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Spherical harmonics and related theory
APPENDIX A
Spherical harmonics and related theory A.1. Scalar spherical harmonics Laplace equation in spherical coordinates The spherical coordinates (r , θ, ϕ) (Fig. A.1) relate to the Cartesian ones (x1 , x2 , x3 ) by x1 + ix2 = r sin θ exp(iϕ), x3 = r cos θ (r ≥ 0, 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π). Separation of variables in Laplace equation written in spherical coordinates as 1 ∂ 1 ∂ 2T ∂ ∂T 2 ∂T T (x) = r =0 + sin θ + 2 ∂r ∂r sin θ ∂θ ∂θ sin θ ∂ϕ 2 2
(A.1)
yields a complete set of partial (“normal”, in Hobson’s [332] terminology) solutions of the form r t Pts (cos θ ) exp(isϕ) = r t χts (θ, ϕ)
(−∞ < t < ∞, |s| ≤ t),
(A.2)
Figure A.1 Spherical coordinates. 519
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Spherical harmonics and related theory
referred to in [16] as scalar solid spherical harmonics of degree t and order s. Here and below, the radius-vector (position-vector field, in Gurtin [91] terminology) is x = xk ik , ik being the unit base vectors of the Cartesian coordinate system. In Eq. (A.2), Pts are the associated Legendre functions of the first kind [332]: Pts (η) = η¯ s
ds Pt (η) (−1)t η¯ s dt+s η¯ 2t = dηs 2t t! dηt+s
(A.3)
for |η| ≤ 1. The relationships Pt−s (η) = (−1)s
(t − s)! s P (η), (t + s)! t
s s P−( t+1) (η) = Pt (η)
Pts (η) ≡ 0
(0 ≤ t < ∞,
(A.4) |s| ≤ t) ,
(|s| > t) ,
extend the definition of the functions Pts of Eq. (A.3) to arbitrary integer indices t and s. It also follows from Eq. (A.3) that Pts (−η) = (−1)t+s Pts (η). In Eq. (A.2), χts (θ, ϕ) = Pts (cos θ ) exp(isϕ)
(A.5)
are the scalar surface spherical harmonics. They possess the orthogonality property [253] 1 S
S
χts χkl dS = αts δtk δsl ,
αts =
1 (t + s)! , 2t + 1 (t − s)!
(A.6)
where the integral is taken over the spherical surface S. The overbar means the complex conjugate and δij is the Kronecker’s delta. Also, in view of Eq. (A.4), χt−s = (−1)s s s χ−( t+1) = χt
χts ≡ 0
(t − s)! s χ, (t + s)! t
(0 ≤ t < ∞, |s| ≤ t) , (|s| > t) .
(A.7)
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Spherical harmonics and related theory
The series expansion of a continuous function f (θ, ϕ) over the set of surface spherical harmonics of Eq. (A.5) is given by the formula [332] f (θ, ϕ) =
t ∞
cts χts (θ, ϕ),
t=0 s=−t
where cts =
2π
1 4παts
π
dϕ f (θ, ϕ) χts (θ, ϕ) sin θ dθ. 0
(A.8)
0
Integration in Eq. (A.8) can be done either analytically or numerically. In the latter case, an appropriate scheme [182] comprises the uniform distribution of integration points in the azimuthal direction ϕ with Gauss– Legendre quadrature rule [253] for integration with respect to η = cos θ : 1 R2
f (η, ϕ) dS = S
0
2π
1
−1
n n 1 wi f ηi , ϕj . n i=1 j=1
f (η, ϕ) dηdϕ ≈
(A.9)
In Eq. (A.9), ϕj = 2π j/n; the integration points ηi are the roots of Legendre polynomials of degree n (Pn (ηi ) = 0) and the weights wi are given by wi =
2
2
1 − ηi
2 .
Pn (ηi )
Note that Eq. (A.9) is exact for the surface spherical harmonics up to degree 2n − 1. Therefore, application of this scheme to Eq. (A.8) yields exact values of the expansion coefficients cts for t ≤ 2n − 1, with a quite moderate numerical effort.
Selected properties of solid spherical harmonics With regard to asymptotic behavior, the whole set of solid spherical harmonics of Eq. (A.2) is divided onto two subsets, consisting of the regular (infinitely growing for r → ∞) and irregular, or singular (tending to zero when r → ∞), functions. We denote them separately as yst and Yts , respectively: yst (x) =
rt (t + s)!
χts (θ, ϕ),
Yts (x) =
(t − s)!
r t+1
χts (θ, ϕ)
(t ≥ 0, |s| ≤ t). (A.10)
Sometimes yst and Yts are referred to as the interior and exterior partial solutions, respectively. Adopted in Eq. (A.10) normalization aims to simplify
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Spherical harmonics and related theory
the subsequent algebra (see, e.g., [107,333]). We have yt−s (x) = (−1)s yst (x),
Yt−s (x) = (−1)s Yts (x),
yst (−x) = (−1)t yst (x),
Yts (−x) = (−1)t Yts (x).
(A.11)
and
An explicit expression of the first order solid harmonics is as follows:
x3 , r3
Y10 (x) =
1 (x1 + ix2 ) , 2 x1 + ix2 Y11 (x) = −Y1−1 (x) = . r3
y11 (x) = −y1−1 (x) =
y01 (x) = x3 ,
(A.12)
We mention also Hobson’s differentiation rules [332] written in our notations as D1 Yts = Yts+−11 ,
D2 Yts = −Yts++11 ,
D3 Yts = −Yts+1 ,
(A.13)
and D1 yst = yts−−11 ,
D2 yst = −yts−+11 ,
D3 yst = yst−1 ,
(A.14)
where Di are the differential operators
D1 =
∂ ∂ x1
−i
∂ ∂ x2
,
D2 = D1 =
∂ ∂ x1
+i
∂ ∂ x2
,
D3 =
∂ ∂ x3
. (A.15)
These operators can be viewed as the directional derivatives along the complex Cartesian base vectors ei defined as 1 e1 = (i1 + ii2 ), 2
e2 = e1 ,
e 3 = i3 ,
(A.16)
where ii are the conventional Cartesian base vectors. The newly introduced orthogonal vectors ei are similar to those used in [334]. Noteworthy is also that D1 = 2 ∂∂z and D2 = 2 ∂∂z where z = x1 + ix2 is the conventional complex variable in the Ox1 x2 plane. The differentiation rules in Eqs. (A.13) and (A.14) are written in compact form as
Dts Ykl (x) = (−1)t Ykl++st (x),
Dts ylk (x) = (−1)s ylk+−st (x),
(A.17)
where Dts = (D2 )s (D3 )t−s . In particular,
Dts
1 = (−1)t Yts (x), r
Dts ylk (x)
x=0
= (−1)s δtk δs,−l .
(A.18)
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Spherical harmonics and related theory
The below formulas provide an efficient recursive evaluation of a whole array of solid scalar harmonics in a given point starting from Y00 = 1/r and y00 = 1: r 2 Yts+1 = (2t + 1)x3 Yts − (t2 − s2 )Yts−1 ,
r 2 Ytt++11 = (2t + 1) (x1 + ix2 ) Ytt ,
(t + 1)2 − s2 yst+1 = (2t + 1)x3 yst − r 2 yst−1 ,
2(t + 1)ytt++11 = (x1 + ix2 ) ytt .
It is sometimes convenient to consider an extended set of functions Eq. (A.10) defined by ys−(t+1) (x) = (−1)t+s Yts (x).
(A.19)
This definition is consistent with Eq. (A.7) provided we formally replace (−1)t+s /(−t − 1 + s)! with (t − s)! in Eq. (A.10). With the aid of Eq. (A.19), the formulas for the regular solid harmonics are readily obtainable from the analogous formulas for the irregular solid harmonics (see, e.g., Eq. (A.13) which transforms into Eq. (A.14)) and vice versa.
Spherical harmonics vs multipole potentials Maxwell [16] has discovered the relationship between the solid spherical harmonics Yts and the potential fields of multipoles. So, the potential surrounding a point charge (being a singular point of the zeroth order, or monopole) is 1/r = Y00 (x). The first order singular point, or dipole, is obtained by pushing two monopoles of equal strength – but of opposite sign – toward each other. The potential of the dipole is to be given (up to rescaling) by the directional derivative ∇n1 (1/r ), where n1 is the direction along which the two monopoles approach one another. Similarly, pushing together two dipoles with opposite signs gives (up to rescaling) a quadrupole with potential ∇n1 ∇n2 (1/r ), where n2 is the direction along which the dipoles approach, and so on. In general, the multipole of order t is constructed with the aid of 2t point charges and has the potential proportional to ∇n1 ∇n2 · · · ∇nt (1/r ). The latter can be expanded into a weighted sum of 2t + 1 solid spherical harmonics of order t, i.e., Yts (x) (−t ≤ s ≤ t). And, vise versa, Yts (x) can be written as ∇n1 ∇n2 · · · ∇nt (1/r ), provided the directions ni are taken in accordance with the formula of Eq. (A.17). This is why the series expansion in terms of solid spherical harmonics Yts is often (incorrectly, however) referred to as multipole expansion. In fact, a one-to-one mapping exists only between the low-order harmonics and multipole potentials. As was mentioned above, the monopole potential is given by Y00 ; three firstorder harmonics Y1−1 , Y10 , and Y11 correspond to the dipoles oriented along
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Spherical harmonics and related theory
the complex Cartesian coordinate axes ei as in Eq. (A.16). Disagreement starts already from the quadrupoles (6 in total, in contrast to 5 second order harmonics, Y2s ) and extends to the higher order multipoles, which means that some of the multipole potentials are linearly dependent. For more details, see [89]. Under these circumstances, using the complete, linearly independent and orthogonal set of solid spherical harmonics Eq. (A.10) is a self-evident preferable option. Below, we use the “multipole expansion” term in a wide sense, as discussed in Section 1.4.
Fourier integral representation The irregular solid spherical harmonics Yts of Eq. (A.10) are written in the form of double Fourier integral as [6,78] Yts (x) = (∓1)t+s
∞
± ξts Eαβ (x)dα dβ
(x3 ≶ 0),
(A.20)
−∞
where the integral density is ξts = γ t−s−1 (β − iα)s . The harmonic functions ± entering Eq. (A.20) are defined by the formula Eαβ
± Eαβ (x) = exp n± αβ · x = exp (±γ x3 ) Fαβ (x1 , x2 ) ,
(A.21)
where Fαβ (x1 , x2 ) = exp i (α x1 + β x2 ) and n±αβ = iα i1 + iβ i2 ±γ i3 . By anal± can be thought as the scalar solid ogy with Eq. (A.2), the functions Eαβ harmonics for the half-spaces x3 ≶ 0. Note that the Faxen [335] integral representation of the fundamental solution Y00 1 = r
∞
± Eαβ (x)
−∞
γ
dα dβ
(x3 ≶ 0)
(A.22)
is a particular case of Eq. (A.20). ± The functions Eαβ of Eq. (A.21) are regular in the half-spaces x3 ≶ 0 and hence can be expanded into a series over a set of the regular solid spherical harmonics yst of Eq. (A.10) as [6,78] ± Eαβ (x) =
t ∞
κts± (α, β)yst (x),
t=0 s=−t
where the series coefficients κts± are given by the formula κts± = (±1)t+s κts = (±1)t+s γ t−s (β + iα)s .
