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Convergent all-order spherical multipole expansion with monopole, dipole and quadrupole terms

Physical scope

The full exterior compact- $L^{1}$ spherical multipole potential theory, its exact monopole/dipole/quadrupole terms, geometric tails and every-order compact derivative convergence, plus a separate source-free radius- $R$ ball-potential branch for independent real $L^{2}$ angular boundary data: bounded radial $L^{2}$ norm, $L^{2}$ boundary convergence and uniqueness, interior smooth Maxwell completion, and continuous-data Poisson representation/continuous boundary extension. No exterior source-radius trace, singular-source expansion, boundary-field trace or empirical assertion is inferred.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ), inherited from the cited Lebesgue, spherical-harmonic and Newtonian interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $a > 0$ and let $ρ \in L^{1} (R^{3})$ be real and vanish $λ_{3}$ -a.e. outside ${\overline{B}}_{a} (0)$ (Open ball, closed ball and sphere in a metric space, Measure-null sets and almost-everywhere statements relative to a measure); let $Q$ , $p$ and $Q_{i j}$ be the total charge, dipole moment and traceless quadrupole of $ρ$ about the chart origin (Total charge, dipole moment, Cartesian moments and traceless quadrupole of a static charge distribution (a)); let $φ : = ε_{0}^{- 1} N ρ$ be the potential of The localized Newtonian potential and the ideal point-charge Coulomb field (c) with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values (Newtonian potential of compactly supported data); and let ${Y_{l m} : l \geq 0, m = 1, \dots, 2 l + 1}$ be a real orthonormal basis of spherical harmonics of $L^{2} (S^{2}, d σ)$ of degree $l$ , with $d σ$ the surface measure of total mass $4 π$ and the addition theorem of Spherical harmonics: orthogonal decomposition, complete basis and addition theorem (Orthonormal families, complete orthonormal systems and Hilbert bases, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). For every $l, m$ define the spherical multipole moment of degree $l$

$q_{l m} : = \int_{R^{3}} ∣ y ∣^{l} Y_{l m} (y / ∣ y ∣) ρ (y) d λ_{3} (y),$

absolutely convergent because $∣ Y_{l m} ∣ \leq ((2 l + 1) / (4 π))^{1 / 2}$ on $S^{2}$ by the addition theorem and $∣ y ∣ \leq a$ $λ_{3}$ -a.e. on the support, so the integrand is dominated by $a^{l} ((2 l + 1) / (4 π))^{1 / 2} ∣ ρ ∣$ (The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral). In every coefficient integral the factor $∣ y ∣^{l} Y_{l m} (y / ∣ y ∣)$ is defined at $y = 0$ by its homogeneous polynomial extension: zero for $l \geq 1$ and the constant degree-zero harmonic for $l = 0$ ; this is a finite assignment at a null singleton and changes no integral. Write $x = r n$ with $r = ∣ x ∣$ and $n \in S^{2}$ . Then:

(a) [Exterior spherical multipole expansion] for every $x$ with $r > a$ ,

$φ (x) = \frac{1}{4 π ε_{0}} \sum_{l = 0}^{\infty} \sum_{m = 1}^{2 l + 1} \frac{4 π}{2 l + 1} \frac{Y_{l m} (n)}{r^{l + 1}} q_{l m},$

the series converging absolutely and uniformly on ${∣ x ∣ \geq r_{0}}$ for every $r_{0} > a$ , with the tail after degree $N$ bounded by

$∥ φ - \frac{1}{4 π ε_{0}} \sum_{l = 0}^{N} \sum_{m = 1}^{2 l + 1} \frac{4 π}{2 l + 1} \frac{Y_{l m} (n)}{r^{l + 1}} q_{l m} ∥_{\infty, {∣ x ∣ \geq r_{0}}} \leq \frac{∥ ρ ∥_{1} a^{N + 1}}{4 π ε_{0} r_{0}^{N + 2} (1 - a / r_{0})};$

