514: 9510: 9487: 9482: 9478: 9451: 9448: 9445: 9441: 9435: 9431: 9426: 9422: 9419: 9414: 9410: 9406: 9401: 9396: 9393: 9390: 9386: 9382: 9377: 9372: 9368: 9348: 9345: 9342: 9338: 9332: 9328: 9324: 9321: 9316: 9312: 9307: 9303: 9300: 9295: 9291: 9287: 9282: 9277: 9274: 9271: 9267: 9259: 9255: 9249: 9244: 9241: 9236: 9232: 9219: 9216: 9212: 9206: 9202: 9198: 9195: 9190: 9186: 9181: 9177: 9174: 9169: 9165: 9161: 9156: 9151: 9148: 9145: 9141: 9133: 9129: 9123: 9120: 9115: 9110: 9106: 9085: 9080: 9076: 9072: 9067: 9062: 9059: 9056: 9052: 9048: 9043: 9038: 9034: 8997: 8992: 8988: 8950: 8947: 8944: 8941: 8936: 8931: 8927: 8923: 8920: 8917: 8914: 8909: 8904: 8900: 8875: 8860:solid harmonic 8847: 8842: 8837: 8832: 8827: 8822: 8818: 8797: 8794: 8789: 8784: 8779: 8774: 8769: 8765: 8758: 8754: 8750: 8745: 8740: 8737: 8734: 8730: 8726: 8721: 8716: 8712: 8697: 8688: 8679: 8670: 8664: 8655: 8634: 8612: 8609: 8597: 8587: 8571: 8564: 8561: 8556: 8552: 8548: 8543: 8539: 8533: 8530: 8526: 8520: 8515: 8512: 8505: 8501: 8493: 8487: 8482: 8478: 8474: 8469: 8465: 8461: 8458: 8451: 8447: 8443: 8438: 8434: 8429: 8419: 8415: 8411: 8406: 8400: 8397: 8392: 8388: 8384: 8381: 8376: 8372: 8368: 8363: 8360: 8354: 8349: 8346: 8341: 8338: 8333: 8328: 8323: 8318: 8314: 8310: 8305: 8301: 8297: 8294: 8287: 8283: 8279: 8274: 8270: 8265: 8250: 8225: 8222: 8217: 8213: 8209: 8204: 8200: 8196: 8191: 8187: 8183: 8178: 8174: 8170: 8165: 8161: 8157: 8152: 8148: 8144: 8139: 8135: 8131: 8126: 8122: 8118: 8110: 8106: 8098: 8094: 8089: 8081: 8077: 8069: 8065: 8060: 8055: 8050: 8047: 8042: 8037: 8030: 8026: 8022: 8017: 8013: 8009: 8002: 7998: 7994: 7989: 7985: 7980: 7972: 7968: 7964: 7959: 7955: 7950: 7946: 7943: 7940: 7933: 7929: 7921: 7917: 7913: 7910: 7905: 7901: 7896: 7888: 7884: 7876: 7872: 7868: 7865: 7860: 7856: 7851: 7847: 7844: 7842: 7840: 7835: 7831: 7827: 7820: 7812: 7808: 7804: 7797: 7793: 7789: 7786: 7781: 7777: 7773: 7767: 7756: 7752: 7747: 7743: 7740: 7737: 7732: 7727: 7724: 7719: 7715: 7710: 7704: 7699: 7696: 7691: 7687: 7682: 7673: 7669: 7665: 7662: 7658: 7653: 7649: 7644: 7641: 7637: 7633: 7632: 7609: 7587: 7584: 7580: 7576: 7571: 7566: 7562: 7556: 7552: 7548: 7545: 7542: 7539: 7536: 7532: 7528: 7525: 7520: 7515: 7511: 7486: 7483: 7480: 7477: 7474: 7469: 7465: 7461: 7458: 7455: 7450: 7446: 7442: 7439: 7436: 7431: 7427: 7423: 7418: 7414: 7410: 7406: 7401: 7398: 7393: 7388: 7381: 7377: 7373: 7370: 7363: 7359: 7355: 7352: 7348: 7344: 7339: 7336: 7331: 7326: 7319: 7315: 7307: 7303: 7298: 7290: 7286: 7278: 7274: 7270: 7267: 7262: 7258: 7253: 7249: 7244: 7240: 7236: 7229: 7221: 7217: 7213: 7208: 7205: 7199: 7188: 7184: 7180: 7177: 7173: 7169: 7166: 7163: 7158: 7153: 7150: 7145: 7141: 7136: 7132: 7129: 7124: 7121: 7116: 7111: 7106: 7103: 7098: 7093: 7088: 7083: 7079: 7048: 7043: 7039: 7016: 7011: 7007: 6986: 6983: 6978: 6975: 6970: 6965: 6960: 6957: 6952: 6947: 6942: 6937: 6933: 6928: 6923: 6920: 6915: 6910: 6905: 6902: 6897: 6893: 6887: 6883: 6879: 6876: 6873: 6867: 6862: 6859: 6856: 6853: 6849: 6843: 6838: 6835: 6832: 6828: 6824: 6817: 6813: 6808: 6805: 6800: 6795: 6790: 6787: 6782: 6777: 6774: 6769: 6766: 6761: 6755: 6750: 6745: 6738: 6732: 6727: 6722: 6717: 6712: 6706: 6701: 6675: 6672: 6669: 6666: 6657: 6651: 6648: 6643: 6638: 6633: 6630: 6625: 6619: 6615: 6611: 6605: 6602: 6597: 6591: 6570: 6565: 6562: 6557: 6552: 6547: 6544: 6539: 6534: 6529: 6526: 6521: 6516: 6511: 6508: 6503: 6496: 6491: 6488: 6483: 6480: 6475: 6470: 6465: 6462: 6457: 6452: 6447: 6444: 6439: 6434: 6429: 6426: 6421: 6396: 6387: 6380: 6367: 6340: 6332: 6329: 6325: 6319: 6315: 6311: 6308: 6301: 6297: 6291: 6287: 6278: 6275: 6272: 6268: 6262: 6259: 6256: 6252: 6248: 6243: 6240: 6236: 6220: 6192: 6175: 6166: 6163: 6150: 6147: 6142: 6137: 6133: 6129: 6124: 6119: 6115: 6111: 6108: 6103: 6100: 6092: 6088: 6082: 6077: 6074: 6071: 6067: 6063: 6056: 6052: 6047: 6016: 6008: 6004: 5998: 5994: 5990: 5987: 5981: 5977: 5971: 5967: 5957: 5949: 5945: 5939: 5935: 5931: 5928: 5922: 5918: 5914: 5907: 5904: 5899: 5895: 5889: 5885: 5881: 5876: 5872: 5866: 5862: 5858: 5853: 5849: 5843: 5839: 5835: 5827: 5823: 5817: 5813: 5809: 5806: 5802: 5797: 5794: 5790: 5786: 5781: 5778: 5775: 5771: 5750: 5747: 5744: 5741: 5738: 5735: 5732: 5729: 5725: 5720: 5717: 5713: 5707: 5703: 5697: 5692: 5689: 5686: 5682: 5678: 5673: 5669: 5656: 5651: 5647: 5643: 5638: 5634: 5630: 5625: 5621: 5617: 5614: 5610: 5605: 5602: 5597: 5593: 5589: 5584: 5580: 5576: 5571: 5567: 5563: 5560: 5556: 5524: 5519: 5515: 5509: 5504: 5501: 5498: 5494: 5490: 5484: 5481: 5478: 5473: 5457: 5452: 5448: 5444: 5441: 5434: 5431: 5428: 5423: 5417: 5414: 5410: 5406: 5401: 5398: 5395: 5391: 5332: 5329: 5326: 5321: 5317: 5314: 5310: 5305: 5300: 5296: 5292: 5286: 5283: 5277: 5272: 5269: 5264: 5260: 5254: 5250: 5246: 5243: 5240: 5235: 5230: 5227: 5224: 5221: 5217: 5209: 5206: 5203: 5199: 5195: 5187: 5183: 5179: 5174: 5168: 5165: 5162: 5159: 5154: 