(A.23)
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Spherical harmonics and related theory
A.2. Reexpansion formulas for Yts and yst Equally oriented coordinate systems Three reexpansion formulas for the scalar solid harmonics are considered. They are singular-to-regular (S2R) Yts (x + X) =
∞ k
(−1)k+l Yts+−kl (X)ylk (x),
x < X ,
(A.24)
k=0 l=−k
regular-to-regular (R2R) yst (x + X) =
t k
yts−−kl (X)ylk (x),
(A.25)
k=0 l=−k
and singular-to-singular (S2S) Yts (x + X) =
∞ k
(−1)t+k+s+l yks−−lt (X)Ykl (x),
x > X
(A.26)
k=t l=−k
reexpansions. In the notations of Eq. (A.10), these formulas take probably the simplest form. The finite formula of Eq. (A.25) is exact and valid for any x and X. The convergence domains of the series in Eqs. (A.24) and (A.26) are shown in Fig. A.2. In [115], Eqs. (A.25) and (A.26) are referred to as a translation of regular and singular solid harmonics, respectively. In [116], they are called a translation of local and multipole expansions whereas Eq. (A.24) is referred to as a conversion of a multipole expansion into a local one. Eqs. (A.24)–(A.26) can be derived in many ways, one of them is based on using Eq. (2.4) of Chapter 2. Probably the most straightforward derivation consists in applying the differentiation rule in Eq. (A.17) to the addition theorem for the fundamental solution [332]. In our notations, this theorem takes a compact form as follows: 1 x − X
= Y00 (x − X) =
⎧ ∞ k ⎪ ⎪ ⎪ (−1)l Yk−l (X)ylk (x), ⎨
x < X ,
⎪ ⎪ ⎪ ⎩
x > X .
k=0 l=−k k ∞
k=0 l=−k
(−1)l yk−l (X)Ykl (x),
(A.27)
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Spherical harmonics and related theory
Figure A.2 Convergence domains of (S2R) in Eq. (A.24) and (S2S) in Eq. (A.26) reexpansions.
In fact, the upper and lower formulas are identical: their differentiation with respect to X yields Eqs. (A.24) and (A.26), respectively.
Arbitrarily oriented coordinate systems In [160], the transformation formula for the spherical surface harmonics due to rotation Skp (O · n) =
k
k+p
k+p,k+l
k+l (−1)l+p C2k /C2k S2k
(w)Skl (n)
(A.28)
l=−k
is derived. Here, n = x/ x 3 , x is a radius-vector, w = {w1 , w3 , w3 , w4 } is the unit four-dimensional vector ( w = w12 + w22 + w32 + w42 = 1) and Cnm are the binomial coefficients. Also, Skp are the scalar spherical harmonics in a three-dimensional space, and Skp, l are the scalar spherical harmonics in a four-dimensional space [160]: Snk, l (w) = (−1)k (w4 + iw1 )n−k−l (w3 + iw2 )k−l × Pl(n−k−l, k−l) (w32 + w22 − w42 − w12 ),
n ≥ k + l,
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Spherical harmonics and related theory
Snk, l (w) = (−1)n−l (w4 − iw1 )k+l−n (w3 − iw2 )l−k × Pn(k−+l l−n,l−k) (w32
+ w22
− w42
− w12 ),
(A.29)
n < k + l,
where Pn(α,β) is the Jacobi polynomial [253] Pn(α,β) = 2−n
n
−m Cnm+α Cnn+β (x − 1)n−m (x + 1)m .
m=0
Eq. (A.28) describes the action of an orthogonal transformation O in the three-dimensional space and gives the coefficients of a linear transformation of the functions Skp in terms of spherical harmonics in the four-dimensional space. An explicit form of the matrix O in terms of wi is ⎛
⎞
w22 − w12 − w32 + w42 2(w2 w3 − w1 w4 ) 2(w1 w2 + w3 w4 ) ⎜ ⎟ O = ⎝ 2(w2 w3 + w1 w4 ) w32 − w12 − w22 + w42 2(w1 w3 − w2 w4 ) ⎠ . 2(w1 w2 − w3 w4 ) 2(w1 w3 + w2 w4 ) w12 − w22 − w32 + w42 (A.30) According to [160], Sts (n) = (−1)t+s
2t t! s 2t t! −s Pt (cos θ ) exp(−isϕ) = (−1)t+s χ (θ, ϕ), (t + s)! (t − s)! t
so from Eq. (A.28) we get χks (θ , ϕ ) =
k (k − l)! k−s, k−l S (w)χkl (θ, ϕ). (k − s)! 2k
(A.31)
l=−k
We employ this result to generalize the formulas of Eqs. (A.24)–(A.26) to the case of arbitrarily positioned and oriented reference systems O1 x11 x21 x31 and O2 x12 x22 x32 . It is common knowledge that a general transformation of coordinates can be split into a translation along vector X (system O2 x12 x22 x32 ) and rotation around a fixed origin O2 of the coordinate system, see Fig. A.3. Let transformation O describe the rotation of the coordinate axes from O2 x12 x22 x32 to O2 x12 x22 x32 . This is an orthogonal transformation with determinant equal to +1, hence Eq. (A.28) applies. Consider, for example, Eq. (A.24). We have Yts (x1 ) =
∞ k
k=0 p=−k
s−p
(−1)k+p Yt+k (X)
r k2 (k + p)!
χk (cos θ , ϕ ). p
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Spherical harmonics and related theory
Figure A.3 Reexpansion: translation plus rotation.
Because n2 = O· n2 and r 2 = r2 are not affected by rotation, we substitute Eq. (A.31) into the last equality to obtain, for x1 = X + O · x2 , Yts (x1 ) =
∞ k
ηtksl (X, O) ylk (x2 ),
(A.32)
k=0 l=−k
where ηtksl (X, O) =
k
(−1)p+l
p=−k
(k − l)!(k + l)! k−p,k−l s−p S (w)Yt+k (X). (k − p)!(k + p)! 2k
An analogous transformation of Eqs. (A.25) and (A.26) gives also yst (x1 ) =
t k
νtksl (X, O) ylk (x2 ),
k=0 l=−k
where νtksl (X, O) =
k
(−1)p+l
p=−k
(k − l)!(k + l)! k−p,k−l s−p S (w)yt−k (X), (k − p)!(k + p)! 2k
and Yts (x1 ) =
∞ k
k=t l=−k
μtksl (X, O) Ykl (x2 ),
(A.33)
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Spherical harmonics and related theory
where μtksl (X, O) = (−1)t+s
k
k−p,k−l
(−1)k+p S2k
s−p
(w)yk−t (X).
p=−k
The geometrical restrictions for the reexpansions in Eqs. (A.32)–(A.33) are the same as for Eqs. (A.24)–(A.26).
A.3. Scalar spherical biharmonics By analogy with Eqs. (A.1) and (A.2), we expand a general solution of the scalar biharmonic equation ∇ 4 g = 0 over a full and orthogonal on a sphere set of scalar surface harmonics χts from Eq. (A.5). This yields the following sets of regular zst and irregular Zts biharmonic functions: zst (x) =
r2 ys (x) 2(2t + 3) t
Zts (x) = zs−(t+1) (x) = −
(t = 0, 1, . . . ; |s| ≤ t) ,
r2 Y s (x) 2(2t − 1) t
(A.34)
(t = 1, 2, . . . ; |s| ≤ t) .
For the irregular functions, index t starts from 1 because Z00 (x) = r /2 is a regular biharmonic. As expected, ∇ 2 Zts = Yts and ∇ 2 zst = yst . Also, it follows directly from Eq. (A.34) that zt−s (x) = (−1)s zst (x),
Zt−s (x) = (−1)s Zts (x).
The formulas (t − s)(t + s) s Yt−1 , 2t + 1 (t − s)(t − s − 1) s+1 (x1 + ix2 )Yts = −2Zts++11 − Yt−1 , 2t + 1 (t + s)(t + s − 1) s−1 Yt−1 , (x1 − ix2 )Yts = 2Zts+−11 + 2t + 1 (t − s + 1)(t + s + 1) s x3 yst = 2zst−1 + yt+1 , 2t + 1 (t + s + 1(t + s + 2) s+1 +1 (x1 + ix2 )yst = −2zts− yt+1 , 1+ 2t + 1 (t − s + 1)(t − s + 2) s−1 −1 (x1 − ix2 )yst = 2zts− yt+1 , 1− 2t + 1
x3 Yts = −2Zts+1 +
(A.35)
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Spherical harmonics and related theory
are valid for all indices t and s. Extending the definition in analogy with Eq. (A.19), namely, zs−(t+1) (x) = (−1)t+s Zts (x), transform the formulas for the regular biharmonics in Eq. (A.35) into the formulas for the irregular biharmonics, and vise versa. Note that Eq. (A.34) is not the only choice. Adding an arbitrary harmonic does not alter the biharmonic nature of these functions but can greatly simplify the relevant formulas. Consider, for example, the biharmonics defined as (t − s)(t − s − 1) s Yt−2 (x), 2(2t − 1) (t + s + 1)(t + s + 2) s s s zˇ st (x) = Zˇ −( yt+2 (x). t+1) (x) = zt (x) − 2(2t + 3)
Zˇ ts (x) = Zts (x) +
(A.36)
A motivation of this particular choice is clear from Eq. (A.35): Zˇ ts (x) =
(x1 + ix2 )
2
Yts−−11 ,
zˇ st (x) =
(x1 + ix2 )
2
yts+−11 .
The newly introduced functions of Eq. (A.36) possess remarkable properties. First, we mention the differential rules D1 Zˇ ts = Zˇ ts+−11 − Yts−−11 , D1 zˇ st = zˇ ts−−11 − yts+−11 ,
D2 Zˇ ts = −Zˇ ts++11 , D2 zˇ st = −zˇ ts−+11 ,
D3 Zˇ ts = −Zˇ ts+1 , D3 zˇ st = zˇ st−1 ,
which can be written in the same compact form as Eq. (A.17) for the spherical harmonics:
Dts Zˇ kl (x) = (−1)t Zˇ kl++st (x),
Dts zˇ lk (x) = (−1)s zˇ lk+−st (x).