(b) [Monopole, dipole and quadrupole terms] the degree- $0$ and degree- $1$ terms of (a) are the monopole and dipole terms $Q / (4 π ε_{0} r)$ and $p \cdot x / (4 π ε_{0} r^{3})$ , and for $r \geq 2 a$ the difference between $φ$ and the truncation after degree $1$ obeys the remainder bound of Uniform Coulomb dipole expansion of a compactly supported charge density with an explicit remainder; the degree- $2$ term equals the traceless-quadrupole term

$\frac{1}{8 π ε_{0} r^{3}} n_{i} Q_{i j} n_{j};$

(c) [Regularity and termwise differentiation] $φ$ is of class $C^{\infty}$ on ${∣ x ∣ > a}$ , the series in (a) may be differentiated termwise to every order, the differentiated series converging uniformly on every compact subset of ${∣ x ∣ > a}$ ; in particular the field $E : = - \nabla φ$ of The electric and magnetic fields as fixed-time spatial vector fields equals $- \frac{1}{4 π ε_{0}} \sum_{l, m} \frac{4 π}{2 l + 1} \nabla_{x} (Y_{l m} (n) r^{- l - 1}) q_{l m}$ on ${∣ x ∣ > a}$ , with the same uniform convergence on compact subsets;

(d) [Scope of exterior clauses (a)-(c)] those clauses concern a real compactly supported $L^{1}$ density and the exterior region $r > a$ ; they make no claim on or inside ${\overline{B}}_{a} (0)$ , no convergence at $r = a$ is asserted, the exterior-density expansion is not asserted for point or singular sources, and no empirical content is involved. The expansion is the exterior expansion of the potential of the static source $ρ$ read in the fixed-time dictionary of The static and pure electrostatic regimes and their fixed-time reading (e);

(e) [Separate source-free ball boundary branch] let $R > 0$ be a radius and let real $f \in L^{2} (S^{2}, d σ)$ be prescribed scalar-potential boundary data (unit V) on the sphere $x = R n$ , with $f_{l m} = \int_{S^{2}} f Y_{l m} d σ$ . Independently of the exterior density $ρ$ , define on $∣ x ∣ < R$

$φ_{f} (r n) = \sum_{l = 0}^{\infty} \sum_{m = 1}^{2 l + 1} (r / R)^{l} f_{l m} Y_{l m} (n),$

with the constant degree-zero value at $r = 0$ . Then this is smooth and harmonic in the ball; its series and every derivative converge uniformly on every smaller closed ball, $\sup_{0 \leq t < 1} ∥ φ_{f} (R t \cdot) ∥_{2} \leq ∥ f ∥_{2}$ , and $φ_{f} (R t \cdot) \to f$ in $L^{2}$ as $t ↑ 1$ ; it is the unique harmonic scalar potential in the class with bounded radial $L^{2}$ norm and that $L^{2}$ boundary limit. With $E_{f} = - \nabla φ_{f}$ , $B = 0$ , $ρ = J = 0$ and time-independent fields, this gives a full source-free electrostatic Maxwell model inside the ball under The microscopic vacuum Maxwell equations: smooth and weak branches. For continuous $f$ the same potential is

$φ_{f} (x) = \int_{S^{2}} \frac{1 - ∣ x ∣^{2} / R^{2}}{4 π ∣ x / R - n^{'} ∣^{3}} f (n^{'}) d σ (n^{'}),$

and it extends continuously to the closed ball with value $f (n)$ at $R n$ ; the angular Poisson kernel is positive of mass one. No boundary value or boundary electric-field trace for the exterior arbitrary- $L^{1}$ density is inferred, and no derivative boundary regularity for arbitrary $f$ is asserted.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $a > 0$ , let $ρ \in L^{1} (R^{3})$ be real and vanish $λ_{3}$ -a.e. outside ${\overline{B}}_{a} (0)$ , let $Q$ , $p$ , $Q_{i j}$ be its source moments about the chart origin, let $φ = ε_{0}^{- 1} N ρ$ be the potential of the Coulomb solution with the n=3 kernel, and let ${Y_{l m}}$ be a real orthonormal basis of spherical harmonics of degree $l$ for the surface measure $d σ$ of total mass $4 π$ , with addition theorem as in Spherical harmonics: orthogonal decomposition, complete basis and addition theorem.

