Misplaced Pages

#949050

102-460: The model consists of a degree and order 12 spherical harmonic expansion of the magnetic scalar potential of the geomagnetic main field generated in the Earth's core. Apart from the 168 spherical-harmonic "Gauss" coefficients, the model also has an equal number of spherical-harmonic secular variation coefficients predicting the temporal evolution of the field over the upcoming five-year epoch. WMM

204-1183: A (smooth) function f : R 3 → C {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } .) In spherical coordinates this is: ∇ 2 f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ f ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ φ 2 = 0. {\displaystyle \nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.} Consider

306-416: A bounded domain Ω . Then for every non-negative harmonic function u , Harnack's inequality sup V u ≤ C inf V u {\displaystyle \sup _{V}u\leq C\inf _{V}u} holds for some constant C that depends only on V and Ω . The following principle of removal of singularities holds for harmonic functions. If f is a harmonic function defined on

408-661: A dotted open subset Ω ∖ { x 0 } {\displaystyle \Omega \smallsetminus \{x_{0}\}} of ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ , which is less singular at x 0 than the fundamental solution (for n > 2 ), that is f ( x ) = o ( | x − x 0 | 2 − n ) , as  x → x 0 , {\displaystyle f(x)=o\left(\vert x-x_{0}\vert ^{2-n}\right),\qquad {\text{as }}x\to x_{0},} then f extends to

510-758: A general fact about elliptic operators , of which the Laplacian is a major example. The uniform limit of a convergent sequence of harmonic functions is still harmonic. This is true because every continuous function satisfying the mean value property is harmonic. Consider the sequence on ⁠ ( − ∞ , 0 ) × R {\displaystyle (-\infty ,0)\times \mathbb {R} } ⁠ defined by f n ( x , y ) = 1 n exp ⁡ ( n x ) cos ⁡ ( n y ) ; {\textstyle f_{n}(x,y)={\frac {1}{n}}\exp(nx)\cos(ny);} this sequence

612-402: A generalized Dirichlet energy functional (this includes harmonic functions as a special case, a result known as Dirichlet principle ). This kind of harmonic map appears in the theory of minimal surfaces. For example, a curve, that is, a map from an interval in ⁠ R {\displaystyle \mathbb {R} } ⁠ to a Riemannian manifold, is a harmonic map if and only if it

714-504: A harmonic function defines a martingale for the Brownian motion. Then a probabilistic coupling argument finishes the proof. A function (or, more generally, a distribution ) is weakly harmonic if it satisfies Laplace's equation Δ f = 0 {\displaystyle \Delta f=0\,} in a weak sense (or, equivalently, in the sense of distributions). A weakly harmonic function coincides almost everywhere with

816-405: A harmonic function on Ω (compare Riemann's theorem for functions of a complex variable). Theorem : If f is a harmonic function defined on all of ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ which is bounded above or bounded below, then f is constant. (Compare Liouville's theorem for functions of a complex variable ). Edward Nelson gave

918-431: A harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution approaches 0 as r approaches infinity. In this case, uniqueness follows by Liouville's theorem . The singular points of the harmonic functions above are expressed as " charges " and " charge densities " using

1020-1428: A homogeneous polynomial of degree ℓ {\displaystyle \ell } with domain and codomain R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } , which happens to be independent of x 3 {\displaystyle x_{3}} . This polynomial is easily seen to be harmonic. If we write p {\displaystyle p} in spherical coordinates ( r , θ , φ ) {\displaystyle (r,\theta ,\varphi )} and then restrict to r = 1 {\displaystyle r=1} , we obtain p ( θ , φ ) = c sin ⁡ ( θ ) ℓ ( cos ⁡ ( φ ) + i sin ⁡ ( φ ) ) ℓ , {\displaystyle p(\theta ,\varphi )=c\sin(\theta )^{\ell }(\cos(\varphi )+i\sin(\varphi ))^{\ell },} which can be rewritten as p ( θ , φ ) = c ( 1 − cos 2 ⁡ ( θ ) ) ℓ e i ℓ φ . {\displaystyle p(\theta ,\varphi )=c\left({\sqrt {1-\cos ^{2}(\theta )}}\right)^{\ell }e^{i\ell \varphi }.} After using