5151: 5145: 5138: 5133: 5130: 5127: 5123: 5114: 5110: 5106: 5103: 5099: 5094: 5091: 5089: 5087: 5082: 5077: 5073: 5069: 5065: 5061: 5056: 5053: 5048: 5044: 5038: 5034: 5030: 5027: 5024: 5019: 5014: 5011: 5008: 5005: 5001: 4995: 4990: 4987: 4984: 4980: 4971: 4967: 4963: 4960: 4956: 4951: 4948: 4946: 4944: 4940: 4936: 4933: 4930: 4929: 4882: 4879: 4876: 4872: 4867: 4861: 4858: 4852: 4847: 4842: 4838: 4827: 4824: 4821: 4818: 4813: 4810: 4803: 4800: 4796: 4792: 4787: 4782: 4778: 4754: 4751: 4717: 4714: 4711: 4708: 4705: 4702: 4699: 4692: 4689: 4684: 4679: 4674: 4669: 4664: 4660: 4654: 4650: 4644: 4639: 4636: 4633: 4629: 4625: 4620: 4615: 4611: 4582: 4578: 4574: 4569: 4564: 4560: 4537: 4534: 4531: 4527: 4511:solid harmonic 4498: 4494: 4490: 4485: 4482: 4477: 4473: 4452: 4449: 4444: 4439: 4434: 4429: 4424: 4420: 4414: 4410: 4404: 4399: 4396: 4393: 4389: 4385: 4381: 4377: 4372: 4369: 4364: 4360: 4354: 4350: 4346: 4343: 4340: 4335: 4330: 4327: 4324: 4321: 4317: 4311: 4306: 4303: 4300: 4296: 4287: 4283: 4279: 4276: 4272: 4267: 4260: 4255: 4251: 4246: 4241: 4235: 4229: 4225: 4221: 4218: 4212: 4208: 4200: 4195: 4192: 4189: 4185: 4181: 4178: 4174: 4170: 4167: 4150: 4121:The potential 4118: 4117:Spherical form 4115: 4098: 4095: 4091: 4087: 4083: 4079: 4071: 4067: 4061: 4057: 4053: 4050: 4046: 4041: 4038: 4034: 4030: 4027: 3979: 3976: 3971: 3968: 3964: 3958: 3954: 3941: 3938: 3933: 3930: 3926: 3920: 3916: 3888: 3885: 3880: 3876: 3870: 3866: 3860: 3857: 3853: 3847: 3844: 3841: 3838: 3835: 3832: 3829: 3826: 3823: 3819: 3810: 3806: 3802: 3798: 3793: 3788: 3784: 3778: 3774: 3768: 3765: 3762: 3759: 3756: 3753: 3750: 3746: 3738: 3734: 3730: 3725: 3720: 3714: 3711: 3708: 3703: 3697: 3694: 3692: 3690: 3687: 3683: 3679: 3674: 3669: 3664: 3661: 3656: 3652: 3646: 3641: 3638: 3635: 3631: 3627: 3624: 3622: 3620: 3616: 3612: 3609: 3604: 3600: 3596: 3593: 3590: 3589: 3565: 3562: 3557: 3552: 3548: 3542: 3539: 3535: 3531: 3526: 3523: 3519: 3513: 3510: 3506: 3502: 3499: 3494: 3490: 3484: 3479: 3476: 3473: 3469: 3465: 3460: 3457: 3453: 3442: 3437: 3434: 3430: 3424: 3420: 3414: 3409: 3406: 3403: 3399: 3395: 3390: 3386: 3381: 3376: 3372: 3366: 3361: 3358: 3355: 3351: 3347: 3341: 3338: 3335: 3330: 3309: 3302: 3298: 3291: 3287: 3281: 3278: 3274: 3270: 3265: 3261: 3255: 3251: 3247: 3241: 3238: 3234: 3230: 3225: 3222: 3218: 3205: 3198: 3194: 3188: 3184: 3178: 3175: 3172: 3168: 3164: 3159: 3155: 3150: 3145: 3142: 3137: 3134: 3130: 3126: 3123: 3099: 3092: 3087: 3083: 3079: 3074: 3069: 3064: 3061: 3057: 3053: 3049: 3045: 3042: 3015: 3012: 2981: 2965: 2962: 2958: 2954: 2949: 2946: 2942: 2937: 2931: 2927: 2921: 2918: 2914: 2910: 2905: 2901: 2895: 2891: 2887: 2883: 2877: 2874: 2871: 2868: 2865: 2862: 2859: 2855: 2849: 2846: 2843: 2840: 2837: 2834: 2831: 2827: 2821: 2818: 2813: 2810: 2806: 2802: 2797: 2794: 2790: 2784: 2780: 2774: 2770: 2764: 2761: 2758: 2755: 2752: 2749: 2746: 2742: 2736: 2733: 2730: 2727: 2724: 2721: 2718: 2714: 2689: 2686: 2683: 2680: 2676: 2672: 2667: 2664: 2660: 2654: 2651: 2648: 2645: 2642: 2639: 2636: 2632: 2628: 2622: 2618: 2614: 2608: 2604: 2600: 2596: 2592: 2588: 2585: 2580: 2576: 2571: 2556:satisfies the 2530: 2524: 2520: 2516: 2510: 2502: 2498: 2494: 2489: 2485: 2481: 2476: 2472: 2468: 2464: 2460: 2457: 2452: 2448: 2441: 2436: 2433: 2429: 2425: 2420: 2417: 2413: 2399: 2395: 2391: 2385: 2377: 2373: 2369: 2364: 2360: 2356: 2352: 2348: 2345: 2342: 2336: 2331: 2328: 2324: 2320: 2315: 2311: 2286: 2283: 2280: 2277: 2274: 2270: 2266: 2261: 2258: 2254: 2248: 2244: 2238: 2234: 2228: 2225: 2222: 2219: 2216: 2213: 2210: 2206: 2200: 2197: 2194: 2191: 2188: 2185: 2182: 2178: 2172: 2169: 2164: 2161: 2157: 2153: 2148: 2144: 2138: 2134: 2128: 2125: 2122: 2119: 2116: 2113: 2110: 2106: 2102: 2099: 2095: 2091: 2088: 2085: 2082: 2080: 2078: 2075: 2072: 2069: 2066: 2063: 2059: 2056: 2052: 2047: 2044: 2040: 2034: 2030: 2024: 2020: 2014: 2011: 2008: 2005: 2002: 1999: 1996: 1992: 1986: 1983: 1980: 1977: 1974: 1971: 1968: 1964: 1958: 1955: 1950: 1947: 1943: 1940: 1936: 1931: 1927: 1921: 1917: 1911: 1908: 1905: 1902: 1899: 1896: 1893: 1889: 1885: 1882: 1878: 1875: 1871: 1868: 1865: 1862: 1860: 1858: 1854: 1850: 1846: 1842: 1839: 1836: 1835: 1775: 1772: 1708: 1675: 1666: 1657: 1642: 1631: 1626:point charges 1619: 1616: 1562:of systems of 1555: 1552: 1540: 1537: 1534: 1531: 1528: 1525: 1520: 1515: 1511: 1502: 1498: 1492: 1487: 1484: 1481: 1477: 1469: 1464: 1461: 1458: 1455: 1451: 1444: 1439: 1436: 1433: 1429: 1422: 1417: 1414: 1411: 1407: 1403: 1400: 1397: 1394: 1391: 1388: 1383: 1378: 1374: 1369: 1366: 1363: 1358: 1353: 1349: 1343: 1338: 1335: 1332: 1329: 1325: 1318: 1313: 1310: 1307: 1303: 1299: 1296: 1293: 1290: 1287: 1284: 1281: 1278: 1275: 1255: 1235: 1224:Laurent series 1211: 1170: 1164: 1159: 1154: 1151: 1148: 1144: 1140: 1135: 1132: 1129: 1125: 1118: 1113: 1108: 1105: 1101: 1097: 1092: 1089: 1085: 1078: 1073: 1068: 1064: 1060: 1055: 1051: 1044: 