In particular, for Zˇ 00 = r /2, we get the formula Dts
r
2
= (D2 )s (D3 )t−s
r
2
ˇ ts (x), = (−1)t Z
quite analogous to a classical Hobson’s result [332] for the solid spherical harmonics. The (S2R) reexpansion formula for the functions Eq. (A.36) is Zˇ ts (x + X) =
∞ k
k=0 l=−k
ˇ s−l (X)ylk (x) . (A.37) (−1)k+l Yts+−kl −2 (X)zˇ lk−2 (x) + Z t+k
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Spherical harmonics and related theory
The other two reexpansions, (R2R) and (S2S), also take similar to Eq. (A.37) simple forms: zˇ st (x) =
t+1 k+1
k=0 l=−k−1
Zˇ ts (x) =
∞ k
l
yts−−kl +1 (X)zˇ k−1 (x) + zˇ ts−−kl (X)ylk (x) ,
ˇ kl (x) + zˇ s−l (X)Ykl (x) . (−1)t+k+s+l yks−−lt (X)Z k−t
k=t−2 l=−k
The functions of Eq. (A.36) will be used in Chapter 5 where their close relationship to the periodic biharmonics [110] will be established.
A.4. Vector spherical surface harmonics The vector surface spherical harmonics S(tsi) (see, e.g., [89]) are defined in terms of their scalar counterpart, χts = χts (θ, ϕ), as
∂ s eϕ ∂ s χ + χ, ∂θ t sin θ ∂ϕ t eθ ∂ s ∂ S(ts2) = r ∇ × er χts = χt − eϕ χts , sin θ ∂ϕ ∂θ
S(ts1) = r ∇ χts = eθ
S(ts3) = er χts
(A.38)
(t ≥ 0, |s| ≤ t).
In order to simplify the subsequent algebra, we deliberately omit here the √ conventional normalizing multiplier t(t + 1). The functions of Eq. (A.38) constitute a complete and orthogonal on a sphere set of vector harmonics. Specifically, 1 S(i) · S(j) dS = αts(i) δtk δsl δij , (A.39) S S ts kl where αts(1) = αts(2) = t(t + 1)αts and αts(3) = αts given by Eq. (A.6). The functions S(tsi) possess remarkable differential, r ∇ · S(ts1) = −t(t + 1)χts , r ∇ × S(ts1) = −S(ts2) ,
∇ · S(ts2) = 0,
r ∇ · S(ts1) = 2χts ,
r ∇ × S(ts2) = S(ts1) + t(t + 1)S(ts3) ,
(A.40)
r ∇ × S(ts3) = S(ts2) ,
and algebraic, er · S(ts1) = 0, (1)
er · S(ts2) = 0, (2)
er × Sts = −Sts ,
(2)
er · S(ts3) = χts , (1)
er × Sts = Sts ,
(A.41) (3)
er × Sts = 0,
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Spherical harmonics and related theory
properties. In the vector problems, the functions of Eq. (A.38) play the same role as the surface harmonics χts of Eq. (A.5) in the scalar potential problems. We mention the following useful relation: i) s+i−1 (i) Sts , S(t,− s = (−1)
(A.42)
(i) (i) and, by analogy with the scalar case, extend the definition as S−( t+1),s = Sts . With the aid of Eq. (A.40), it is readily proven (see, e.g., [189]) that a transformation of the vector spherical surface harmonics Eq. (A.38) due to a rotation of the coordinate system is given by the formula
t t − l ! t−s,t−l Sts (O · x) = S (w) S(tli) (x) , (t − s)! 2t (i)
(A.43)
l=−t
where Sts,l of Eq. (A.29) are the spherical harmonics in the four-dimensional space [160] and w = {w1 , w3 , w3 , w4 }T of Eq. (A.30) is the unit vector determining uniquely the rotation matrix O. Eq. (A.43) generalizes Eq. (A.28) to the case of vector surface harmonics. In this and all the subsequent analogous formulas, we keep in mind that the vector functions on the opposite sides of an equality are written in the variables and unit vectors of the local (different, in general) coordinate systems.
A.5. Partial solutions of Lamé equation Definition The above-obtained results are sufficient to find a complete set of the partial solutions of Lamé equation, Eq. (1.7) Their expression in terms of vector spherical harmonics from Eq. (A.38) is as follows (see, e.g., [136]): r t−1 (1) 1 rt Sts + tS(ts3) , u(ts2) = − S(2) , (t + s)! (t + 1) (t + s)! ts r t+1 βt (ν)S(ts1) + γt (ν)S(ts3) , u(ts3) = (t + s)!
u(ts1) =
where the coefficients βt (ν) =
t + 5 − 4ν t − 2 + 4ν and γt (ν) = (t + 1)(2t + 3) (2t + 3)
(A.44)
533
Spherical harmonics and related theory
are related by γt + (t + 1)βt ≡ 1. In complex Cartesian projections, u(ts1) = e1 yts−−11 − e2 yts−+11 + e3 yst−1 , (A.45) i u(ts2) = [−e1 (t − s + 1)yts−1 − e2 (t + s + 1)yts+1 + e3 syst ], (t + 1) (t − s + 1)(t − s + 2) s−1 (3) s−1 uts = e1 2zt−1 + εt (ν) yt+1 (t + 1) (2t + 3) (t + s + 1)(t + s + 2) s+1 s+1 + e2 −2zt−1 − εt (ν) yt+1 (t + 1) (2t + 3) (t − s + 1)(t + s + 1) s s + e3 2zt−1 − εt (ν) yt+1 , (t + 1) (2t + 3) where εt (ν) = 3 − 4ν −
1 (2t + 1)
.
The irregular (infinitely growing as r → 0 and vanishing at infinity) (i) functions U(tsi) = u−( t+1), s are given by Eq. (A.44) with t replaced by −(t + 1). Eqs. (A.44) to (A.48) are also valid for U(tsi) with t replaced by −(t + 1). At the spherical surface S : r = R, u(tsj) and U(tsj) are written in compact form as
u(tsj) = S
3
UMtsij (R, ν) S(tsi) ,
i=1
U(tsj) =
3
S
UGtsij (R, ν) S(tsi) ,
(A.46)
i=1
where UMt = UMtij is a (3 × 3) matrix of the form ⎛
1 0 t−1 ⎜ UMt (r , ν) = r ⎝ 0 − t+r 1 t 0
⎞
r 2 βt (ν) ⎟ 0 ⎠ r 2 γt (ν)
(A.47)
and UGt = UGtij = UM−(t+1) .
Explicit expressions The functions u(tsi) (x) obey the differential relations: u(ts1) = ∇ yst , (3)
∇ · uts = 2(2ν
∇ · u(ts1) = ∇ · u(ts2) = ∇ × u(ts1) = 0, − 1)yst ,
(2)
(1)
∇ × uts = −uts ,
(A.48) (3)
∇ × uts = 4(1 − ν)u(ts2) .
The formulas the irregular solutions are the same, with u(tsi) replaced by U(tsi) (x) and yst by Y st . The formulas in Eq. (A.48) are particularly convenient
534
Spherical harmonics and related theory
because the differential operators are invariant of the specific choice of the coordinate frame. The functions with positive and negative s index are related by i) s+i−1 (i) u(t,− uts , s = (−1)
i) s+i−1 (i) U(t,− Uts . s = (−1)
(A.49)
Now, we consider the low order functions in more detail. Note first that, by definition, u(001) = u(002) = 0 and the only zeroth-order regular function is u(003) = γ0 (ν)x. Recall that u(ts3) for t > 0 are biharmonic functions whereas u(003) is the only vector harmonics with constant nonzero divergence ∇ · u(003) = 2(2ν − 1), describing a uniform dilatation (volumetric expansion) of a solid. The next three functions, u(1s1) , represent rigid body translations in three orthogonal directions, along the introduced by Eq. (A.16) complex Cartesian axes: u(101) = e3 ,
u(111) = e1 ,
u(11,−) 1 = −e2 .
The rigid body rotation x × is also described by three functions, this time by u(1s2) . For u = W · x, where W = Wij ii ij is the antisymmetric, or skew (Wij = −Wji ), tensor, one finds
u = 2W12 u(102) + 2W13 i u(112) − u(12,−) 1 + 2W23 u(112) + u(12,−) 1 .
(A.50)
The linear displacement u = E · x, where E = Eij ii ij is the symmetric (Eij = Eji ) tensor, is expressed by means of functions u(2s1) plus the alreadymentioned function u(003) . Specifically, (3) (3) E · x = c00 u00 +
2
(1) (1) c2s u2s ,
(A.51)
s=−2
where (3) c00 =
(E11 + E22 + E33 ) , 3γ0 (ν)
(1) c21 = −c2(1,−) 1 = E13 − iE23 ,
(1) c20 =
(2E33 − E11 − E22 )
3
,
(1) c22 = c2(1,−) 2 = E11 − E22 − 2iE12 .
Eqs. (A.50) and (A.51) provide an expansion of u = A · x where A = E + W is an arbitrary (not necessarily symmetric) tensor.
535
Spherical harmonics and related theory
Normal traction At the spherical surface S : r = R, we have n = er , and so the traction vector Tn = σ · n takes the form 1 1 ν ∂ Tn (u) = er (∇ · u) + u+ er × (∇ × u) . 2μ 1 − 2ν ∂r 2 Application to the vector functions u(tsi) yields [28] 1 1 (t − 1) (1) (t − 1) (2) Tn (u(ts1) ) = Tn (u(ts2) ) = uts , uts , 2μ r 2μ 2r rt
1 Tn (u(ts3) ) = bt (ν)S(ts1) + gt (ν)S(ts3) , 2μ (t + s)!
(A.52)
where bt (ν) = (t + 1)βt − 2(1 − ν)/(t + 1) and gt (ν) = (t + 1)γt − 2ν . Again, Eq. (A.52) holds true for the irregular solutions, with u(tsi) replaced by U(tsi) (x) and t by −(t + 1). In view of Eq. (A.44), a representation of Tn (u(tsi) ) in terms of vector spherical harmonics from Eq. (A.38) is obvious. As expected, Tn ≡ 0 for u(1s1) and u(1s2) representing rigid body translation and rotation, respectively. At the spherical surface S : r = R, Tn (u(tsi) ) and Tn (U(tsi) ) are written in compact form as
3
3
1 Tn (u(tsi) ) = TMtsji (R, ν)S(tsj) , 2μ S j=1 1 Tn (U(tsi) ) = TGtsji (R, ν) S(tsj) , 2μ S j=1
TMt = TMtij being the (3 × 3) matrix ⎞
⎛
t−1 0 ⎜ 0 − 2r ((tt−+11)) TMt (r , ν) = r t−2 ⎝ t(t − 1) 0
r 2 bt (ν) 0 ⎟ ⎠. 2 r gt (ν)
(A.53)
Here, gt and bt are defined by Eq. (A.52); TGt = TGtij = TM−(t+1) .