1122-528: A number of properties typical of holomorphic functions. They are (real) analytic; they have a maximum principle and a mean-value principle; a theorem of removal of singularities as well as a Liouville theorem holds for them in analogy to the corresponding theorems in complex functions theory. Some important properties of harmonic functions can be deduced from Laplace's equation. Harmonic functions are infinitely differentiable in open sets. In fact, harmonic functions are real analytic . Harmonic functions satisfy

SECTION 10

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1224-536: A pair of harmonic conjugate functions). Conversely, any harmonic function u on an open subset Ω of ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠ is locally the real part of a holomorphic function. This is immediately seen observing that, writing z = x + i y , {\displaystyle z=x+iy,} the complex function g ( z ) := u x − i u y {\displaystyle g(z):=u_{x}-iu_{y}}

1326-441: A particularly short proof of this theorem for the case of bounded functions, using the mean value property mentioned above: Given two points, choose two balls with the given points as centers and of equal radius. If the radius is large enough, the two balls will coincide except for an arbitrarily small proportion of their volume. Since f is bounded, the averages of it over the two balls are arbitrarily close, and so f assumes

1428-1012: A product of trigonometric functions , here represented as a complex exponential , and associated Legendre polynomials: Y ℓ m ( θ , φ ) = N e i m φ P ℓ m ( cos ⁡ θ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })} which fulfill r 2 ∇ 2 Y ℓ m ( θ , φ ) = − ℓ ( ℓ + 1 ) Y ℓ m ( θ , φ ) . {\displaystyle r^{2}\nabla ^{2}Y_{\ell }^{m}(\theta ,\varphi )=-\ell (\ell +1)Y_{\ell }^{m}(\theta ,\varphi ).} Here Y ℓ m : S 2 → C {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} }

1530-469: A role in a wide variety of topics including indirect lighting ( ambient occlusion , global illumination , precomputed radiance transfer , etc.) and modelling of 3D shapes. Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his Mécanique Céleste , determined that

1632-459: A strongly harmonic function, and is in particular smooth. A weakly harmonic distribution is precisely the distribution associated to a strongly harmonic function, and so also is smooth. This is Weyl's lemma . There are other weak formulations of Laplace's equation that are often useful. One of which is Dirichlet's principle , representing harmonic functions in the Sobolev space H (Ω) as

1734-460: A wavelength of 51 km as opposed to the 3000 km of WMM. At this resolution, it is not only able to model the Earth's magnetic field at the core-mantle boundary ("main field"), but also take into account magnetic anomalies caused by the minerals in the Earth's crust. Spherical harmonic In mathematics and physical science , spherical harmonics are special functions defined on

1836-870: Is harmonic if Δ p = 0 , {\displaystyle \Delta p=0,} where Δ {\displaystyle \Delta } is the Laplacian . Then for each ℓ {\displaystyle \ell } , we define A ℓ = { harmonic polynomials  R 3 → C  that are homogeneous of degree  ℓ } . {\displaystyle \mathbf {A} _{\ell }=\left\{{\text{harmonic polynomials }}\mathbb {R} ^{3}\to \mathbb {C} {\text{ that are homogeneous of degree }}\ell \right\}.} For example, when ℓ = 1 {\displaystyle \ell =1} , A 1 {\displaystyle \mathbf {A} _{1}}

1938-563: Is homogeneous of degree ℓ {\displaystyle \ell } if p ( λ x ) = λ ℓ p ( x ) {\displaystyle p(\lambda \mathbf {x} )=\lambda ^{\ell }p(\mathbf {x} )} for all real numbers λ ∈ R {\displaystyle \lambda \in \mathbb {R} } and all x ∈ R 3 {\displaystyle x\in \mathbb {R} ^{3}} . We say that p {\displaystyle p}

2040-492: Is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ ( ℓ + 1) for some non-negative integer with ℓ ≥ | m | ; this is also explained below in terms of the orbital angular momentum . Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation , whose solution is a multiple of the associated Legendre polynomial P ℓ (cos θ ) . Finally,

2142-628: Is a geodesic . If M and N are two Riemannian manifolds, then a harmonic map u : M → N {\displaystyle u:M\to N} is defined to be a critical point of the Dirichlet energy D [ u ] = 1 2 ∫ M ‖ d u ‖ 2 d Vol {\displaystyle D[u]={\frac {1}{2}}\int _{M}\left\|du\right\|^{2}\,d\operatorname {Vol} } in which d u : T M → T N {\displaystyle du:TM\to TN}