[F1]

For $ρ \in L^{1} (R^{3})$ vanishing $λ_{3}$ -a.e. outside ${\overline{B}}_{a} (0)$ , the moments $Q = \int ρ d λ_{3}$ , $p = \int y ρ (y) d λ_{3} (y)$ and $Q_{i j} = \int (3 y_{i} y_{j} - ∣ y ∣^{2} δ_{i j}) ρ d λ_{3}$ are absolutely convergent and $\sum_{i} Q_{i i} = 0$ (Total charge, dipole moment, Cartesian moments and traceless quadrupole of a static charge distribution).

[F2]

Let $N f$ be the Newtonian potential of a measurable compactly supported $f$ , defined by $N f (x) = \int Φ (x - y) f (y) d y$ at each $x$ where the integral is absolutely finite; the kernel $Φ$ may be assigned an arbitrary finite value at $0$ , which changes no integral (Newtonian potential of compactly supported data).

[F3]

In dimension three the fundamental solution is $Φ (z) = 1 / (4 π ∣ z ∣)$ away from the pole; for compactly supported real $L^{1}$ data $ρ$ the potential $N ρ$ is finite $λ_{3}$ -a.e., locally integrable, $C^{\infty}$ and harmonic off the support, and $- Δ T_{ε_{0}^{- 1} N ρ} = T_{ρ / ε_{0}}$ in $D^{'} (R^{3})$ (The localized Newtonian potential and the ideal point-charge Coulomb field).

[F4]

The framework's constants satisfy $ε_{0} = (μ_{0} c^{2})^{- 1} > 0$ with $μ_{0} > 0$ and $c > 0$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F5]

In the pure electrostatic regime the smooth Maxwell equations reduce to the spatial pair and the fixed-time dictionary identifies spatial and four-dimensional weak readings (The static and pure electrostatic regimes and their fixed-time reading).

[F6]

At fixed time an electric field is the map $E (\cdot, t) : Ω \to R^{3}$ , read in the frame basis; the field $E : = - \nabla φ$ of a smooth potential is of this type (The electric and magnetic fields as fixed-time spatial vector fields).

[F7]

Let $a > 0$ and let $ρ \in L^{1} (R^{3}; R)$ vanish outside ${\overline{B}}_{a} (0)$ ; with the orthonormal spherical-harmonic basis and addition theorem of Spherical harmonics: orthogonal decomposition, complete basis and addition theorem, for every $x = r n$ with $r > a$ the exterior multipole expansion of the integral $\int ρ (y) ∣ x - y ∣^{- 1} d y$ holds with coefficients $\int ∣ y ∣^{l} Y_{l m} (y / ∣ y ∣) ρ d y$ , converges absolutely and uniformly on ${∣ x ∣ \geq r_{0}}$ for $r_{0} > a$ with tail after degree $N$ bounded by $∥ ρ ∥_{1} a^{N + 1} / (r_{0}^{N + 2} (1 - a / r_{0}))$ , has degree-zero and degree-one terms equal to the Coulomb and dipole terms with the remainder bound of Monopole-dipole expansion with an explicit remainder for compactly supported $L^{1}$ densities, may be differentiated termwise to every order with uniform convergence on compact sets with $\inf_{K} ∣ x ∣ > a$ , and its degree- $l$ coefficient function is the $L^{2} (S^{2}, d σ)$ angular projection of the potential on the sphere $r_{*} > a$ (Exterior spherical multipole expansion and the ball boundary series).

[F8]

The union over $l$ of the spaces of degree- $l$ spherical harmonics admits a complete orthonormal basis of $L^{2} (S^{2}, d σ)$ ; for any real orthonormal degree- $l$ basis the addition theorem $\sum_{m = 1}^{2 l + 1} Y_{l m} (n) Y_{l m} (n^{'}) = \frac{2 l + 1}{4 π} P_{l} (n \cdot n^{'})$ holds (Spherical harmonics: orthogonal decomposition, complete basis and addition theorem).