536
Spherical harmonics and related theory
Net force and torque The net force T and torque M acting on the surface S enclosing the point x = 0 are given by the formulas
T=
Tn dS,
M=
S
x × Tn dS.
(A.54)
S
It is readily found that T = M = 0 for all regular functions u(tsi) . The irregular functions with nonzero net force are
T U(103) = 16μπ(ν − 1)e3 ,
T U(113) = −T U(13,−) 1 = 32μπ(ν − 1)e1 . (A.55)
By analogy with Y00 , U(1s3) can be regarded as the vector monopoles. The net torque is zero for all U(tsi) but U(1s2) for which we get
M U(102) = −8μπ e3 ,
M U(112) = M U(12,−) 1 = −16μπ e1 .
(A.56)
Eqs. (A.55) and (A.56) exhibit the physical meaning of the irregular vector functions U(1s2) and U(1s3) , being the displacements due to point torque and point force, respectively, applied in the point x = 0.
A.6. Partial solutions for a half-space Cartesian vector surface harmonics (i) Following [42], we introduce the Cartesian vector surface harmonics Lαβ :
L(αβ1) (x1 , x2 )= L(αβ2) (x1 , x2 )
=
1 γ
∇ × Fαβ =
1 γ
iα γ
∇ × i3 Fαβ =
i1 +
iβ γ
iβ γ
i2 Fαβ ,
i1 −
iα γ
(A.57)
i2 Fαβ ,
L(αβ3) (x1 , x2 ) = i3 Fαβ , where Fαβ (x1 , x2 ) is the scalar surface (double Fourier) harmonic, see Eq. (2.36). These harmonics obey the orthogonality property
537
Spherical harmonics and related theory
∞
(j) Lα(i)β · Lαβ dx1 dx2 = αi δij δ α − α δ β − β ,
(A.58)
−∞
where α1 = α2 = αα + ββ = γ 2 and α3 = 1. These harmonics are appropriate for fulfilling the boundary conditions on the planes x3 = X3 = const [42]. In view of Eq. (A.58), one can expand an arbitrary, sufficiently smooth vector function f(x) as
f(x) =
∞ −∞
∞
3
−∞ j=1
(j) aj (α, β)Lαβ exp(±γ X3 )dα dβ,
(A.59)
where the integral densities aj (α, β) are given by the inverse double Fourier transform exp(∓γ X3 ) αi
aj (α, β) =
∞ −∞
∞
−∞
(j) f · Lαβ dx1 dx2 .
Vector solutions of Lamé equation for a half-space The vector functions 1
1
2)± ± ± , h(αβ = ), ∇ Eαβ ∇ × (i3 Eαβ (±γ ) (±γ ) ± ± − 4 (1 − ν) i3 Eαβ h(αβ3)± = ∇ x3 Eαβ
h(αβ1)± =
(A.60)
constitute a set of the partial vector solutions of Lamé equation for a halfspace. Note that h(αβ3)± is written in the standard Papkovich–Neuber form (see, e.g., [137]). A general solution of Lamé equation bounded in the homogeneous half-spaces x3 ≶ 0 can be represented by the double Fourier integral
u(x) =
∞
−∞
∞
3
−∞ j=1
(j)± Gj (α, β)hαβ (x)dα dβ
(x3 ≶ 0) ,
(A.61)
where Gi are the integral densities (complex, in general). Fulfilling the displacement boundary conditions on the plane x3 = const conveniently uses an expression of these functions in terms of the Cartesian vector surface harmonics from Eq. (A.57): (j)± hαβ
=
3 i=1
(i)(j)± (i) UZαβ Lαβ exp(±γ x3 ),
538
Spherical harmonics and related theory
where
⎛
±1
⎜
(i)(j)± UZ±αβ (x3 ) = UZαβ =⎝ 0 1
⎞
0 γ x3 ⎟ ±1 0 ⎠. 0 (4ν − 3 ± γ x3 )
(A.62)
Note that det UZ±αβ (x3 ) = 4ν − 3 = 0 for all x3 , γ , and 0 < ν < 0.5, guaranteeing invertibility of UZ±αβ matrix. The stress boundary conditions involve the normal traction vector Tn = σ · n taking on the plane x3 = const (for the bottom half-space, an outer normal unit vector n = i3 ) the form 1 1 ν ∂ Tn (u) = i3 (∇ · u) + u + i3 × (∇ × u) . 2μ 1 − 2ν ∂ x3 2
(A.63)
(j)± Its application to hαβ yields
1 (j)± (i)(j)± (i) Tn (hαβ )= TZαβ Lαβ exp(±γ x3 ), 2μ i=1 3
(A.64)
where
(i)(j)± TZ±αβ (x3 ) = TZαβ
⎛
⎞
1 0 (2ν − 1) ± γ x3 ⎜ ⎟ 1 =γ ⎝ 0 0 ⎠. 2 ±1 0 ±2(ν − 1) + γ x3
(A.65)
Note that det γ1 TZ±αβ (x3 ) = ∓ 12 = 0, which means that this matrix is invertible.
Integral transforms and series expansions The double vector Fourier transform for the irregular functions U(tsi) is [78] (i)
t+s
Uts (x) = (∓1)
∞
−∞
∞
3
−∞ j=1
(i)(j)± (j)±
ξtsαβ hαβ (x)dα dβ
(x3 ≶ 0) ,
where ξts(1αβ)(1)± = ±ξt+1,s ,
is t
ξts(2αβ)(1)± = − ξts ,
ξts(2αβ)(2)± = −ξts ,
ξts(3αβ)(3)± = ±ξt−1,s ,
is t
ξts(3αβ)(2)± = ∓4 (1 − ν) ξt−1,s ,
(A.66)
539
Spherical harmonics and related theory
ξts(3αβ)(1)± = ±ξt−1,s (3 − 4ν) − C−(t+1),s ,
ξts = γ t−s−1 (β − iα)s .
(j)± The series expansion of hαβ in terms of u(tsi) is
(j)± hαβ (x)
=
t 3 ∞
(±1)t+s κtsαβ u(tsi) (x), (j)(i)±
(A.67)
i=1 t=0 s=−t
where )(1)± κts(1αβ = ±κt−1,s ,
)(2)± κts(2αβ = −κts ,
)(3)± κts(3αβ = ±κt+1,s ,
is is )(2)± κts(3αβ = 4 (1 − ν) κts , t t (3)(1)± t−s κtsαβ = ∓C−(t+1),s κt−1,s , κts = γ (β + iα)s .
)(1)± κts(2αβ = ± κt−1,s ,
A.7. Reexpansion formulas for U(tsi) and u(tsi) Translation The irregular-to-regular (S2R) reexpansion formulas for the irregular vector functions U(tsi) [28] are: U(tsi) (x + X) =
3 ∞ k
(−1)k+l ηtksl (X)ukl (x) (i)(j)
(j)
( x < X ) ,
(A.68)
j=1 k=0 l=−k
where (i)(j) (1)(1) (2)(2) (3)(3) ηtksl = 0 j > l , ηtksl = ηtksl = ηtksl = Yts+−kl , s l (2)(1) (3)(2) (2)(1) ηtksl =i + Y s−l , ηtksl = −4 (1 − ν) ηtksl , t k t+k−1 ls (3)(1) ˇ s−l + Y s−l = 2Z ηtksl t+k t+k−2 4 (1 − ν) 1 + kt − C−(t+1),s − C−(k+1),l − (t + k − s + l) .
(A.69)
The regular-to-regular (S2R) reexpansions of u(tsi) are u(tsi) (x + X) =
+i−j k i t
j=1 k=0 l=−k
(−1)k+l νtksl (X)ukl (x), (i)(j)
(j)
(A.70)
540
Spherical harmonics and related theory
where (i)(j) νtksl = 0 j > l , s (2)(1) νtksl = i − t+1
(1)(1) (2)(2) (3)(3) −l νtksl = νtksl = νtksl = yts− k, l s−l (3)(2) (2)(1) y , νtksl = −4 (1 − ν) νtksl , k t−k+1 l s (3)(1) −l s−l = 2zˇ ts− νtksl k + yt−k+2 4 (1 − ν) 1 − kt+1 − Cts − C−(k+1),l + (t − k + s − l + 1) .
The irregular-to-irregular (S2S) reexpansions of U(tsi) are U(tsi) (x + X) =
k 3 ∞
(−1)t+k+s+l μtksl (X)Ukl (x) (i)(j)
(j)
( x > X ) ,
j=1 k=t−i+jl=−k
(A.71) where (i)(j) 1)(1) 2)(2) 3)(3) μtksl = 0 j > l , μ(tksl = μ(tksl = μ(tksl = yks−−lt , s l 2)(1) 3)(2) 2)(1) μ(tksl =i − ys−l , μ(tksl = −4 (1 − ν) μ(tksl , t k + 1 k−t+1 l s 3)(1) = 2zˇ ks−−lt + yks−−lt+2 4 (1 − ν) 1 − μ(tksl k+1t − C−(t+1),s − Ckl + k − t + s − l + 1 .
In these formulas, Cts = (t + 1)2 − s2 βt . These formulas are the vector counterparts of Eqs. (A.24)–(A.26). Being incorporated into the fast multipole [116] scheme, they provide a very efficient algorithm for elastic interactions in the large-scale clusters of inhomogeneities. For the Stokes interactions in the fluid suspension of spherical particles, such a work has been done in [115].
Rotation The reexpansion formulas of the vector functions u(tsi) and U(tsi) in the case of arbitrarily oriented local coordinate systems can be found by decomposition of the general orthogonal transformation of the coordinates into a sum of parallel translation of vector X and rotation by O around the fixed origin of the coordinate system. The first step (translation) is done already. The
541
Spherical harmonics and related theory
regular vector functions u(tsi) are reexpanded due to rotation as u(tsi) (O · x) =
t (t − l)!(t + l)! t−s,t−l S (w)u(tli) (x). (t − s)!(t + s)! 2t
(A.72)
l=−t
An analogous formula for the irregular vector functions U(tsi) is U(tsi) (O · x) =
t
t−s,t−l S2t (w)U(tli) (x).
(A.73)
l=−t
Both these formulas follow directly from Eq. (A.43), they are of finite form and exact.