SECTION 20

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2244-429: Is a linear combination of Y ℓ m : S 2 → C {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } . In fact, for any such solution, r Y ( θ , φ ) is the expression in spherical coordinates of a homogeneous polynomial R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } that

2346-453: Is a positive operator . If Y is a joint eigenfunction of L and L z , then by definition L 2 Y = λ Y L z Y = m Y {\displaystyle {\begin{aligned}\mathbf {L} ^{2}Y&=\lambda Y\\L_{z}Y&=mY\end{aligned}}} for some real numbers m and λ . Here m must in fact be an integer, for Y must be periodic in

2448-1077: Is a "raising operator") and L −  : E λ , m → E λ , m −1 (it is a "lowering operator"). In particular, L +  : E λ , m → E λ , m + k must be zero for k sufficiently large, because the inequality λ ≥ m must hold in each of the nontrivial joint eigenspaces. Let Y ∈ E λ , m be a nonzero joint eigenfunction, and let k be the least integer such that L + k Y = 0. {\displaystyle L_{+}^{k}Y=0.} Then, since L − L + = L 2 − L z 2 − L z {\displaystyle L_{-}L_{+}=\mathbf {L} ^{2}-L_{z}^{2}-L_{z}} it follows that 0 = L − L + k Y = ( λ − ( m + k ) 2 − ( m + k ) ) Y . {\displaystyle 0=L_{-}L_{+}^{k}Y=(\lambda -(m+k)^{2}-(m+k))Y.} Thus λ = ℓ ( ℓ + 1) for

2550-408: Is a complex constant, but because Φ must be a periodic function whose period evenly divides 2 π , m is necessarily an integer and Φ is a linear combination of the complex exponentials e . The solution function Y ( θ , φ ) is regular at the poles of the sphere, where θ = 0, π . Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain

2652-400: Is a harmonic function on U , then all partial derivatives of f are also harmonic functions on U . The Laplace operator Δ and the partial derivative operator will commute on this class of functions. In several ways, the harmonic functions are real analogues to holomorphic functions . All harmonic functions are analytic , that is, they can be locally expressed as power series . This is

2754-674: Is an open subset of ⁠ R n , {\displaystyle \mathbb {R} ^{n},} ⁠ that satisfies Laplace's equation , that is, ∂ 2 f ∂ x 1 2 + ∂ 2 f ∂ x 2 2 + ⋯ + ∂ 2 f ∂ x n 2 = 0 {\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0} everywhere on U . This

2856-411: Is called a spherical harmonic function of degree ℓ and order m , P ℓ m : [ − 1 , 1 ] → R {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } is an associated Legendre polynomial , N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular,

2958-405: Is completely contained in the open set Ω ⊂ R n , {\displaystyle \Omega \subset \mathbb {R} ^{n},} then the value u ( x ) of a harmonic function u : Ω → R {\displaystyle u:\Omega \to \mathbb {R} } at the center of the ball is given by the average value of u on the surface of

3060-623: Is continuous in Ω , u ∗ χ s {\displaystyle u*\chi _{s}} converges to u as s → 0 showing the mean value property for u in Ω . Conversely, if u is any L l o c 1 {\displaystyle L_{\mathrm {loc} }^{1}\;} function satisfying the mean-value property in Ω , that is, u ∗ χ r = u ∗ χ s {\displaystyle u*\chi _{r}=u*\chi _{s}\;} holds in Ω r for all 0 < s < r then, iterating m times

3162-1211: Is conventional in quantum mechanics; it is convenient to work in units in which ħ = 1 . The spherical harmonics are eigenfunctions of the square of the orbital angular momentum L 2 = − r 2 ∇ 2 + ( r ∂ ∂ r + 1 ) r ∂ ∂ r = − 1 sin ⁡ θ ∂ ∂ θ sin ⁡ θ ∂ ∂ θ − 1 sin 2 ⁡ θ ∂ 2 ∂ φ 2 . {\displaystyle {\begin{aligned}\mathbf {L} ^{2}&=-r^{2}\nabla ^{2}+\left(r{\frac {\partial }{\partial r}}+1\right)r{\frac {\partial }{\partial r}}\\&=-{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial }{\partial \theta }}-{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}.\end{aligned}}} Laplace's spherical harmonics are

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3264-399: Is harmonic (see below ), and so counting dimensions shows that there are 2 ℓ + 1 linearly independent such polynomials. The general solution f : R 3 → C {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } to Laplace's equation Δ f = 0 {\displaystyle \Delta f=0} in a ball centered at