1039: 1035: 1031: 1028: 1008: 1002: 999: 994: 990: 984: 980: 976: 973: 970: 967: 962: 959: 954: 950: 916: 912: 891: 867: 847: 821: 817: 796: 793: 788: 784: 778: 774: 768: 764: 758: 754: 748: 745: 742: 739: 735: 731: 726: 722: 716: 712: 706: 702: 696: 693: 690: 686: 682: 677: 673: 667: 663: 657: 654: 650: 646: 641: 637: 631: 627: 623: 620: 617: 614: 611: 608: 605: 602: 599: 577: 572: 568: 564: 559: 554: 550: 546: 541: 538: 533: 529: 506: 501: 497: 474: 469: 465: 444: 441: 438: 435: 432: 427: 422: 418: 397: 394: 391: 388: 385: 380: 375: 371: 364: 359: 355: 348: 343: 340: 337: 334: 330: 323: 318: 315: 312: 308: 304: 301: 298: 295: 292: 289: 286: 266: 263: 260: 257: 254: 251: 235: 232: 156: 133: 128: 104: 99: 63: 58: 15: 9: 6: 4: 3: 2: 10845: 10834: 10831: 10829: 10826: 10824: 10821: 10820: 10818: 10806: 10800: 10793: 10787: 10779: 10773: 10768: 10767: 10758: 10756: 10747: 10743: 10739: 10735: 10731: 10727: 10720: 10713: 10705: 10698: 10690: 10686: 10682: 10678: 10674: 10670: 10665: 10660: 10656: 10652: 10645: 10637: 10633: 10629: 10625: 10621: 10617: 10613: 10609: 10602: 10595: 10587: 10585:9780199565122 10581: 10577: 10570: 10562: 10560:9780691079127 10556: 10551: 10550: 10541: 10537: 10528: 10525: 10523: 10520: 10518: 10515: 10513: 10509: 10506: 10504: 10501: 10499: 10496: 10494: 10491: 10489: 10486: 10485: 10479: 10477: 10473: 10469: 10463: 10461: 10457: 10453: 10449: 10445: 10440: 10438: 10434: 10430: 10426: 10422: 10418: 10408: 10406: 10401: 10395: 10391: 10387: 10382: 10371: 10365: 10362: 10358: 10352: 10349: 10345: 10339: 10336: 10335: 10334: 10332: 10328: 10324: 10320: 10310: 10307: 10305: 10300: 10296: 10292: 10288: 10284: 10280: 10270: 10251: 10247: 10241: 10237: 10230: 10226: 10222: 10217: 10212: 10209: 10206: 10202: 10196: 10189: 10186: 10181: 10179: 10172: 10167: 10163: 10153: 10149: 10143: 10139: 10132: 10128: 10124: 10119: 10114: 10111: 10108: 10104: 10098: 10093: 10091: 10084: 10079: 10075: 10062: 10057: 10053: 10049: 10044: 10039: 10035: 10025: 10021: 10017: 10012: 10007: 10004: 10001: 9997: 9991: 9984: 9981: 9976: 9974: 9967: 9962: 9958: 9948: 9944: 9938: 9934: 9927: 9923: 9919: 9914: 9909: 9906: 9903: 9899: 9893: 9888: 9886: 9879: 9874: 9870: 9857: 9852: 9848: 9844: 9839: 9834: 9830: 9826: 9817: 9813: 9809: 9804: 9799: 9796: 9793: 9789: 9783: 9780: 9775: 9773: 9766: 9761: 9757: 9747: 9743: 9736: 9732: 9728: 9723: 9718: 9715: 9712: 9708: 9704: 9702: 9695: 9690: 9686: 9676: 9672: 9665: 9661: 9657: 9652: 9647: 9644: 9641: 9637: 9633: 9631: 9624: 9619: 9615: 9605: 9601: 9594: 9590: 9586: 9581: 9576: 9573: 9570: 9566: 9562: 9560: 9553: 9548: 9544: 9517: 9512: 9508: 9500:or sine type 9485: 9480: 9476: 9467: 9462: 9449: 9433: 9429: 9412: 9408: 9404: 9399: 9394: 9391: 9388: 9384: 9380: 9375: 9370: 9366: 9346: 9330: 9326: 9322: 9319: 9314: 9310: 9293: 9289: 9285: 9280: 9275: 9272: 9269: 9265: 9257: 9253: 9247: 9242: 9239: 9234: 9230: 9204: 9200: 9196: 9193: 9188: 9184: 9167: 9163: 9159: 9154: 9149: 9146: 9143: 9139: 9131: 9127: 9121: 9118: 9113: 9108: 9104: 9083: 9078: 9074: 9070: 9065: 9060: 9057: 9054: 9050: 9046: 9041: 9036: 9032: 9023: 9018: 9014: 8995: 8990: 8986: 8976: 8972: 8968: 8964: 8948: 8939: 8934: 8929: 8925: 8921: 8912: 8907: 8902: 8898: 8889: 8873: 8865: 8861: 8858:is a regular 8840: 8825: 8820: 8816: 8795: 8787: 8772: 8767: 8763: 8756: 8752: 8748: 8743: 8738: 8735: 8732: 8728: 8724: 8719: 8714: 8710: 8700: 8696: 8691: 8687: 8682: 8678: 8673: 8667: 8663: 8658: 8654: 8650: 8646: 8645:atomic number 8642: 8637: 8633: 8629: 8624: 8621: 8619: 8608: 8606: 8600: 8596: 8590: 8569: 8562: 8559: 8554: 8550: 8546: 8541: 8537: 8531: 8528: 8524: 8518: 8513: 8510: 8503: 8491: 8480: 8476: 8472: 8467: 8463: 8456: 8449: 8445: 8441: 8436: 8432: 8427: 8417: 8413: 8409: 8404: 8398: 8395: 8390: 8386: 8382: 8379: 8374: 8370: 8366: 8361: 8358: 8352: 8347: 8339: 8336: 8316: 8312: 8308: 8303: 8299: 8292: 8285: 8281: 8277: 8272: 8268: 8263: 8255:apart. Since 8253: 8249: 8245: 8240: 8223: 8215: 8211: 8207: 8202: 8198: 8194: 8189: 8185: 8181: 8176: 8172: 8168: 8163: 8159: 8155: 8150: 8146: 8142: 8137: 8133: 8129: 8124: 8120: 8108: 8104: 8096: 8092: 8087: 8079: 8075: 8067: 8063: 8058: 8048: 8045: 8028: 8024: 8020: 8015: 8011: 8007: 8000: 7996: 7992: 7987: 7983: 7978: 7970: 7966: 7962: 7957: 7953: 7944: 7941: 7931: 7927: 7919: 7915: 7911: 7908: 7903: 7899: 7894: 7886: 7882: 7874: 7870: 7866: 7863: 7858: 7854: 7849: 7845: 7843: 7833: 7829: 7825: 7810: 7806: 7802: 7795: 7791: 7787: 7784: 7779: 7775: 7771: 7754: 7750: 7741: 7738: 7725: 7722: 7717: 7713: 7708: 7697: 7694: 7689: 7685: 7680: 7671: 7667: 7663: 7660: 7656: 7647: 7642: 7639: 7635: 7622: 7612: 7605: 7601: 7585: 7569: 7564: 7560: 7554: 7546: 7543: 7537: 7526: 7518: 7513: 7509: 7500: 7484: 7478: 7475: 7472: 7467: 7463: 7459: 7456: 7453: 7448: 7444: 7440: 7437: 7434: 7429: 7425: 7421: 7416: 7412: 7399: 7396: 7379: 7375: 7371: 7368: 7361: 7357: 7353: 7350: 7346: 7337: 