Spherical harmonics and related theory A.1. Scalar spherical harmonics Laplace equation in spherical coordinates The spherical coordinates (r , θ, ϕ) (Fig. A.1) relate to the Cartesian ones (x1 , x2 , x3 ) by x1 + ix2 = r sin θ exp(iϕ), x3 = r cos θ (r ≥ 0, 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π). Separation of variables in Laplace equation written in spherical coordinates as 1 ∂ 1 ∂ 2T ∂ ∂T 2 ∂T T (x) = r =0 + sin θ + 2 ∂r ∂r sin θ ∂θ ∂θ sin θ ∂ϕ 2 2
(A.1)
yields a complete set of partial (“normal”, in Hobson’s [332] terminology) solutions of the form r t Pts (cos θ ) exp(isϕ) = r t χts (θ, ϕ)
(−∞ < t < ∞, |s| ≤ t),
(A.2)
Figure A.1 Spherical coordinates. 519
520
Spherical harmonics and related theory
referred to in [16] as scalar solid spherical harmonics of degree t and order s. Here and below, the radius-vector (position-vector field, in Gurtin [91] terminology) is x = xk ik , ik being the unit base vectors of the Cartesian coordinate system. In Eq. (A.2), Pts are the associated Legendre functions of the first kind [332]: Pts (η) = η¯ s
ds Pt (η) (−1)t η¯ s dt+s η¯ 2t = dηs 2t t! dηt+s
(A.3)
for |η| ≤ 1. The relationships Pt−s (η) = (−1)s
(t − s)! s P (η), (t + s)! t
s s P−( t+1) (η) = Pt (η)
Pts (η) ≡ 0
(0 ≤ t < ∞,
(A.4) |s| ≤ t) ,
(|s| > t) ,
extend the definition of the functions Pts of Eq. (A.3) to arbitrary integer indices t and s. It also follows from Eq. (A.3) that Pts (−η) = (−1)t+s Pts (η). In Eq. (A.2), χts (θ, ϕ) = Pts (cos θ ) exp(isϕ)
(A.5)
are the scalar surface spherical harmonics. They possess the orthogonality property [253] 1 S
S
χts χkl dS = αts δtk δsl ,
αts =
1 (t + s)! , 2t + 1 (t − s)!
(A.6)
where the integral is taken over the spherical surface S. The overbar means the complex conjugate and δij is the Kronecker’s delta. Also, in view of Eq. (A.4), χt−s = (−1)s s s χ−( t+1) = χt
χts ≡ 0
(t − s)! s χ, (t + s)! t
(0 ≤ t < ∞, |s| ≤ t) , (|s| > t) .
(A.7)
521
Spherical harmonics and related theory
The series expansion of a continuous function f (θ, ϕ) over the set of surface spherical harmonics of Eq. (A.5) is given by the formula [332] f (θ, ϕ) =
t ∞
cts χts (θ, ϕ),
t=0 s=−t
where cts =
2π
1 4παts
π
dϕ f (θ, ϕ) χts (θ, ϕ) sin θ dθ. 0
(A.8)
0
Integration in Eq. (A.8) can be done either analytically or numerically. In the latter case, an appropriate scheme [182] comprises the uniform distribution of integration points in the azimuthal direction ϕ with Gauss– Legendre quadrature rule [253] for integration with respect to η = cos θ : 1 R2
f (η, ϕ) dS = S
0
2π
1
−1
n n 1 wi f ηi , ϕj . n i=1 j=1
f (η, ϕ) dηdϕ ≈
(A.9)
In Eq. (A.9), ϕj = 2π j/n; the integration points ηi are the roots of Legendre polynomials of degree n (Pn (ηi ) = 0) and the weights wi are given by wi =
2
2
1 − ηi
2 .
Pn (ηi )
Note that Eq. (A.9) is exact for the surface spherical harmonics up to degree 2n − 1. Therefore, application of this scheme to Eq. (A.8) yields exact values of the expansion coefficients cts for t ≤ 2n − 1, with a quite moderate numerical effort.
Selected properties of solid spherical harmonics With regard to asymptotic behavior, the whole set of solid spherical harmonics of Eq. (A.2) is divided onto two subsets, consisting of the regular (infinitely growing for r → ∞) and irregular, or singular (tending to zero when r → ∞), functions. We denote them separately as yst and Yts , respectively: yst (x) =
rt (t + s)!
χts (θ, ϕ),
Yts (x) =
(t − s)!
r t+1
χts (θ, ϕ)
(t ≥ 0, |s| ≤ t). (A.10)
Sometimes yst and Yts are referred to as the interior and exterior partial solutions, respectively. Adopted in Eq. (A.10) normalization aims to simplify
522
Spherical harmonics and related theory
the subsequent algebra (see, e.g., [107,333]). We have yt−s (x) = (−1)s yst (x),
Yt−s (x) = (−1)s Yts (x),
yst (−x) = (−1)t yst (x),
Yts (−x) = (−1)t Yts (x).
(A.11)
and
An explicit expression of the first order solid harmonics is as follows:
x3 , r3
Y10 (x) =
1 (x1 + ix2 ) , 2 x1 + ix2 Y11 (x) = −Y1−1 (x) = . r3
y11 (x) = −y1−1 (x) =
y01 (x) = x3 ,
(A.12)
We mention also Hobson’s differentiation rules [332] written in our notations as D1 Yts = Yts+−11 ,
D2 Yts = −Yts++11 ,
D3 Yts = −Yts+1 ,
(A.13)
and D1 yst = yts−−11 ,
D2 yst = −yts−+11 ,
D3 yst = yst−1 ,
(A.14)
where Di are the differential operators
D1 =
∂ ∂ x1
−i
∂ ∂ x2
,
D2 = D1 =
∂ ∂ x1
+i
∂ ∂ x2
,
D3 =
∂ ∂ x3
. (A.15)
These operators can be viewed as the directional derivatives along the complex Cartesian base vectors ei defined as 1 e1 = (i1 + ii2 ), 2
e2 = e1 ,
e 3 = i3 ,
(A.16)
where ii are the conventional Cartesian base vectors. The newly introduced orthogonal vectors ei are similar to those used in [334]. Noteworthy is also that D1 = 2 ∂∂z and D2 = 2 ∂∂z where z = x1 + ix2 is the conventional complex variable in the Ox1 x2 plane. The differentiation rules in Eqs. (A.13) and (A.14) are written in compact form as
Dts Ykl (x) = (−1)t Ykl++st (x),
Dts ylk (x) = (−1)s ylk+−st (x),
(A.17)
where Dts = (D2 )s (D3 )t−s . In particular,
Dts
1 = (−1)t Yts (x), r
Dts ylk (x)
x=0
= (−1)s δtk δs,−l .
(A.18)
523
Spherical harmonics and related theory
The below formulas provide an efficient recursive evaluation of a whole array of solid scalar harmonics in a given point starting from Y00 = 1/r and y00 = 1: r 2 Yts+1 = (2t + 1)x3 Yts − (t2 − s2 )Yts−1 ,
r 2 Ytt++11 = (2t + 1) (x1 + ix2 ) Ytt ,
(t + 1)2 − s2 yst+1 = (2t + 1)x3 yst − r 2 yst−1 ,
2(t + 1)ytt++11 = (x1 + ix2 ) ytt .
It is sometimes convenient to consider an extended set of functions Eq. (A.10) defined by ys−(t+1) (x) = (−1)t+s Yts (x).
(A.19)
This definition is consistent with Eq. (A.7) provided we formally replace (−1)t+s /(−t − 1 + s)! with (t − s)! in Eq. (A.10). With the aid of Eq. (A.19), the formulas for the regular solid harmonics are readily obtainable from the analogous formulas for the irregular solid harmonics (see, e.g., Eq. (A.13) which transforms into Eq. (A.14)) and vice versa.
Spherical harmonics vs multipole potentials Maxwell [16] has discovered the relationship between the solid spherical harmonics Yts and the potential fields of multipoles. So, the potential surrounding a point charge (being a singular point of the zeroth order, or monopole) is 1/r = Y00 (x). The first order singular point, or dipole, is obtained by pushing two monopoles of equal strength – but of opposite sign – toward each other. The potential of the dipole is to be given (up to rescaling) by the directional derivative ∇n1 (1/r ), where n1 is the direction along which the two monopoles approach one another. Similarly, pushing together two dipoles with opposite signs gives (up to rescaling) a quadrupole with potential ∇n1 ∇n2 (1/r ), where n2 is the direction along which the dipoles approach, and so on. In general, the multipole of order t is constructed with the aid of 2t point charges and has the potential proportional to ∇n1 ∇n2 · · · ∇nt (1/r ). The latter can be expanded into a weighted sum of 2t + 1 solid spherical harmonics of order t, i.e., Yts (x) (−t ≤ s ≤ t). And, vise versa, Yts (x) can be written as ∇n1 ∇n2 · · · ∇nt (1/r ), provided the directions ni are taken in accordance with the formula of Eq. (A.17). This is why the series expansion in terms of solid spherical harmonics Yts is often (incorrectly, however) referred to as multipole expansion. In fact, a one-to-one mapping exists only between the low-order harmonics and multipole potentials. As was mentioned above, the monopole potential is given by Y00 ; three firstorder harmonics Y1−1 , Y10 , and Y11 correspond to the dipoles oriented along
524
Spherical harmonics and related theory
the complex Cartesian coordinate axes ei as in Eq. (A.16). Disagreement starts already from the quadrupoles (6 in total, in contrast to 5 second order harmonics, Y2s ) and extends to the higher order multipoles, which means that some of the multipole potentials are linearly dependent. For more details, see [89]. Under these circumstances, using the complete, linearly independent and orthogonal set of solid spherical harmonics Eq. (A.10) is a self-evident preferable option. Below, we use the “multipole expansion” term in a wide sense, as discussed in Section 1.4.
Fourier integral representation The irregular solid spherical harmonics Yts of Eq. (A.10) are written in the form of double Fourier integral as [6,78] Yts (x) = (∓1)t+s
∞
± ξts Eαβ (x)dα dβ
(x3 ≶ 0),
(A.20)
−∞
where the integral density is ξts = γ t−s−1 (β − iα)s . The harmonic functions ± entering Eq. (A.20) are defined by the formula Eαβ
± Eαβ (x) = exp n± αβ · x = exp (±γ x3 ) Fαβ (x1 , x2 ) ,
(A.21)
where Fαβ (x1 , x2 ) = exp i (α x1 + β x2 ) and n±αβ = iα i1 + iβ i2 ±γ i3 . By anal± can be thought as the scalar solid ogy with Eq. (A.2), the functions Eαβ harmonics for the half-spaces x3 ≶ 0. Note that the Faxen [335] integral representation of the fundamental solution Y00 1 = r
∞
± Eαβ (x)
−∞
γ
dα dβ
(x3 ≶ 0)
(A.22)
is a particular case of Eq. (A.20). ± The functions Eαβ of Eq. (A.21) are regular in the half-spaces x3 ≶ 0 and hence can be expanded into a series over a set of the regular solid spherical harmonics yst of Eq. (A.10) as [6,78] ± Eαβ (x) =
t ∞
κts± (α, β)yst (x),
t=0 s=−t
where the series coefficients κts± are given by the formula κts± = (±1)t+s κts = (±1)t+s γ t−s (β + iα)s .