3366-515: Is harmonic and converges uniformly to the zero function; however note that the partial derivatives are not uniformly convergent to the zero function (the derivative of the zero function). This example shows the importance of relying on the mean value property and continuity to argue that the limit is harmonic. The real and imaginary part of any holomorphic function yield harmonic functions on ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠ (these are said to be

3468-600: Is harmonic in Ω 0 = Δ u ∗ w r , s = u ∗ Δ w r , s = u ∗ χ r − u ∗ χ s {\displaystyle 0=\Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi _{r}-u*\chi _{s}\;} holds in the set Ω r of all points x in Ω with dist ⁡ ( x , ∂ Ω ) > r . {\displaystyle \operatorname {dist} (x,\partial \Omega )>r.} Since u

3570-499: Is holomorphic in Ω because it satisfies the Cauchy–Riemann equations . Therefore, g locally has a primitive f , and u is the real part of f up to a constant, as u x is the real part of f ′ = g . {\displaystyle f'=g.} Although the above correspondence with holomorphic functions only holds for functions of two real variables, harmonic functions in n variables still enjoy

3672-564: Is independent of x , we denote it merely as vol B R .) In the last expression, we may multiply and divide by vol B r and use the averaging property again, to obtain f ( x ) ≤ vol ⁡ ( B r ) vol ⁡ ( B R ) f ( y ) . {\displaystyle f(x)\leq {\frac {\operatorname {vol} (B_{r})}{\operatorname {vol} (B_{R})}}f(y).} But as R → ∞ , {\displaystyle R\rightarrow \infty ,}

3774-935: Is just the 3-dimensional space of all linear functions R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } , since any such function is automatically harmonic. Meanwhile, when ℓ = 2 {\displaystyle \ell =2} , we have a 5-dimensional space: A 2 = span C ⁡ ( x 1 x 2 , x 1 x 3 , x 2 x 3 , x 1 2 − x 2 2 , x 1 2 − x 3 2 ) . {\displaystyle \mathbf {A} _{2}=\operatorname {span} _{\mathbb {C} }(x_{1}x_{2},\,x_{1}x_{3},\,x_{2}x_{3},\,x_{1}^{2}-x_{2}^{2},\,x_{1}^{2}-x_{3}^{2}).} For any ℓ {\displaystyle \ell } ,

3876-752: Is pre-installed in Android and iOS devices to correct for the magnetic declination . The WMM is produced by the U.S. National Geophysical Data Center (NGDC) in collaboration with the British Geological Survey (BGS). The model, associated software, and documentation are distributed by the NGDC on behalf of National Geospatial-Intelligence Agency (NGA). Updated model coefficients are released at 5-year intervals, with WMM2015 (released Dec 15, 2014) supposed to last until December 31, 2019. However, due to extraordinarily large and erratic movements of

3978-491: Is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series . Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency , as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3) ,

4080-482: Is sometimes used as well, and is named Racah's normalization after Giulio Racah . It can be shown that all of the above normalized spherical harmonic functions satisfy Harmonic function In mathematics , mathematical physics and the theory of stochastic processes , a harmonic function is a twice continuously differentiable function f : U → R , {\displaystyle f\colon U\to \mathbb {R} ,} where U

4182-475: Is subharmonic if and only if, in the interior of any ball in its domain, its graph lies below that of the harmonic function interpolating its boundary values on the ball. One generalization of the study of harmonic functions is the study of harmonic forms on Riemannian manifolds , and it is related to the study of cohomology . Also, it is possible to define harmonic vector-valued functions, or harmonic maps of two Riemannian manifolds, which are critical points of

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4284-2205: Is the Kronecker delta and d Ω = sin( θ ) dφ dθ . This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e., ∫ | Y ℓ m | 2 d Ω = 1. {\displaystyle \int {|Y_{\ell }^{m}|^{2}d\Omega }=1.} The disciplines of geodesy and spectral analysis use Y ℓ m ( θ , φ ) = ( 2 ℓ + 1 ) ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos ⁡ θ ) e i m φ {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{(2\ell +1)}{\frac {(\ell -m)!}{(\ell +m)!}}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }} which possess unit power 1 4 π ∫ θ = 0 π ∫ φ = 0 2 π Y ℓ m Y ℓ ′ m ′ ∗ d Ω = δ ℓ ℓ ′ δ m m ′ . {\displaystyle {\frac {1}{4\pi }}\int _{\theta =0}^{\pi }\int _{\varphi =0}^{2\pi }Y_{\ell }^{m}\,Y_{\ell '}^{m'}{}^{*}d\Omega =\delta _{\ell \ell '}\,\delta _{mm'}.} The magnetics community, in contrast, uses Schmidt semi-normalized harmonics Y ℓ m ( θ , φ ) = ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos ⁡ θ ) e i m φ {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }} which have