7334: 7317: 7313: 7305: 7301: 7296: 7288: 7284: 7276: 7272: 7268: 7265: 7260: 7256: 7251: 7247: 7242: 7238: 7234: 7219: 7215: 7211: 7206: 7203: 7186: 7182: 7178: 7175: 7167: 7164: 7156: 7151: 7148: 7143: 7139: 7134: 7130: 7122: 7119: 7109: 7104: 7101: 7086: 7081: 7077: 7068: 7064: 7046: 7041: 7037: 7014: 7009: 7005: 6984: 6976: 6973: 6963: 6958: 6955: 6940: 6935: 6931: 6921: 6918: 6903: 6900: 6895: 6891: 6885: 6877: 6874: 6865: 6860: 6857: 6854: 6851: 6847: 6836: 6833: 6830: 6826: 6822: 6806: 6803: 6793: 6788: 6785: 6772: 6767: 6764: 6748: 6743: 6730: 6720: 6715: 6699: 6689: 6673: 6670: 6667: 6664: 6649: 6646: 6636: 6631: 6628: 6613: 6603: 6600: 6568: 6563: 6560: 6550: 6545: 6542: 6532: 6527: 6524: 6514: 6509: 6506: 6489: 6486: 6481: 6478: 6468: 6463: 6460: 6450: 6445: 6442: 6432: 6427: 6424: 6399: 6395: 6390: 6386: 6379: 6373: 6370: 6366: 6362: 6338: 6330: 6327: 6323: 6317: 6313: 6309: 6306: 6299: 6295: 6289: 6285: 6276: 6273: 6270: 6266: 6260: 6257: 6254: 6250: 6246: 6241: 6238: 6234: 6223: 6219: 6214: 6210: 6206: 6195: 6191: 6178: 6174: 6162: 6148: 6140: 6135: 6131: 6127: 6122: 6117: 6113: 6109: 6101: 6098: 6090: 6086: 6080: 6075: 6072: 6069: 6065: 6061: 6054: 6050: 6045: 6034: 6028: 6014: 6006: 6002: 5996: 5992: 5988: 5985: 5975: 5955: 5947: 5943: 5937: 5933: 5929: 5926: 5916: 5905: 5897: 5893: 5887: 5883: 5879: 5874: 5870: 5864: 5860: 5856: 5851: 5847: 5841: 5837: 5825: 5821: 5815: 5811: 5807: 5804: 5800: 5795: 5779: 5776: 5773: 5769: 5748: 5745: 5742: 5739: 5736: 5733: 5730: 5727: 5723: 5718: 5715: 5711: 5705: 5701: 5695: 5690: 5687: 5684: 5680: 5676: 5671: 5667: 5649: 5645: 5641: 5636: 5632: 5628: 5623: 5619: 5612: 5603: 5595: 5591: 5587: 5582: 5578: 5574: 5569: 5565: 5558: 5543: 5538: 5537:Coulomb's law 5522: 5517: 5513: 5507: 5502: 5499: 5496: 5492: 5488: 5471: 5455: 5450: 5446: 5442: 5439: 5421: 5415: 5399: 5396: 5393: 5389: 5381:term becomes 5378: 5373: 5369: 5364: 5361: 5355: 5350: 5319: 5315: 5312: 5308: 5303: 5298: 5294: 5281: 5270: 5267: 5262: 5258: 5252: 5244: 5241: 5233: 5228: 5225: 5222: 5219: 5215: 5207: 5204: 5201: 5197: 5193: 5185: 5181: 5177: 5172: 5166: 5163: 5160: 5157: 5152: 5149: 5143: 5131: 5128: 5125: 5121: 5112: 5108: 5104: 5101: 5097: 5092: 5090: 5080: 5075: 5071: 5054: 5051: 5046: 5042: 5036: 5028: 5025: 5017: 5012: 5009: 5006: 5003: 4999: 4988: 4985: 4982: 4978: 4969: 4965: 4961: 4958: 4954: 4949: 4947: 4931: 4919: 4916: 4912:at the point 4909: 4905: 4901:of the field 4900: 4880: 4877: 4874: 4870: 4856: 4845: 4840: 4836: 4825: 4822: 4819: 4816: 4811: 4808: 4801: 4785: 4780: 4776: 4749: 4738: 4733: 4731: 4715: 4712: 4709: 4706: 4703: 4700: 4697: 4690: 4682: 4667: 4662: 4658: 4652: 4648: 4642: 4637: 4634: 4631: 4627: 4623: 4618: 4613: 4609: 4600: 4567: 4562: 4558: 4535: 4532: 4529: 4525: 4516: 4512: 4483: 4480: 4475: 4471: 4450: 4442: 4427: 4422: 4418: 4412: 4408: 4402: 4397: 4394: 4391: 4387: 4370: 4367: 4362: 4358: 4352: 4344: 4341: 4333: 4328: 4325: 4322: 4319: 4315: 4304: 4301: 4298: 4294: 4285: 4281: 4277: 4274: 4270: 4265: 4249: 4244: 4227: 4223: 4219: 4216: 4210: 4206: 4198: 4193: 4190: 4187: 4183: 4179: 4165: 4157: 4149: 4144: 4136: 4129: 4125: 4114: 4112: 4109:the electric 4096: 4085: 4069: 4065: 4059: 4055: 4051: 4048: 4044: 4039: 4025: 4015: 4011: 4007: 4003: 3994: 3990: 3977: 3974: 3969: 3966: 3962: 3956: 3952: 3939: 3936: 3931: 3928: 3924: 3918: 3914: 3903: 3886: 3883: 3878: 3874: 3868: 3864: 3858: 3855: 3851: 3845: 3842: 3839: 3836: 3833: 3830: 3827: 3824: 3821: 3817: 3808: 3804: 3800: 3796: 3791: 3786: 3782: 3776: 3772: 3766: 3763: 3760: 3757: 3754: 3751: 3748: 3744: 3736: 3732: 3728: 3723: 3718: 3701: 3695: 3693: 3677: 3672: 3659: 3654: 3650: 3644: 3639: 3636: 3633: 3629: 3625: 3623: 3607: 3602: 3598: 3594: 3591: 3579: 3563: 3555: 3550: 3546: 3540: 3537: 3533: 3529: 3524: 3521: 3517: 3511: 3508: 3504: 3500: 3492: 3488: 3482: 3477: 3474: 3471: 3467: 3463: 3458: 3455: 3451: 3440: 3435: 3432: 3428: 3422: 3418: 3412: 3407: 3404: 3401: 3397: 3393: 3388: 3384: 3379: 3374: 3370: 3364: 3359: 3356: 3353: 3349: 3345: 3328: 3307: 3300: 3296: 3289: 3285: 3279: 3276: 3272: 3268: 3263: 3259: 3253: 3249: 3245: 3239: 3223: 3220: 3216: 3203: 3196: 3192: 3186: 3182: 3176: 3173: 3157: 3153: 3148: 3143: 3140: 3135: 3121: 3113: 3097: 3081: 3067: 3062: 3051: 3040: 3030: 3026: 3022: 3011: 3008: 3000: 2995: 2990: 2984: 2980: 2963: 2947: 2944: 2940: 2935: 2929: 2925: 2919: 2916: 2912: 2908: 2903: 2899: 2893: 2889: 2885: 2881: 2875: 2872: 2869: 2866: 2863: 2860: 2857: 2853: 2847: 2844: 2841: 2838: 2835: 2832: 2829: 2825: 2819: 2816: 2811: 2795: 2792: 2788: 2782: 2778: 2772: 2768: 2762: 2759: 2756: 2753: 2750: 2747: 2744: 2740: 2734: 2731: 2728: 2725: 2722: 2719: 2716: 2712: 2703: 2687: 2684: 2681: 2665: 2662: 2658: 2652: 2649: 2646: 2643: 2640: 2637: 2634: 2630: 2626: 2616: 2606: 2594: 2583: 2578: 2569: 2559: 2553: 2549: 2545: 2528: 2518: 2508: 2500: 2496: 2487: 2483: 2466: 2455: 2450: 