[F9]

For $z \in [- 1, 1]$ the polynomials $P_{l} (z) = π^{- 1} \int_{0}^{π} [z + i \sqrt{1 - z^{2}} \cos θ]^{l} d θ$ satisfy $∣ P_{l} (z) ∣ \leq 1$ , $P_{l} (1) = 1$ , and $(1 - 2 q z + q^{2})^{- 1 / 2} = \sum_{l \geq 0} q^{l} P_{l} (z)$ for $∣ q ∣ < 1$ (Legendre polynomials from a convergent zonal kernel: generating function and exterior expansion).

[F10]

Taylor coefficients of a real function at a point are unique: if a function has a Taylor polynomial of order $k$ with remainder, the coefficient of $q^{k}$ is fixed by the derivatives at the base point (Taylor polynomials and their remainders, Taylor polynomials match the prescribed derivatives at the centre). A smooth generating function has an order-two Taylor remainder $O (q^{3})$ on a sufficiently small compact interval (A uniform derivative bound gives a uniform Taylor remainder bound); two degree-two polynomials differing by $o (q^{2})$ as $q \to 0$ have identical coefficients, successively by division by $1, q, q^{2}$ .

[F11]

For $f \in L^{2} (S^{2}, d σ)$ the ball series $u (r n) = \sum_{l, m} r^{l} f_{l m} Y_{l m} (n)$ and all its derivatives converge uniformly on every closed smaller ball, $u$ is harmonic there, $u (r \cdot) \to f$ in $L^{2}$ as $r ↑ 1$ , $u$ is the unique harmonic function on the unit ball with finite $\sup_{r < 1} ∥ u (r \cdot) ∥_{2}$ and that boundary limit, and for continuous $f$ it equals the Poisson integral and extends continuously to the boundary (Ball Poisson series: interior convergence, boundary data and the Poisson kernel).

[F12]

If $f \in L^{1}$ and $α \in R$ , then $\int α f = α \int f$ ; if $f, g \in L^{1}$ agree a.e. their integrals agree; $∣ \int f ∣ \leq \int ∣ f ∣$ and nonnegative domination gives monotonicity of the integral (The Lebesgue integral is linear on $L^{1} (μ)$ , Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree, The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F13]

${\overline{B}}_{a} (0) = {y : ∣ y ∣ \leq a}$ , and for $x = r n$ the unit vector is $n = x / ∣ x ∣$ with $∣ x ∣ = r$ in the Euclidean structure (Open ball, closed ball and sphere in a metric space, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F14]

For $φ$ of class $C^{2}$ , $- Δ φ = div (- \nabla φ)$ and $curl \nabla φ = 0$ ; the smooth vacuum Maxwell equations are $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ for jointly $C^{1}$ fields with the additional derivatives required by each equation (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, The curl of the gradient of a $C^{2}$ function vanishes, The microscopic vacuum Maxwell equations: smooth and weak branches).

Proof

technique · direct

1.1F8F12F13given

Establish the absolute convergence of the coefficients and the null-singleton convention. On $S^{2}$ the diagonal addition theorem gives $\sum_{m = 1}^{2 l + 1} Y_{l m} (n)^{2} = (2 l + 1) / (4 π)$ [F8], so $∣ Y_{l m} (n) ∣ \leq ((2 l + 1) / (4 π))^{1 / 2}$ for every $n$ ; for $λ_{3}$ -a.e. $y$ with $ρ (y) \neq 0$ one has $y \in {\overline{B}}_{a} (0)$ and $∣ y ∣ \leq a$ [F13], so the integrand of $q_{l m}$ is dominated $λ_{3}$ -a.e. by $a^{l} ((2 l + 1) / (4 π))^{1 / 2} ∣ ρ ∣ \in L^{1}$ , and $q_{l m}$ is absolutely convergent by [F12]; the definition at $y = 0$ affects a null singleton only and changes no integral by [F12].