(A.23)
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Spherical harmonics and related theory
A.2. Reexpansion formulas for Yts and yst Equally oriented coordinate systems Three reexpansion formulas for the scalar solid harmonics are considered. They are singular-to-regular (S2R) Yts (x + X) =
∞ k
(−1)k+l Yts+−kl (X)ylk (x),
x < X ,
(A.24)
k=0 l=−k
regular-to-regular (R2R) yst (x + X) =
t k
yts−−kl (X)ylk (x),
(A.25)
k=0 l=−k
and singular-to-singular (S2S) Yts (x + X) =
∞ k
(−1)t+k+s+l yks−−lt (X)Ykl (x),
x > X
(A.26)
k=t l=−k
reexpansions. In the notations of Eq. (A.10), these formulas take probably the simplest form. The finite formula of Eq. (A.25) is exact and valid for any x and X. The convergence domains of the series in Eqs. (A.24) and (A.26) are shown in Fig. A.2. In [115], Eqs. (A.25) and (A.26) are referred to as a translation of regular and singular solid harmonics, respectively. In [116], they are called a translation of local and multipole expansions whereas Eq. (A.24) is referred to as a conversion of a multipole expansion into a local one. Eqs. (A.24)–(A.26) can be derived in many ways, one of them is based on using Eq. (2.4) of Chapter 2. Probably the most straightforward derivation consists in applying the differentiation rule in Eq. (A.17) to the addition theorem for the fundamental solution [332]. In our notations, this theorem takes a compact form as follows: 1 x − X
= Y00 (x − X) =
⎧ ∞ k ⎪ ⎪ ⎪ (−1)l Yk−l (X)ylk (x), ⎨
x < X ,
⎪ ⎪ ⎪ ⎩
x > X .
k=0 l=−k k ∞
k=0 l=−k
(−1)l yk−l (X)Ykl (x),
(A.27)
526
Spherical harmonics and related theory
Figure A.2 Convergence domains of (S2R) in Eq. (A.24) and (S2S) in Eq. (A.26) reexpansions.
In fact, the upper and lower formulas are identical: their differentiation with respect to X yields Eqs. (A.24) and (A.26), respectively.
Arbitrarily oriented coordinate systems In [160], the transformation formula for the spherical surface harmonics due to rotation Skp (O · n) =
k
k+p
k+p,k+l
k+l (−1)l+p C2k /C2k S2k
(w)Skl (n)
(A.28)
l=−k
is derived. Here, n = x/ x 3 , x is a radius-vector, w = {w1 , w3 , w3 , w4 } is the unit four-dimensional vector ( w = w12 + w22 + w32 + w42 = 1) and Cnm are the binomial coefficients. Also, Skp are the scalar spherical harmonics in a three-dimensional space, and Skp, l are the scalar spherical harmonics in a four-dimensional space [160]: Snk, l (w) = (−1)k (w4 + iw1 )n−k−l (w3 + iw2 )k−l × Pl(n−k−l, k−l) (w32 + w22 − w42 − w12 ),
n ≥ k + l,
527
Spherical harmonics and related theory
Snk, l (w) = (−1)n−l (w4 − iw1 )k+l−n (w3 − iw2 )l−k × Pn(k−+l l−n,l−k) (w32
+ w22
− w42
− w12 ),
(A.29)
n < k + l,
where Pn(α,β) is the Jacobi polynomial [253] Pn(α,β) = 2−n
n
−m Cnm+α Cnn+β (x − 1)n−m (x + 1)m .
m=0
Eq. (A.28) describes the action of an orthogonal transformation O in the three-dimensional space and gives the coefficients of a linear transformation of the functions Skp in terms of spherical harmonics in the four-dimensional space. An explicit form of the matrix O in terms of wi is ⎛
⎞
w22 − w12 − w32 + w42 2(w2 w3 − w1 w4 ) 2(w1 w2 + w3 w4 ) ⎜ ⎟ O = ⎝ 2(w2 w3 + w1 w4 ) w32 − w12 − w22 + w42 2(w1 w3 − w2 w4 ) ⎠ . 2(w1 w2 − w3 w4 ) 2(w1 w3 + w2 w4 ) w12 − w22 − w32 + w42 (A.30) According to [160], Sts (n) = (−1)t+s
2t t! s 2t t! −s Pt (cos θ ) exp(−isϕ) = (−1)t+s χ (θ, ϕ), (t + s)! (t − s)! t
so from Eq. (A.28) we get χks (θ , ϕ ) =
k (k − l)! k−s, k−l S (w)χkl (θ, ϕ). (k − s)! 2k
(A.31)
l=−k
We employ this result to generalize the formulas of Eqs. (A.24)–(A.26) to the case of arbitrarily positioned and oriented reference systems O1 x11 x21 x31 and O2 x12 x22 x32 . It is common knowledge that a general transformation of coordinates can be split into a translation along vector X (system O2 x12 x22 x32 ) and rotation around a fixed origin O2 of the coordinate system, see Fig. A.3. Let transformation O describe the rotation of the coordinate axes from O2 x12 x22 x32 to O2 x12 x22 x32 . This is an orthogonal transformation with determinant equal to +1, hence Eq. (A.28) applies. Consider, for example, Eq. (A.24). We have Yts (x1 ) =
∞ k
k=0 p=−k
s−p
(−1)k+p Yt+k (X)
r k2 (k + p)!
χk (cos θ , ϕ ). p
528
Spherical harmonics and related theory
Figure A.3 Reexpansion: translation plus rotation.
Because n2 = O· n2 and r 2 = r2 are not affected by rotation, we substitute Eq. (A.31) into the last equality to obtain, for x1 = X + O · x2 , Yts (x1 ) =
∞ k
ηtksl (X, O) ylk (x2 ),
(A.32)
k=0 l=−k
where ηtksl (X, O) =
k
(−1)p+l
p=−k
(k − l)!(k + l)! k−p,k−l s−p S (w)Yt+k (X). (k − p)!(k + p)! 2k
An analogous transformation of Eqs. (A.25) and (A.26) gives also yst (x1 ) =
t k
νtksl (X, O) ylk (x2 ),
k=0 l=−k
where νtksl (X, O) =
k
(−1)p+l
p=−k
(k − l)!(k + l)! k−p,k−l s−p S (w)yt−k (X), (k − p)!(k + p)! 2k
and Yts (x1 ) =
∞ k
k=t l=−k
μtksl (X, O) Ykl (x2 ),
(A.33)
529
Spherical harmonics and related theory
where μtksl (X, O) = (−1)t+s
k
k−p,k−l
(−1)k+p S2k
s−p
(w)yk−t (X).
p=−k
The geometrical restrictions for the reexpansions in Eqs. (A.32)–(A.33) are the same as for Eqs. (A.24)–(A.26).
A.3. Scalar spherical biharmonics By analogy with Eqs. (A.1) and (A.2), we expand a general solution of the scalar biharmonic equation ∇ 4 g = 0 over a full and orthogonal on a sphere set of scalar surface harmonics χts from Eq. (A.5). This yields the following sets of regular zst and irregular Zts biharmonic functions: zst (x) =
r2 ys (x) 2(2t + 3) t
Zts (x) = zs−(t+1) (x) = −
(t = 0, 1, . . . ; |s| ≤ t) ,
r2 Y s (x) 2(2t − 1) t
(A.34)
(t = 1, 2, . . . ; |s| ≤ t) .
For the irregular functions, index t starts from 1 because Z00 (x) = r /2 is a regular biharmonic. As expected, ∇ 2 Zts = Yts and ∇ 2 zst = yst . Also, it follows directly from Eq. (A.34) that zt−s (x) = (−1)s zst (x),
Zt−s (x) = (−1)s Zts (x).
The formulas (t − s)(t + s) s Yt−1 , 2t + 1 (t − s)(t − s − 1) s+1 (x1 + ix2 )Yts = −2Zts++11 − Yt−1 , 2t + 1 (t + s)(t + s − 1) s−1 Yt−1 , (x1 − ix2 )Yts = 2Zts+−11 + 2t + 1 (t − s + 1)(t + s + 1) s x3 yst = 2zst−1 + yt+1 , 2t + 1 (t + s + 1(t + s + 2) s+1 +1 (x1 + ix2 )yst = −2zts− yt+1 , 1+ 2t + 1 (t − s + 1)(t − s + 2) s−1 −1 (x1 − ix2 )yst = 2zts− yt+1 , 1− 2t + 1
x3 Yts = −2Zts+1 +
(A.35)
530
Spherical harmonics and related theory
are valid for all indices t and s. Extending the definition in analogy with Eq. (A.19), namely, zs−(t+1) (x) = (−1)t+s Zts (x), transform the formulas for the regular biharmonics in Eq. (A.35) into the formulas for the irregular biharmonics, and vise versa. Note that Eq. (A.34) is not the only choice. Adding an arbitrary harmonic does not alter the biharmonic nature of these functions but can greatly simplify the relevant formulas. Consider, for example, the biharmonics defined as (t − s)(t − s − 1) s Yt−2 (x), 2(2t − 1) (t + s + 1)(t + s + 2) s s s zˇ st (x) = Zˇ −( yt+2 (x). t+1) (x) = zt (x) − 2(2t + 3)
Zˇ ts (x) = Zts (x) +
(A.36)
A motivation of this particular choice is clear from Eq. (A.35): Zˇ ts (x) =
(x1 + ix2 )
2
Yts−−11 ,
zˇ st (x) =
(x1 + ix2 )
2
yts+−11 .
The newly introduced functions of Eq. (A.36) possess remarkable properties. First, we mention the differential rules D1 Zˇ ts = Zˇ ts+−11 − Yts−−11 , D1 zˇ st = zˇ ts−−11 − yts+−11 ,
D2 Zˇ ts = −Zˇ ts++11 , D2 zˇ st = −zˇ ts−+11 ,
D3 Zˇ ts = −Zˇ ts+1 , D3 zˇ st = zˇ st−1 ,
which can be written in the same compact form as Eq. (A.17) for the spherical harmonics:
Dts Zˇ kl (x) = (−1)t Zˇ kl++st (x),
Dts zˇ lk (x) = (−1)s zˇ lk+−st (x).