2439: 2434: 2418: 2415: 2411: 2393: 2383: 2375: 2371: 2354: 2343: 2334: 2329: 2313: 2309: 2284: 2281: 2278: 2275: 2259: 2256: 2252: 2246: 2242: 2236: 2232: 2226: 2223: 2220: 2217: 2214: 2211: 2208: 2204: 2198: 2195: 2192: 2189: 2186: 2183: 2180: 2176: 2170: 2167: 2162: 2146: 2142: 2136: 2132: 2126: 2123: 2120: 2117: 2114: 2111: 2108: 2104: 2100: 2086: 2083: 2081: 2073: 2070: 2067: 2064: 2045: 2042: 2038: 2032: 2028: 2022: 2018: 2012: 2009: 2006: 2003: 2000: 1997: 1994: 1990: 1984: 1981: 1978: 1975: 1972: 1969: 1966: 1962: 1956: 1953: 1948: 1929: 1925: 1919: 1915: 1909: 1906: 1903: 1900: 1897: 1894: 1891: 1887: 1883: 1866: 1863: 1861: 1848: 1837: 1824: 1820: 1813: 1809: 1805: 1800: 1794: 1790: 1786: 1782: 1771: 1769: 1762: 1758: 1751: 1747: 1742: 1736: 1730: 1726: 1722: 1721:Taylor series 1717: 1707: 1702: 1694: 1687: 1683: 1674: 1665: 1660: 1656: 1651: 1645: 1641: 1634: 1630: 1615: 1613: 1609: 1605: 1601: 1597: 1592: 1589: 1585: 1584:atomic nuclei 1581: 1577: 1573: 1569: 1565: 1561: 1551: 1538: 1532: 1529: 1526: 1518: 1513: 1509: 1500: 1496: 1490: 1485: 1482: 1479: 1475: 1467: 1462: 1459: 1456: 1453: 1449: 1437: 1434: 1431: 1427: 1415: 1412: 1409: 1405: 1401: 1395: 1392: 1389: 1381: 1376: 1372: 1364: 1356: 1351: 1347: 1341: 1336: 1333: 1330: 1327: 1323: 1311: 1308: 1305: 1301: 1297: 1291: 1288: 1285: 1282: 1279: 1273: 1253: 1233: 1226:in powers of 1225: 1209: 1200: 1198: 1194: 1190: 1186: 1181: 1168: 1162: 1157: 1152: 1149: 1146: 1142: 1138: 1133: 1130: 1127: 1123: 1116: 1111: 1106: 1103: 1099: 1095: 1090: 1087: 1083: 1076: 1071: 1066: 1062: 1058: 1053: 1049: 1042: 1037: 1033: 1029: 1026: 1006: 1000: 997: 992: 988: 982: 974: 971: 965: 960: 957: 952: 948: 939: 935: 930: 914: 910: 889: 881: 865: 845: 837: 819: 815: 794: 791: 786: 782: 776: 772: 766: 762: 756: 752: 746: 743: 740: 737: 733: 729: 724: 720: 714: 710: 704: 700: 694: 691: 688: 684: 680: 675: 671: 665: 661: 655: 652: 648: 644: 639: 635: 629: 625: 621: 618: 615: 609: 606: 603: 597: 575: 570: 566: 562: 557: 552: 548: 544: 539: 536: 531: 527: 504: 499: 495: 472: 467: 463: 439: 436: 433: 425: 420: 416: 392: 389: 386: 378: 373: 369: 362: 357: 353: 346: 341: 338: 335: 332: 328: 316: 313: 310: 306: 302: 296: 293: 290: 284: 261: 258: 255: 249: 241: 231: 229: 225: 221: 216: 211: 209: 205: 201: 197: 193: 189: 184: 182: 178: 174: 169: 154: 131: 102: 87: 83: 79: 78:Taylor series 61: 46: 42: 38: 34: 30: 26: 22: 10804: 10799: 10791: 10786: 10765: 10729: 10725: 10712: 10703: 10697: 10654: 10650: 10644: 10614:(1): 32–35. 10611: 10607: 10594: 10575: 10569: 10548: 10540: 10510:are used in 10464: 10441: 10414: 10399: 10380: 10377:Examples of 10376: 10369: 10356: 10343: 10316: 10308: 10303: 10290: 10283:this article 10276: 9463: 9019: 9012: 8862:function in 8698: 8694: 8689: 8685: 8680: 8676: 8671: 8665: 8661: 8656: 8652: 8648: 8647:). Particle 8640: 8635: 8631: 8627: 8625: 8617: 8614: 8598: 8594: 8585: 8251: 8247: 8243: 8242:This is the 8241: 7623: 7610: 7603: 6410:. Note that 6397: 6393: 6388: 6384: 6377: 6374: 6368: 6364: 6360: 6221: 6217: 6193: 6189: 6186:and one set 6176: 6172: 6168: 6032: 6029: 5541: 5376: 5367: 5365: 5359: 5351: 4920: 4914: 4907: 4903: 4898: 4897:so that the 4734: 4729: 4598: 4147: 4146:| > 4142: 4134: 4127: 4123: 4120: 4013: 4009: 4005: 4001: 3992: 3991: 3904: 3577: 3028: 3024: 3020: 3017: 3006: 2998: 2993: 2982: 2978: 2551: 2547: 2543: 1822: 1818: 1811: 1807: 1803: 1792: 1788: 1784: 1780: 1777: 1760: 1756: 1740: 1734: 1728: 1715: 1705: 1704:| > 1700: 1692: 1685: 1681: 1672: 1663: 1658: 1654: 1649: 1643: 1639: 1632: 1628: 1621: 1593: 1587: 1579: 1557: 1554:Applications 1201: 1182: 931: 237: 227: 223: 219: 212: 185: 170: 20: 18: 10417:mathematics 10327:coordinates 9015:= −1, 0, 1) 9010:vanish for 4132:at a point 836:unit vector 277:as the sum 10817:Categories 10777:047143132X 10533:References 10456:orthogonal 10347:potential; 10319:potentials 10293:-particle 4732:charges). 1754:1 / | 807:where the 222:or simply 200:quadrupole 10689:119108102 10664:1204.4016 10398:ln 10373:potential 10368:ln 10203:∑ 10105:∑ 10050:− 9998:∑ 9900:∑ 9845:− 9790:∑ 9709:∑ 9638:∑ 9567:∑ 9513:ℓ 9481:ℓ 9447:⟩ 9444:Ψ 9421:Ψ 9418:⟨ 9385:∑ 9344:⟩ 9341:Ψ 9320:− 9302:Ψ 9299:⟨ 9266:∑ 9240:− 9218:⟩ 9215:Ψ 9176:Ψ 9173:⟨ 9140:∑ 9122:− 9051:∑ 8969:λ of the 8946:⟩ 8943:Ψ 8940:∣ 8930:ℓ 8922:∣ 8919:Ψ 8916:⟨ 8913:≡ 8903:ℓ 8874:ℓ 8821:ℓ 8768:ℓ 8729:∑ 8725:≡ 8715:ℓ 8551:ℓ 8538:ℓ 8504:^ 8457:− 8446:ℓ 8433:ℓ 8387:ℓ 8371:ℓ 8362:π 8348:≡ 8293:− 8282:ℓ 8269:ℓ 8221:⟩ 8186:ℓ 8173:ℓ 8169:∣ 8147:ℓ 8121:ℓ 8117:⟨ 8093:ℓ 8064:ℓ 8021:− 8008:− 7997:ℓ 7984:ℓ 7942:− 7928:ℓ 7916:ℓ 7912:− 7895:∑ 7883:ℓ 7871:ℓ 7867:− 7850:∑ 7846:× 7807:ℓ 7792:ℓ 7776:ℓ 7751:ℓ 7739:− 7731:∞ 7714:ℓ 7709:∑ 7703:∞ 7686:ℓ 7681:∑ 7668:ε 7664:π 7565:ℓ 7555:ℓ 7544:− 7527:− 7514:ℓ 7482:⟩ 7473:∣ 7460:− 7445:ℓ 7441:− 7413:ℓ 7409:⟨ 7372:− 7358:ℓ 7354:− 7302:ℓ 7285:ℓ 7273:ℓ 7269:− 7252:∑ 7248:× 7216:ℓ 7183:ℓ 7179:− 7165:− 7140:ℓ 7135:∑ 7110:− 