2.1step 1.1F2F3F4F7F12

Prove the exterior expansion and its tail bound (a). By [F3] and [F2] the potential satisfies $φ (x) = ε_{0}^{- 1} N ρ (x) = (4 π ε_{0})^{- 1} \int_{R^{3}} ρ (y) ∣ x - y ∣^{- 1} d λ_{3} (y)$ for $r > a$ , and [F7] gives $\int ρ (y) ∣ x - y ∣^{- 1} d y = \sum_{l, m} \frac{4 π}{2 l + 1} \frac{Y_{l m} (n)}{r^{l + 1}} \int ∣ y ∣^{l} Y_{l m} (y / ∣ y ∣) ρ d y$ with absolute and uniform convergence on ${∣ x ∣ \geq r_{0}}$ for every $r_{0} > a$ and the stated tail bound; multiplying the identity and its tail bound by the finite positive constant $(4 π ε_{0})^{- 1}$ [F4] and using scaling [F12] gives (a), with the coefficients exactly the $q_{l m}$ of step 1.1.

3.1step 2.1F1F7

Identify the degree-zero and degree-one terms and the exterior remainder, proving the first part of (b). By [F7] the degree-zero and degree-one terms of the parenthesised series are the Coulomb and dipole terms with the explicit remainder bound of Monopole-dipole expansion with an explicit remainder for compactly supported $L^{1}$ densities, namely $Q / r + p \cdot x / r^{3}$ with $∣ R_{0} (x) ∣ \leq 16 a^{2} ∥ ρ ∥_{1} / r^{3}$ for $r \geq 2 a$ ; scaling by $(4 π ε_{0})^{- 1}$ as in step 2.1 turns the degree-zero term into $Q / (4 π ε_{0} r)$ , the degree-one term into $p \cdot x / (4 π ε_{0} r^{3})$ , and the remainder bound into the bound of Uniform Coulomb dipole expansion of a compactly supported charge density with an explicit remainder (a), which is therefore reproduced.

4.1step 3.1F1F8F9F10F12

Identify the degree-two term with the traceless quadrupole, completing (b). Apply the addition theorem [F8] at $l = 2$ : $\sum_{m = 1}^{5} Y_{2 m} (n) Y_{2 m} (y / ∣ y ∣) = \frac{5}{4 π} P_{2} (n \cdot y / ∣ y ∣)$ , so by finite linearity [F12] $\sum_{m = 1}^{5} Y_{2 m} (n) q_{2 m} = \frac{5}{4 π} \int ∣ y ∣^{2} P_{2} (n \cdot y / ∣ y ∣) ρ (y) d λ_{3} (y)$ . By [F9] the generating identity $(1 - 2 q z + q^{2})^{- 1 / 2} = \sum_{l} q^{l} P_{l} (z)$ holds, and the bound $∣ P_{l} (z) ∣ \leq 1$ in [F9] gives a tail after degree two of modulus at most $∣ q ∣^{3} / (1 - ∣ q ∣) = o (q^{2})$ as real $q \to 0$ . Thus the degree-two partial sum is the Taylor polynomial of the generating function; by [F10] its coefficient of $q^{2}$ is both $P_{2} (z)$ and, by the chain rule applied to the power function, $\frac{1}{2} \frac{d^{2}}{d q^{2}} (1 - 2 q z + q^{2})^{- 1 / 2} ∣_{q = 0} = \frac{3 z^{2} - 1}{2}$ ; hence $P_{2} (z) = (3 z^{2} - 1) / 2$ and $∣ y ∣^{2} P_{2} (n \cdot y / ∣ y ∣) = (3 (n \cdot y)^{2} - ∣ y ∣^{2}) / 2$ wherever $y \neq 0$ . Therefore $\sum_{m} Y_{2 m} (n) q_{2 m} = \frac{5}{8 π} \int (3 (n \cdot y)^{2} - ∣ y ∣^{2}) ρ d λ_{3} = \frac{5}{8 π} n_{i} Q_{i j} n_{j}$ by [F1] and [F12], and the degree-two term of (a) is $\frac{1}{4 π ε_{0}} \cdot \frac{4 π}{5} \cdot \frac{1}{r^{3}} \cdot \frac{5}{8 π} n_{i} Q_{i j} n_{j} = \frac{n_{i} Q_{i j} n_{j}}{8 π ε_{0} r^{3}}$ .