In particular, for Zˇ 00 = r /2, we get the formula Dts
r
2
= (D2 )s (D3 )t−s
r
2
ˇ ts (x), = (−1)t Z
quite analogous to a classical Hobson’s result [332] for the solid spherical harmonics. The (S2R) reexpansion formula for the functions Eq. (A.36) is Zˇ ts (x + X) =
∞ k
k=0 l=−k
ˇ s−l (X)ylk (x) . (A.37) (−1)k+l Yts+−kl −2 (X)zˇ lk−2 (x) + Z t+k
531
Spherical harmonics and related theory
The other two reexpansions, (R2R) and (S2S), also take similar to Eq. (A.37) simple forms: zˇ st (x) =
t+1 k+1
k=0 l=−k−1
Zˇ ts (x) =
∞ k
l
yts−−kl +1 (X)zˇ k−1 (x) + zˇ ts−−kl (X)ylk (x) ,
ˇ kl (x) + zˇ s−l (X)Ykl (x) . (−1)t+k+s+l yks−−lt (X)Z k−t
k=t−2 l=−k
The functions of Eq. (A.36) will be used in Chapter 5 where their close relationship to the periodic biharmonics [110] will be established.
A.4. Vector spherical surface harmonics The vector surface spherical harmonics S(tsi) (see, e.g., [89]) are defined in terms of their scalar counterpart, χts = χts (θ, ϕ), as
∂ s eϕ ∂ s χ + χ, ∂θ t sin θ ∂ϕ t eθ ∂ s ∂ S(ts2) = r ∇ × er χts = χt − eϕ χts , sin θ ∂ϕ ∂θ
S(ts1) = r ∇ χts = eθ
S(ts3) = er χts
(A.38)
(t ≥ 0, |s| ≤ t).
In order to simplify the subsequent algebra, we deliberately omit here the √ conventional normalizing multiplier t(t + 1). The functions of Eq. (A.38) constitute a complete and orthogonal on a sphere set of vector harmonics. Specifically, 1 S(i) · S(j) dS = αts(i) δtk δsl δij , (A.39) S S ts kl where αts(1) = αts(2) = t(t + 1)αts and αts(3) = αts given by Eq. (A.6). The functions S(tsi) possess remarkable differential, r ∇ · S(ts1) = −t(t + 1)χts , r ∇ × S(ts1) = −S(ts2) ,
∇ · S(ts2) = 0,
r ∇ · S(ts1) = 2χts ,
r ∇ × S(ts2) = S(ts1) + t(t + 1)S(ts3) ,
(A.40)
r ∇ × S(ts3) = S(ts2) ,
and algebraic, er · S(ts1) = 0, (1)
er · S(ts2) = 0, (2)
er × Sts = −Sts ,
(2)
er · S(ts3) = χts , (1)
er × Sts = Sts ,
(A.41) (3)
er × Sts = 0,
532
Spherical harmonics and related theory
properties. In the vector problems, the functions of Eq. (A.38) play the same role as the surface harmonics χts of Eq. (A.5) in the scalar potential problems. We mention the following useful relation: i) s+i−1 (i) Sts , S(t,− s = (−1)
(A.42)
(i) (i) and, by analogy with the scalar case, extend the definition as S−( t+1),s = Sts . With the aid of Eq. (A.40), it is readily proven (see, e.g., [189]) that a transformation of the vector spherical surface harmonics Eq. (A.38) due to a rotation of the coordinate system is given by the formula
t t − l ! t−s,t−l Sts (O · x) = S (w) S(tli) (x) , (t − s)! 2t (i)
(A.43)
l=−t
where Sts,l of Eq. (A.29) are the spherical harmonics in the four-dimensional space [160] and w = {w1 , w3 , w3 , w4 }T of Eq. (A.30) is the unit vector determining uniquely the rotation matrix O. Eq. (A.43) generalizes Eq. (A.28) to the case of vector surface harmonics. In this and all the subsequent analogous formulas, we keep in mind that the vector functions on the opposite sides of an equality are written in the variables and unit vectors of the local (different, in general) coordinate systems.
A.5. Partial solutions of Lamé equation Definition The above-obtained results are sufficient to find a complete set of the partial solutions of Lamé equation, Eq. (1.7) Their expression in terms of vector spherical harmonics from Eq. (A.38) is as follows (see, e.g., [136]): r t−1 (1) 1 rt Sts + tS(ts3) , u(ts2) = − S(2) , (t + s)! (t + 1) (t + s)! ts r t+1 βt (ν)S(ts1) + γt (ν)S(ts3) , u(ts3) = (t + s)!
u(ts1) =
where the coefficients βt (ν) =
t + 5 − 4ν t − 2 + 4ν and γt (ν) = (t + 1)(2t + 3) (2t + 3)
(A.44)
533
Spherical harmonics and related theory
are related by γt + (t + 1)βt ≡ 1. In complex Cartesian projections, u(ts1) = e1 yts−−11 − e2 yts−+11 + e3 yst−1 , (A.45) i u(ts2) = [−e1 (t − s + 1)yts−1 − e2 (t + s + 1)yts+1 + e3 syst ], (t + 1) (t − s + 1)(t − s + 2) s−1 (3) s−1 uts = e1 2zt−1 + εt (ν) yt+1 (t + 1) (2t + 3) (t + s + 1)(t + s + 2) s+1 s+1 + e2 −2zt−1 − εt (ν) yt+1 (t + 1) (2t + 3) (t − s + 1)(t + s + 1) s s + e3 2zt−1 − εt (ν) yt+1 , (t + 1) (2t + 3) where εt (ν) = 3 − 4ν −
1 (2t + 1)
.
The irregular (infinitely growing as r → 0 and vanishing at infinity) (i) functions U(tsi) = u−( t+1), s are given by Eq. (A.44) with t replaced by −(t + 1). Eqs. (A.44) to (A.48) are also valid for U(tsi) with t replaced by −(t + 1). At the spherical surface S : r = R, u(tsj) and U(tsj) are written in compact form as
u(tsj) = S
3
UMtsij (R, ν) S(tsi) ,
i=1
U(tsj) =
3
S
UGtsij (R, ν) S(tsi) ,
(A.46)
i=1
where UMt = UMtij is a (3 × 3) matrix of the form ⎛
1 0 t−1 ⎜ UMt (r , ν) = r ⎝ 0 − t+r 1 t 0
⎞
r 2 βt (ν) ⎟ 0 ⎠ r 2 γt (ν)
(A.47)
and UGt = UGtij = UM−(t+1) .
Explicit expressions The functions u(tsi) (x) obey the differential relations: u(ts1) = ∇ yst , (3)
∇ · uts = 2(2ν
∇ · u(ts1) = ∇ · u(ts2) = ∇ × u(ts1) = 0, − 1)yst ,
(2)
(1)
∇ × uts = −uts ,
(A.48) (3)
∇ × uts = 4(1 − ν)u(ts2) .
The formulas the irregular solutions are the same, with u(tsi) replaced by U(tsi) (x) and yst by Y st . The formulas in Eq. (A.48) are particularly convenient
534
Spherical harmonics and related theory
because the differential operators are invariant of the specific choice of the coordinate frame. The functions with positive and negative s index are related by i) s+i−1 (i) u(t,− uts , s = (−1)
i) s+i−1 (i) U(t,− Uts . s = (−1)
(A.49)
Now, we consider the low order functions in more detail. Note first that, by definition, u(001) = u(002) = 0 and the only zeroth-order regular function is u(003) = γ0 (ν)x. Recall that u(ts3) for t > 0 are biharmonic functions whereas u(003) is the only vector harmonics with constant nonzero divergence ∇ · u(003) = 2(2ν − 1), describing a uniform dilatation (volumetric expansion) of a solid. The next three functions, u(1s1) , represent rigid body translations in three orthogonal directions, along the introduced by Eq. (A.16) complex Cartesian axes: u(101) = e3 ,
u(111) = e1 ,
u(11,−) 1 = −e2 .
The rigid body rotation x × is also described by three functions, this time by u(1s2) . For u = W · x, where W = Wij ii ij is the antisymmetric, or skew (Wij = −Wji ), tensor, one finds
u = 2W12 u(102) + 2W13 i u(112) − u(12,−) 1 + 2W23 u(112) + u(12,−) 1 .
(A.50)
The linear displacement u = E · x, where E = Eij ii ij is the symmetric (Eij = Eji ) tensor, is expressed by means of functions u(2s1) plus the alreadymentioned function u(003) . Specifically, (3) (3) E · x = c00 u00 +
2
(1) (1) c2s u2s ,
(A.51)
s=−2
where (3) c00 =
(E11 + E22 + E33 ) , 3γ0 (ν)
(1) c21 = −c2(1,−) 1 = E13 − iE23 ,
(1) c20 =
(2E33 − E11 − E22 )
3
,
(1) c22 = c2(1,−) 2 = E11 − E22 − 2iE12 .
Eqs. (A.50) and (A.51) provide an expansion of u = A · x where A = E + W is an arbitrary (not necessarily symmetric) tensor.
535
Spherical harmonics and related theory
Normal traction At the spherical surface S : r = R, we have n = er , and so the traction vector Tn = σ · n takes the form 1 1 ν ∂ Tn (u) = er (∇ · u) + u+ er × (∇ × u) . 2μ 1 − 2ν ∂r 2 Application to the vector functions u(tsi) yields [28] 1 1 (t − 1) (1) (t − 1) (2) Tn (u(ts1) ) = Tn (u(ts2) ) = uts , uts , 2μ r 2μ 2r rt
1 Tn (u(ts3) ) = bt (ν)S(ts1) + gt (ν)S(ts3) , 2μ (t + s)!
(A.52)
where bt (ν) = (t + 1)βt − 2(1 − ν)/(t + 1) and gt (ν) = (t + 1)γt − 2ν . Again, Eq. (A.52) holds true for the irregular solutions, with u(tsi) replaced by U(tsi) (x) and t by −(t + 1). In view of Eq. (A.44), a representation of Tn (u(tsi) ) in terms of vector spherical harmonics from Eq. (A.38) is obvious. As expected, Tn ≡ 0 for u(1s1) and u(1s2) representing rigid body translation and rotation, respectively. At the spherical surface S : r = R, Tn (u(tsi) ) and Tn (U(tsi) ) are written in compact form as
3
3
1 Tn (u(tsi) ) = TMtsji (R, ν)S(tsj) , 2μ S j=1 1 Tn (U(tsi) ) = TGtsji (R, ν) S(tsj) , 2μ S j=1
TMt = TMtij being the (3 × 3) matrix ⎞
⎛
t−1 0 ⎜ 0 − 2r ((tt−+11)) TMt (r , ν) = r t−2 ⎝ t(t − 1) 0
r 2 bt (ν) 0 ⎟ ⎠. 2 r gt (ν)
(A.53)
Here, gt and bt are defined by Eq. (A.52); TGt = TGtij = TM−(t+1) .