6964:− 6901:− 6875:− 6858:− 6848:∑ 6842:∞ 6827:∑ 6794:− 6773:− 6721:− 6637:− 6533:− 6495:⟺ 6314:ε 6310:π 6274:∈ 6267:∑ 6258:∈ 6251:∑ 6209:electrons 6205:molecules 6128:− 6066:∑ 6062:≡ 6035:= 2 5993:ε 5989:π 5976:⋅ 5970:^ 5934:ε 5930:π 5917:⋅ 5812:ε 5808:π 5774:ℓ 5728:α 5719:α 5681:∑ 5677:≡ 5672:α 5544:= 1 5493:∑ 5489:≡ 5447:ε 5443:π 5394:ℓ 5379:= 0 5299:ℓ 5285:^ 5268:− 5263:ℓ 5242:− 5234:ℓ 5229:ℓ 5226:− 5216:∑ 5202:ℓ 5161:ℓ 5153:π 5137:∞ 5126:ℓ 5122:∑ 5109:ε 5105:π 5076:ℓ 5052:− 5047:ℓ 5026:− 5018:ℓ 5013:ℓ 5010:− 5000:∑ 4994:∞ 4983:ℓ 4979:∑ 4966:ε 4962:π 4875:ℓ 4860:^ 4841:ℓ 4820:ℓ 4812:π 4802:≡ 4781:ℓ 4753:^ 4713:ℓ 4710:≤ 4704:≤ 4701:ℓ 4698:− 4663:ℓ 4628:∑ 4624:≡ 4614:ℓ 4563:ℓ 4530:ℓ 4481:− 4476:ℓ 4423:ℓ 4388:∑ 4368:− 4363:ℓ 4342:− 4334:ℓ 4329:ℓ 4326:− 4316:∑ 4310:∞ 4299:ℓ 4295:∑ 4282:ε 4278:π 4250:− 4224:ε 4220:π 4184:∑ 4180:≡ 4086:⋅ 4056:ε 4052:π 3970:α 3967:α 3957:α 3953:∑ 3932:α 3929:α 3919:α 3915:∑ 3887:⋯ 3879:β 3869:α 3859:β 3856:α 3828:β 3822:α 3818:∑ 3787:α 3777:α 3749:α 3745:∑ 3678:− 3630:∑ 3626:≡ 3599:ε 3595:π 3541:β 3538:α 3534:δ 3530:− 3525:β 3512:α 3468:∑ 3464:≡ 3459:β 3456:α 3436:α 3398:∑ 3394:≡ 3389:α 3350:∑ 3346:≡ 3280:β 3277:α 3273:δ 3269:− 3264:β 3254:α 3224:β 3221:α 3187:α 3177:− 3158:α 3082:− 3063:≡ 3052:− 2948:β 2945:α 2920:β 2917:α 2913:δ 2909:− 2904:β 2894:α 2858:β 2854:∑ 2830:α 2826:∑ 2796:β 2793:α 2783:β 2773:α 2745:β 2741:∑ 2717:α 2713:∑ 2666:α 2663:α 2635:α 2631:∑ 2595:− 2575:∇ 2501:β 2493:∂ 2488:α 2480:∂ 2467:− 2447:∂ 2435:≡ 2419:β 2416:α 2376:α 2368:∂ 2355:− 2341:∂ 2330:≡ 2314:α 2285:⋯ 2279:⋯ 2276:− 2260:β 2257:α 2247:β 2237:α 2209:β 2205:∑ 2181:α 2177:∑ 2147:α 2137:α 2109:α 2105:∑ 2101:− 2074:⋯ 2068:⋯ 2055:− 2046:β 2043:α 2033:β 2023:α 1995:β 1991:∑ 1967:α 1963:∑ 1939:− 1930:α 1920:α 1892:α 1888:∑ 1874:− 1849:− 1608:particles 1600:Greengard 1533:φ 1527:θ 1514:ℓ 1480:ℓ 1468:ℓ 1463:ℓ 1460:− 1450:∑ 1443:∞ 1432:ℓ 1428:∑ 1421:∞ 1406:∑ 1396:φ 1390:θ 1377:ℓ 1352:ℓ 1342:ℓ 1337:ℓ 1334:− 1324:∑ 1317:∞ 1306:ℓ 1302:∑ 1292:φ 1286:θ 1169:… 1158:∗ 1112:∗ 1072:∗ 1038:∗ 1001:∗ 993:ℓ 972:− 958:− 953:ℓ 866:φ 846:θ 795:⋯ 787:ℓ 747:ℓ 610:φ 604:θ 537:− 468:ℓ 440:φ 434:θ 421:ℓ 393:φ 387:θ 374:ℓ 358:ℓ 347:ℓ 342:ℓ 339:− 329:∑ 322:∞ 311:ℓ 307:∑ 297:φ 291:θ 262:φ 256:θ 215:converges 41:azimuthal 10636:10039541 10482:See also 10468:monopole 10423:form an 10392:and the 10331:symmetry 10329:and the 10313:Examples 10299:integral 9024:) give: 6406:towards 2996:≡ | 1768:Legendre 1670:, where 1588:interior 1580:exterior 1568:electric 192:monopole 29:function 10734:Bibcode 10669:Bibcode 10616:Bibcode 8669:, and φ 3014:Example 2987:is the 1778:Assume 1723:in the 1604:Rokhlin 1189:tensors 938:complex 188:moments 86:complex 10774: 10687: 10634: 10582: 10557: 10472:dipole 10388:, the 8808:where 8693:, and 8620:-state 6997:where 6213:nuclei 4695: 4550:) and 4463:where 4140:| 3002:| 2976:where 2702:tensor 1764:| 1738:, and 1698:| 1564:masses 1185:scalar 1166: 1120: 1080: 1046: 408:where 208:powers 196:dipole 181:radius 33:angles 10722:(PDF) 10685:S2CID 10659:arXiv 10604:(PDF) 10435:, as 10429:field 10366:of a 10353:of a 10340:of a 8971:group 5761:Then 3993:Note: 1662:< 84:- or 23:is a 10772:ISBN 10632:PMID 10580:ISBN 10555:ISBN 10470:and 10419:and 8584:1 / 7029:and 6614:> 6355:and 5661:with 5465:with 5316:> 2991:and 1787:) = 1602:and 1570:and 934:real 858:and 175:and 82:real 10742:doi 10677:doi 10624:doi 9224:and 6363:of 4151:max 4008:≫ ( 3946:and 3446:and 3210:and 2541:If 2406:and 1801:of 1709:max 1676:max 1667:max 1598:of 936:or 10819:: 10754:^ 10740:. 10730:52 10728:. 10724:. 10683:. 10675:. 10667:. 10653:. 10630:. 10622:. 10612:62 10610:. 10606:. 10439:. 10379:1/ 10355:1/ 10342:1/ 8890:: 8684:, 8660:, 8632:eZ 8607:. 8588:AB 8252:AB 6392:− 6383:= 6381:XY 6372:. 6369:AB 6222:AB 5358:1/ 4735:A 4158:: 4113:. 3978:0. 3033:: 3027:− 2983:αβ 2704:: 2550:− 1821:= 1810:− 1791:(− 1759:− 1732:, 1714:1/ 1652:: 1566:, 1199:. 230:. 47:, 19:A 10780:. 10748:. 10744:: 10736:: 10691:. 10679:: 10671:: 10661:: 10655:8 10638:. 10626:: 10618:: 10588:. 10563:. 10400:R 10381:R 10370:R 10357:R 10344:R 10304:N 10291:N 10252:i 10248:y 10242:i 10238:x 10231:i 10227:Z 10223:e 10218:N 10213:1 10210:= 10207:i 10197:3 10190:3 10187:2 10182:= 10173:2 10168:2 10164:S 10154:i 10150:y 10144:i 10140:z 10133:i 10129:Z 10125:e 10120:N 10115:1 10112:= 10109:i 10099:3 10094:= 10085:1 10080:2 10076:S 10068:) 10063:2 10058:i 10054:y 10045:2 10040:i 10036:x 10032:( 10026:i 10022:Z 10018:e 10013:N 10008:1 10005:= 10002:i 9992:3 9985:3 9982:1 9977:= 9968:2 9963:2 9959:C 9949:i 9945:x 9939:i 9935:z 9928:i 9924:Z 9920:e 9915:N 9910:1 9907:= 9904:i 9894:3 9889:= 9880:1 9875:2 9871:C 9863:) 9858:2 9853:i 9849:r 9840:2 9835:i 9831:z 9827:3 9824:( 9818:i 9814:Z 9810:e 9805:N 9800:1 9797:= 9794:i 9784:2 9781:1 9776:= 9767:0 9762:2 9758:C 9748:i 9744:y 9737:i 9733:Z 9729:e 9724:N 9719:1 9716:= 9713:i 9705:= 9696:1 9691:1 9687:S 9677:i 9673:x 9666:i 9662:Z 9658:e 9653:N 9648:1 9645:= 9642:i 9634:= 9625:1 9620:1 9616:C 9606:i 9602:z 9595:i 9591:Z 9587:e 9582:N 9577:1 9574:= 9571:i 9563:= 9554:0 9549:1 9545:C 9518:m 9509:S 9486:m 9477:C 9450:. 