5.1step 4.1F4F6F7F12

Prove the regularity and termwise differentiation (c). By [F7] the series may be differentiated termwise to every order with uniform convergence on every compact $K$ with $\inf_{K} ∣ x ∣ > a$ , and the potential is $C^{\infty}$ on ${∣ x ∣ > a}$ ; scaling by the constant $(4 π ε_{0})^{- 1}$ preserves termwise differentiation and uniform convergence [F4, F12]. The field $E : = - \nabla φ$ of [F6] is therefore the termwise gradient $- \frac{1}{4 π ε_{0}} \sum_{l, m} \frac{4 π}{2 l + 1} \nabla_{x} (Y_{l m} (n) r^{- l - 1}) q_{l m}$ on ${∣ x ∣ > a}$ , with the same uniform convergence on compact subsets.

6.1step 5.1F4F5

Record the scope of the exterior clauses (d). Steps 2.1, 3.1, 4.1 and 5.1 used only the real $L^{1}$ density vanishing $λ_{3}$ -a.e. off ${\overline{B}}_{a} (0)$ , the region $r > a$ (with the quantitative bounds on $r \geq r_{0} > a$ and $r \geq 2 a$ ), and the framework constants [F4]; no claim on or inside ${\overline{B}}_{a} (0)$ , no convergence at $r = a$ and no point-source or empirical assertion was made, and the potential is the fixed-time reading of the static source through [F5].

7.1step 6.1F4F11F14

Prove the ball boundary branch (e). Write $g (u n) : = \sum_{l, m} u^{l} f_{l m} Y_{l m} (n)$ for $u \in [0, 1)$ and $n \in S^{2}$ as in [F11] with $u = r / R$ and $n = x / ∣ x ∣$ ; then $φ_{f} (r n) = g ((r / R) n)$ is smooth and harmonic on $∣ x ∣ < R$ , its series and every derivative converge uniformly on every smaller closed ball because the substitution $x = R z$ multiplies a derivative of order $α$ by the finite factor $R^{- ∣ α ∣}$ , and the bounds $\sup_{0 \leq t < 1} ∥ φ_{f} (R t \cdot) ∥_{2} = \sup_{t < 1} ∥ g (t \cdot) ∥_{2} \leq ∥ f ∥_{2}$ and $φ_{f} (R t \cdot) \to f$ in $L^{2} (S^{2})$ as $t ↑ 1$ are exactly [F11] in the unit variable; the uniqueness statement carries over because any competitor $v$ on $∣ x ∣ < R$ gives the competitor $u (z) : = v (R z)$ on the unit ball with the same finite radial $L^{2}$ bound and boundary limit, so [F11] forces $u = g$ and hence $v = φ_{f}$ . For continuous $f$ the Poisson integral of [F11] with $r = ∣ x ∣ / R$ is the displayed formula, and the kernel is positive of mass one. Finally, define $E_{f} : = - \nabla φ_{f}$ , $B : = 0$ , $ρ : = J : = 0$ and time-independent fields on the open ball; then $φ_{f}$ is $C^{\infty}$ , so $E_{f}$ is a $C^{1}$ field and $B, ρ, J$ are continuous, and [F14] gives $div E_{f} = - Δ φ_{f} = 0$ and $curl E_{f} = - curl \nabla φ_{f} = 0$ , while $div B = 0$ and $curl B = 0$ ; all four adopted smooth vacuum Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches hold pointwise with vanishing sources and time derivatives, so this is a full source-free electrostatic Maxwell model inside the ball. No boundary trace for the exterior density and no derivative boundary regularity for arbitrary $f$ is used or claimed.

8.1step 7.1∎

Assemble the theorem and close. Step 2.1 gives (a), steps 3.1 and 4.1 give (b), step 5.1 gives (c), step 6.1 records the scope (d), and step 7.1 gives the independent ball boundary branch (e) with its source-free Maxwell completion; the only nonconstructive assumptions are the declared Countable Choice inherited from the Lebesgue, spherical-harmonic and ball-series interfaces, and no exterior trace, singular-source expansion or empirical content is asserted.

Convergent all-order spherical multipole expansion with monopole, dipole and quadrupole terms

Statement

Facts & Assumptions

Proof

Depends on

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