536
Spherical harmonics and related theory
Net force and torque The net force T and torque M acting on the surface S enclosing the point x = 0 are given by the formulas
T=
Tn dS,
M=
S
x × Tn dS.
(A.54)
S
It is readily found that T = M = 0 for all regular functions u(tsi) . The irregular functions with nonzero net force are
T U(103) = 16μπ(ν − 1)e3 ,
T U(113) = −T U(13,−) 1 = 32μπ(ν − 1)e1 . (A.55)
By analogy with Y00 , U(1s3) can be regarded as the vector monopoles. The net torque is zero for all U(tsi) but U(1s2) for which we get
M U(102) = −8μπ e3 ,
M U(112) = M U(12,−) 1 = −16μπ e1 .
(A.56)
Eqs. (A.55) and (A.56) exhibit the physical meaning of the irregular vector functions U(1s2) and U(1s3) , being the displacements due to point torque and point force, respectively, applied in the point x = 0.
A.6. Partial solutions for a half-space Cartesian vector surface harmonics (i) Following [42], we introduce the Cartesian vector surface harmonics Lαβ :
L(αβ1) (x1 , x2 )= L(αβ2) (x1 , x2 )
=
1 γ
∇ × Fαβ =
1 γ
iα γ
∇ × i3 Fαβ =
i1 +
iβ γ
iβ γ
i2 Fαβ ,
i1 −
iα γ
(A.57)
i2 Fαβ ,
L(αβ3) (x1 , x2 ) = i3 Fαβ , where Fαβ (x1 , x2 ) is the scalar surface (double Fourier) harmonic, see Eq. (2.36). These harmonics obey the orthogonality property
537
Spherical harmonics and related theory
∞
(j) Lα(i)β · Lαβ dx1 dx2 = αi δij δ α − α δ β − β ,
(A.58)
−∞
where α1 = α2 = αα + ββ = γ 2 and α3 = 1. These harmonics are appropriate for fulfilling the boundary conditions on the planes x3 = X3 = const [42]. In view of Eq. (A.58), one can expand an arbitrary, sufficiently smooth vector function f(x) as
f(x) =
∞ −∞
∞
3
−∞ j=1
(j) aj (α, β)Lαβ exp(±γ X3 )dα dβ,
(A.59)
where the integral densities aj (α, β) are given by the inverse double Fourier transform exp(∓γ X3 ) αi
aj (α, β) =
∞ −∞
∞
−∞
(j) f · Lαβ dx1 dx2 .
Vector solutions of Lamé equation for a half-space The vector functions 1
1
2)± ± ± , h(αβ = ), ∇ Eαβ ∇ × (i3 Eαβ (±γ ) (±γ ) ± ± − 4 (1 − ν) i3 Eαβ h(αβ3)± = ∇ x3 Eαβ
h(αβ1)± =
(A.60)
constitute a set of the partial vector solutions of Lamé equation for a halfspace. Note that h(αβ3)± is written in the standard Papkovich–Neuber form (see, e.g., [137]). A general solution of Lamé equation bounded in the homogeneous half-spaces x3 ≶ 0 can be represented by the double Fourier integral
u(x) =
∞
−∞
∞
3
−∞ j=1
(j)± Gj (α, β)hαβ (x)dα dβ
(x3 ≶ 0) ,
(A.61)
where Gi are the integral densities (complex, in general). Fulfilling the displacement boundary conditions on the plane x3 = const conveniently uses an expression of these functions in terms of the Cartesian vector surface harmonics from Eq. (A.57): (j)± hαβ
=
3 i=1
(i)(j)± (i) UZαβ Lαβ exp(±γ x3 ),
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Spherical harmonics and related theory
where
⎛
±1
⎜
(i)(j)± UZ±αβ (x3 ) = UZαβ =⎝ 0 1
⎞
0 γ x3 ⎟ ±1 0 ⎠. 0 (4ν − 3 ± γ x3 )
(A.62)
Note that det UZ±αβ (x3 ) = 4ν − 3 = 0 for all x3 , γ , and 0 < ν < 0.5, guaranteeing invertibility of UZ±αβ matrix. The stress boundary conditions involve the normal traction vector Tn = σ · n taking on the plane x3 = const (for the bottom half-space, an outer normal unit vector n = i3 ) the form 1 1 ν ∂ Tn (u) = i3 (∇ · u) + u + i3 × (∇ × u) . 2μ 1 − 2ν ∂ x3 2
(A.63)
(j)± Its application to hαβ yields
1 (j)± (i)(j)± (i) Tn (hαβ )= TZαβ Lαβ exp(±γ x3 ), 2μ i=1 3
(A.64)
where
(i)(j)± TZ±αβ (x3 ) = TZαβ
⎛
⎞
1 0 (2ν − 1) ± γ x3 ⎜ ⎟ 1 =γ ⎝ 0 0 ⎠. 2 ±1 0 ±2(ν − 1) + γ x3
(A.65)
Note that det γ1 TZ±αβ (x3 ) = ∓ 12 = 0, which means that this matrix is invertible.
Integral transforms and series expansions The double vector Fourier transform for the irregular functions U(tsi) is [78] (i)
t+s
Uts (x) = (∓1)
∞
−∞
∞
3
−∞ j=1
(i)(j)± (j)±
ξtsαβ hαβ (x)dα dβ
(x3 ≶ 0) ,
where ξts(1αβ)(1)± = ±ξt+1,s ,
is t
ξts(2αβ)(1)± = − ξts ,
ξts(2αβ)(2)± = −ξts ,
ξts(3αβ)(3)± = ±ξt−1,s ,
is t
ξts(3αβ)(2)± = ∓4 (1 − ν) ξt−1,s ,
(A.66)
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Spherical harmonics and related theory
ξts(3αβ)(1)± = ±ξt−1,s (3 − 4ν) − C−(t+1),s ,
ξts = γ t−s−1 (β − iα)s .
(j)± The series expansion of hαβ in terms of u(tsi) is
(j)± hαβ (x)
=
t 3 ∞
(±1)t+s κtsαβ u(tsi) (x), (j)(i)±
(A.67)
i=1 t=0 s=−t
where )(1)± κts(1αβ = ±κt−1,s ,
)(2)± κts(2αβ = −κts ,
)(3)± κts(3αβ = ±κt+1,s ,
is is )(2)± κts(3αβ = 4 (1 − ν) κts , t t (3)(1)± t−s κtsαβ = ∓C−(t+1),s κt−1,s , κts = γ (β + iα)s .
)(1)± κts(2αβ = ± κt−1,s ,
A.7. Reexpansion formulas for U(tsi) and u(tsi) Translation The irregular-to-regular (S2R) reexpansion formulas for the irregular vector functions U(tsi) [28] are: U(tsi) (x + X) =
3 ∞ k
(−1)k+l ηtksl (X)ukl (x) (i)(j)
(j)
( x < X ) ,
(A.68)
j=1 k=0 l=−k
where (i)(j) (1)(1) (2)(2) (3)(3) ηtksl = 0 j > l , ηtksl = ηtksl = ηtksl = Yts+−kl , s l (2)(1) (3)(2) (2)(1) ηtksl =i + Y s−l , ηtksl = −4 (1 − ν) ηtksl , t k t+k−1 ls (3)(1) ˇ s−l + Y s−l = 2Z ηtksl t+k t+k−2 4 (1 − ν) 1 + kt − C−(t+1),s − C−(k+1),l − (t + k − s + l) .
(A.69)
The regular-to-regular (S2R) reexpansions of u(tsi) are u(tsi) (x + X) =
+i−j k i t
j=1 k=0 l=−k
(−1)k+l νtksl (X)ukl (x), (i)(j)
(j)
(A.70)
540
Spherical harmonics and related theory
where (i)(j) νtksl = 0 j > l , s (2)(1) νtksl = i − t+1
(1)(1) (2)(2) (3)(3) −l νtksl = νtksl = νtksl = yts− k, l s−l (3)(2) (2)(1) y , νtksl = −4 (1 − ν) νtksl , k t−k+1 l s (3)(1) −l s−l = 2zˇ ts− νtksl k + yt−k+2 4 (1 − ν) 1 − kt+1 − Cts − C−(k+1),l + (t − k + s − l + 1) .
The irregular-to-irregular (S2S) reexpansions of U(tsi) are U(tsi) (x + X) =
k 3 ∞
(−1)t+k+s+l μtksl (X)Ukl (x) (i)(j)
(j)
( x > X ) ,
j=1 k=t−i+jl=−k
(A.71) where (i)(j) 1)(1) 2)(2) 3)(3) μtksl = 0 j > l , μ(tksl = μ(tksl = μ(tksl = yks−−lt , s l 2)(1) 3)(2) 2)(1) μ(tksl =i − ys−l , μ(tksl = −4 (1 − ν) μ(tksl , t k + 1 k−t+1 l s 3)(1) = 2zˇ ks−−lt + yks−−lt+2 4 (1 − ν) 1 − μ(tksl k+1t − C−(t+1),s − Ckl + k − t + s − l + 1 .
In these formulas, Cts = (t + 1)2 − s2 βt . These formulas are the vector counterparts of Eqs. (A.24)–(A.26). Being incorporated into the fast multipole [116] scheme, they provide a very efficient algorithm for elastic interactions in the large-scale clusters of inhomogeneities. For the Stokes interactions in the fluid suspension of spherical particles, such a work has been done in [115].
Rotation The reexpansion formulas of the vector functions u(tsi) and U(tsi) in the case of arbitrarily oriented local coordinate systems can be found by decomposition of the general orthogonal transformation of the coordinates into a sum of parallel translation of vector X and rotation by O around the fixed origin of the coordinate system. The first step (translation) is done already. The
541
Spherical harmonics and related theory
regular vector functions u(tsi) are reexpanded due to rotation as u(tsi) (O · x) =
t (t − l)!(t + l)! t−s,t−l S (w)u(tli) (x). (t − s)!(t + s)! 2t
(A.72)
l=−t
An analogous formula for the irregular vector functions U(tsi) is U(tsi) (O · x) =
t
t−s,t−l S2t (w)U(tli) (x).
(A.73)
l=−t
Both these formulas follow directly from Eq. (A.43), they are of finite form and exact.