9440:| 9434:i 9430:z 9425:| 9413:i 9409:Z 9405:e 9400:N 9395:1 9392:= 9389:i 9381:= 9376:0 9371:1 9367:M 9347:. 9337:| 9331:i 9327:y 9323:i 9315:i 9311:x 9306:| 9294:i 9290:Z 9286:e 9281:N 9276:1 9273:= 9270:i 9258:2 9254:1 9248:= 9243:1 9235:1 9231:M 9211:| 9205:i 9201:y 9197:i 9194:+ 9189:i 9185:x 9180:| 9168:i 9164:Z 9160:e 9155:N 9150:1 9147:= 9144:i 9132:2 9128:1 9119:= 9114:1 9109:1 9105:M 9084:, 9079:i 9075:Z 9071:e 9066:N 9061:1 9058:= 9055:i 9047:= 9042:0 9037:0 9033:M 9013:m 8996:m 8991:1 8987:Q 8949:. 8935:m 8926:Q 8908:m 8899:M 8846:) 8841:i 8836:r 8831:( 8826:m 8817:R 8796:, 8793:) 8788:i 8783:r 8778:( 8773:m 8764:R 8757:i 8753:Z 8749:e 8744:N 8739:1 8736:= 8733:i 8720:m 8711:Q 8699:i 8695:z 8690:i 8686:y 8681:i 8677:x 8672:i 8666:i 8662:θ 8657:i 8653:r 8649:i 8641:Z 8636:i 8628:N 8618:S 8599:l 8595:Y 8586:R 8570:, 8563:1 8560:+ 8555:B 8547:+ 8542:A 8532:B 8529:A 8525:R 8519:) 8514:B 8511:A 8500:R 8492:( 8486:) 8481:B 8477:m 8473:+ 8468:A 8464:m 8460:( 8450:B 8442:+ 8437:A 8428:Y 8418:2 8414:/ 8410:1 8405:] 8399:1 8396:+ 8391:B 8383:2 8380:+ 8375:A 8367:2 8359:4 8353:[ 8345:) 8340:B 8337:A 8332:R 8327:( 8322:) 8317:B 8313:m 8309:+ 8304:A 8300:m 8296:( 8286:B 8278:+ 8273:A 8264:I 8248:R 8224:. 8216:B 8212:m 8208:+ 8203:A 8199:m 8195:, 8190:B 8182:+ 8177:A 8164:B 8160:m 8156:, 8151:B 8143:; 8138:A 8134:m 8130:, 8125:A 8109:B 8105:m 8097:B 8088:Q 8080:A 8076:m 8068:A 8059:Q 8054:) 8049:B 8046:A 8041:R 8036:( 8029:B 8025:m 8016:A 8012:m 8001:B 7993:+ 7988:A 7979:I 7971:B 7967:m 7963:+ 7958:A 7954:m 7949:) 7945:1 7939:( 7932:B 7920:B 7909:= 7904:B 7900:m 7887:A 7875:A 7864:= 7859:A 7855:m 7834:2 7830:/ 7826:1 7819:) 7811:A 7803:2 7796:B 7788:2 7785:+ 7780:A 7772:2 7766:( 7755:B 7746:) 7742:1 7736:( 7726:0 7723:= 7718:B 7698:0 7695:= 7690:A 7672:0 7661:4 7657:1 7648:= 7643:B 7640:A 7636:U 7619:L 7611:ℓ 7604:Q 7586:. 7583:) 7579:r 7575:( 7570:m 7561:R 7551:) 7547:1 7541:( 7538:= 7535:) 7531:r 7524:( 7519:m 7510:R 7485:, 7479:M 7476:L 7468:A 7464:m 7457:M 7454:, 7449:A 7438:L 7435:; 7430:A 7426:m 7422:, 7417:A 7405:) 7400:j 7397:B 7392:r 7387:( 7380:A 7376:m 7369:M 7362:A 7351:L 7347:R 7343:) 7338:i 7335:A 7330:r 7325:( 7318:A 7314:m 7306:A 7297:R 7289:A 7277:A 7266:= 7261:A 7257:m 7243:2 7239:/ 7235:1 7228:) 7220:A 7212:2 7207:L 7204:2 7198:( 7187:A 7176:L 7172:) 7168:1 7162:( 7157:L 7152:0 7149:= 7144:A 7131:= 7128:) 7123:j 7120:B 7115:r 7105:i 7102:A 7097:r 7092:( 7087:M 7082:L 7078:R 7047:M 7042:L 7038:R 7015:M 7010:L 7006:I 6985:, 6982:) 6977:j 6974:B 6969:r 6959:i 6956:A 6951:r 6946:( 6941:M 6936:L 6932:R 6927:) 6922:B 6919:A 6914:R 6909:( 6904:M 6896:L 6892:I 6886:M 6882:) 6878:1 6872:( 6866:L 6861:L 6855:= 6852:M 6837:0 6834:= 6831:L 6823:= 6816:| 6812:) 6807:j 6804:B 6799:r 6789:i 6786:A 6781:r 6776:( 6768:B 6765:A 6760:R 6754:| 6749:1 6744:= 6737:| 6731:i 6726:r 6716:j 6711:r 6705:| 6700:1 6674:. 6671:j 6668:, 6665:i 6656:| 6650:i 6647:A 6642:r 6632:j 6629:B 6624:r 6618:| 6610:| 6604:B 6601:A 6596:R 6590:| 6569:. 6564:j 6561:B 6556:r 6551:+ 6546:i 6543:A 6538:r 6528:B 6525:A 6520:R 6515:= 6510:j 6507:i 6502:r 6490:0 6487:= 6482:A 6479:i 6474:r 6469:+ 6464:i 6461:j 6456:r 6451:+ 6446:j 6443:B 6438:r 6433:+ 6428:B 6425:A 6420:R 6408:Y 6404:X 6398:X 6394:r 6389:Y 6385:r 6378:r 6365:U 6357:B 6353:A 6339:. 6331:j 6328:i 6324:r 6318:0 6307:4 6300:j 6296:q 6290:i 6286:q 6277:B 6271:j 6261:A 6255:i 6247:= 6242:B 6239:A 6235:U 6218:U 6201:B 6197:} 6194:j 6190:q 6188:{ 6184:A 6180:} 6177:i 6173:q 6171:{ 6149:, 6146:) 6141:2 6136:i 6132:r 6123:2 6118:i 6114:z 6110:3 6107:( 6102:2 6099:1 6091:i 6087:q 6081:N 6076:1 6073:= 6070:i 6055:2 6051:z 6046:Q 6033:ℓ 6015:. 6007:2 6003:R 5997:0 5986:4 5980:P 5966:R 5956:= 5948:3 5944:R 5938:0 5927:4 5921:P 5913:R 5906:= 5903:) 5898:z 5894:P 5888:z 5884:R 5880:+ 5875:y 5871:P 5865:y 5861:R 5857:+ 5852:x 5848:P 5842:x 5838:R 5834:( 5826:3 5822:R 5816:0 5805:4 5801:1 5796:= 5793:) 5789:R 5785:( 5780:1 5777:= 5770:V 5749:. 5746:z 5743:, 5740:y 5737:, 5734:x 5731:= 5724:, 5716:i 5712:r 5706:i 5702:q 5696:N 5691:1 5688:= 5685:i 5668:P 5655:) 5650:z 5646:P 5642:, 5637:y 5633:P 5629:, 5624:x 5620:P 5616:( 5613:= 5609:P 5604:, 5601:) 5596:z 5592:R 5588:, 5583:y 5579:R 5575:, 5570:x 5566:R 5562:( 5559:= 5555:R 5542:ℓ 5523:. 5518:i 5514:q 5508:N 5503:1 5500:= 5497:i 5483:t 5480:o 5477:t 5472:q 5456:R 5451:0 5440:4 5433:t 5430:o 5427:t 5422:q 5416:= 5413:) 5409:R 5405:( 5400:0 5397:= 5390:V 5377:ℓ 5368:m 5360:R 5331:x 5328:a 5325:m 5320:r 5313:R 5309:, 5304:m 5295:Q 5291:) 5282:R 5276:( 5271:m 5259:Y 5253:m 5249:) 5245:1 5239:( 5223:= 5220:m 5208:1 5205:+ 5198:R 5194:1 5186:2 5182:/ 5178:1 5173:] 5167:1 5164:+ 5158:2 5150:4 5144:[ 5132:0 5129:= 5113:0 5102:4 5098:1 5093:= 5081:m 5072:Q 5068:) 5064:R 5060:( 5055:m 5043:I 5037:m 5033:) 5029:1 5023:( 5007:= 5004:m 4989:0 4986:= 4970:0 4959:4 4955:1 4950:= 4943:) 4939:R 4935:( 4932:V 4915:R 4910:) 4908:R 4906:( 4904:V 4881:1 4878:+ 4871:R 4866:) 4857:R 4851:( 4846:m 4837:Y 4826:1 4823:+ 4817:2 4809:4 4799:) 4795:R 4791:( 4786:m 4777:I 4750:R 4730:N 4716:. 4707:m 4691:, 4688:) 4683:i 4678:r 4673:( 4668:m 4659:R 4653:i 4649:q 4643:N 4638:1 4635:= 4632:i 4619:m 4610:Q 4595:r 4581:) 4577:r 4573:( 4568:m 4559:R 4536:1 4533:+ 4526:R 4497:) 4493:R 4489:( 4484:m 4472:I 4451:, 4448:) 4443:i 4438:r 4433:( 4428:m 4419:R 4413:i 4409:q 4403:N 4398:1 4395:= 4392:i 4384:) 4380:R 4376:( 4371:m 4359:I 4353:m 4349:) 4345:1 4339:( 4323:= 4320:m 4305:0 4302:= 4286:0 4275:4 4271:1 4266:= 4259:| 4254:R 4245:i 4240:r 4234:| 4228:0 4217:4 4211:i 4207:q 4199:N 4194:1 4191:= 4188:i 4177:) 4173:R 4169:( 4166:V 4148:r 4143:R 4135:R 4130:) 4128:R 4126:( 4124:V 4097:, 4094:) 4090:R 4082:P 4078:( 4070:3 4066:R 4060:0 4049:4 4045:1 4040:= 4037:) 4033:R 4029:( 4026:V 4016:) 4014:R 4012:/ 4010:d 4006:R 4004:/ 4002:d 3997:d 3975:= 3963:Q 3940:0 3937:= 3925:v 3884:+ 3875:R 3865:R 3852:Q 3846:z 3843:, 3840:y 3837:, 3834:x 3831:= 3825:, 3809:5 3805:R 3801:2 3797:1 3792:+ 3783:R 3773:P 3767:z 3764:, 3761:y 3758:, 3755:x 3752:= 3737:3 3733:R 3729:1 3724:+ 3719:R 3713:t 3710:o 3707:t 3702:q 3696:= 3686:) 3682:R 3673:i 3668:r 3663:( 3660:v 3655:i 3651:q 3645:N 3640:1 3637:= 3634:i 3619:) 3615:R 3611:( 3608:V 3603:0 3592:4 3564:, 3561:) 3556:2 3551:i 3547:r 3522:i 3518:r 3509:i 3505:r 3501:3 3498:( 3493:i 3489:q 3483:N 3478:1 3475:= 3472:i 3452:Q 3441:, 3433:i 3429:r 3423:i 3419:q 3413:N 3408:1 3405:= 3402:i 3385:P 3380:, 3375:i 3371:q 3365:N 3360:1 3357:= 3354:i 3340:t 3337:o 3334:t 3329:q 3308:. 3301:5 3297:R 3290:2 3286:R 3260:R 3250:R 3246:3 3240:= 3237:) 3233:R 3229:( 3217:v 3204:, 3197:3 3193:R 3183:R 3174:= 3171:) 3167:R 3163:( 3154:v 3149:, 3144:R 3141:1 3136:= 3133:) 3129:R 3125:( 3122:v 3098:. 3091:| 3086:R 3078:r 3073:| 3068:1 3060:) 3056:R 3048:r 3044:( 3041:v 3031:) 3029:R 3025:r 3023:( 3021:v 3007:r 2999:r 2994:r 2979:δ 2964:, 2961:) 2957:R 2953:( 2941:v 2936:) 2930:2 2926:r 2900:r 2890:r 2886:3 2882:( 2876:z 2873:, 2870:y 2867:, 2864:x 2861:= 2848:z 2845:, 2842:y 2839:, 2836:x 2833:= 2820:3 2817:1 2812:= 2809:) 2805:R 2801:( 2789:v 2779:r 2769:r 2763:z 2760:, 2757:y 2754:, 2751:x 2748:= 2735:z 2732:, 2729:y 2726:, 2723:x 2720:= 2688:, 2685:0 2682:= 2679:) 2675:R 2671:( 2659:v 2653:z 2650:, 2647:y 2644:, 2641:x 2638:= 2627:= 2621:0 2617:= 2613:r 2607:) 2603:) 2599:R 2591:r 2587:( 2584:v 2579:2 2570:( 2554:) 2552:R 2548:r 2546:( 2544:v 2529:. 2523:0 2519:= 2515:r 2509:) 2497:r 2484:r 2475:) 2471:R 2463:r 2459:( 2456:v 2451:2 2440:( 2432:) 2428:R 2424:( 2412:v 2398:0 2394:= 2390:r 2384:) 2372:r 2363:) 2359:R 2351:r 2347:( 2344:v 2335:( 2327:) 2323:R 2319:( 2310:v 2282:+ 2273:) 2269:R 2265:( 2253:v 2243:r 2233:r 2227:z 2224:, 2221:y 2218:, 2215:x 2212:= 2199:z 2196:, 2193:y 2190:, 2187:x 2184:= 2171:2 2168:1 2163:+ 2160:) 2156:R 2152:( 2143:v 2133:r 2127:z 2124:, 2121:y 2118:, 2115:x 2112:= 2098:) 2094:R 2090:( 2087:v 2084:= 2071:+ 2065:+ 2062:) 2058:R 2051:( 2039:v 2029:r 2019:r 2013:z 2010:, 2007:y 2004:, 2001:x 1998:= 1985:z 1982:, 1979:y 1976:, 1973:x 1970:= 1957:2 1954:1 1949:+ 1946:) 1942:R 1935:( 1926:v 1916:r 1910:z 1907:, 1904:y 1901:, 1898:x 1895:= 1884:+ 1881:) 1877:R 1870:( 1867:v 1864:= 1857:) 1853:R 1845:r 1841:( 1838:v 1823:0 1819:r 1814:) 1812:R 1808:r 1806:( 1804:v 1795:) 1793:r 1789:v 1785:r 1783:( 1781:v 1761:R 1757:r 1741:z 1735:y 1729:x 1716:R 1706:r 1701:R 1693:R 1688:) 1686:R 1684:( 1682:V 1673:r 1664:r 1659:i 1655:r 1650:i 1644:i 1640:r 1633:i 1629:q 1624:N 1539:. 1536:) 1530:, 1524:( 1519:m 1510:Y 1501:j 1497:r 1491:m 1486:j 1483:, 1476:D 1457:= 1454:m 1438:0 1435:= 1416:1 1413:= 1410:j 1402:= 1399:) 1393:, 1387:( 1382:m 1373:Y 1368:) 1365:r 1362:( 1357:m 1348:C 1331:= 1328:m 1312:0 1309:= 1298:= 1295:) 1289:, 1283:, 1280:r 1277:( 1274:V 1254:V 1234:r 1210:r 1163:; 1153:k 1150:j 1147:i 1143:C 1139:= 1134:k 1131:j 1128:i 1124:C 1117:; 1107:j 1104:i 1100:C 1096:= 1091:j 1088:i 1084:C 1077:; 1067:i 1063:C 1059:= 1054:i 1050:C 1043:; 1034:C 1030:= 1027:C 1007:. 998:m 989:C 983:m 979:) 975:1 969:( 966:= 961:m 949:C 915:i 911:C 890:C 820:i 816:n 792:+ 783:n 777:k 773:n 767:j 763:n 757:i 753:n 744:k 741:j 738:i 734:C 730:+ 725:k 721:n 715:j 711:n 705:i 701:n 695:k 692:j 689:i 685:C 681:+ 676:j 672:n 666:i 662:n 656:j 653:i 649:C 645:+ 640:i 636:n 630:i 626:C 622:+ 619:C 616:= 613:) 607:, 601:( 598:f 576:1 571:1 567:C 563:, 558:0 553:1 549:C 545:, 540:1 532:1 528:C 505:0 500:0 496:C 473:m 464:C 443:) 437:, 431:( 426:m 417:Y 396:) 390:, 384:( 379:m 370:Y 363:m 354:C 336:= 333:m 317:0 314:= 303:= 300:) 294:, 288:( 285:f 265:) 259:, 253:( 250:f 167:. 155:n 132:n 127:R 103:3 98:R 62:3 57:R

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Knowledge

Multipole expansion

Index