4386-432: Is the angle between the vectors x and x 1 . The functions P i : [ − 1 , 1 ] → R {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } are the Legendre polynomials , and they can be derived as a special case of spherical harmonics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent

4488-415: Is the differential of u , and the norm is that induced by the metric on M and that on N on the tensor product bundle T ∗ M ⊗ u − 1 T N . {\displaystyle T^{\ast }M\otimes u^{-1}TN.} Important special cases of harmonic maps between manifolds include minimal surfaces , which are precisely the harmonic immersions of

4590-1595: Is the natural normalization given by Rodrigues' formula. In acoustics , the Laplace spherical harmonics are generally defined as (this is the convention used in this article) Y ℓ m ( θ , φ ) = ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos ⁡ θ ) e i m φ {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{\frac {(2\ell +1)}{4\pi }}{\frac {(\ell -m)!}{(\ell +m)!}}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }} while in quantum mechanics : Y ℓ m ( θ , φ ) = ( − 1 ) m ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos ⁡ θ ) e i m φ {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=(-1)^{m}{\sqrt {{\frac {(2\ell +1)}{4\pi }}{\frac {(\ell -m)!}{(\ell +m)!}}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }} where P ℓ m {\displaystyle P_{\ell }^{m}} are associated Legendre polynomials without

4692-891: Is the standard geomagnetic model of the United States Department of Defense (DoD), the Ministry of Defence (United Kingdom) , the North Atlantic Treaty Organization (NATO), the World Hydrographic Office (WHO) navigation and attitude/heading reference, and the Federal Aviation Administration (FAA). It is also used widely in civilian navigation systems as the magnetic model of the World Geodetic System . For example, WMM

4794-679: Is the volume of the unit ball in n dimensions and σ is the ( n − 1) -dimensional surface measure. Conversely, all locally integrable functions satisfying the (volume) mean-value property are both infinitely differentiable and harmonic. In terms of convolutions , if χ r := 1 | B ( 0 , r ) | χ B ( 0 , r ) = n ω n r n χ B ( 0 , r ) {\displaystyle \chi _{r}:={\frac {1}{|B(0,r)|}}\chi _{B(0,r)}={\frac {n}{\omega _{n}r^{n}}}\chi _{B(0,r)}} denotes

4896-549: Is usually written as ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} or Δ f = 0 {\displaystyle \Delta f=0} The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion . The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics . Fourier analysis involves expanding functions on

4998-771: The f ℓ m ∈ C {\displaystyle f_{\ell }^{m}\in \mathbb {C} } are constants and the factors r Y ℓ are known as ( regular ) solid harmonics R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } . Such an expansion is valid in the ball r < R = 1 lim sup ℓ → ∞ | f ℓ m | 1 / ℓ . {\displaystyle r<R={\frac {1}{\limsup _{\ell \to \infty }|f_{\ell }^{m}|^{{1}/{\ell }}}}.} For r > R {\displaystyle r>R} ,

5100-1032: The Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle . Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see Higher dimensions ). A specific set of spherical harmonics, denoted Y ℓ m ( θ , φ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} or Y ℓ m ( r ) {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functions form an orthogonal system, and are thus basic to

5202-581: The Laplace–Beltrami operator Δ . In this context, a function is called harmonic if   Δ f = 0. {\displaystyle \ \Delta f=0.} Many of the properties of harmonic functions on domains in Euclidean space carry over to this more general setting, including the mean value theorem (over geodesic balls), the maximum principle, and the Harnack inequality. With

SECTION 50

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5304-804: The Taylor series (about r = 0 {\displaystyle r=0} ) used above, to match the terms and find series expansion coefficients f ℓ m ∈ C {\displaystyle f_{\ell }^{m}\in \mathbb {C} } . In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum L = − i ℏ ( x × ∇ ) = L x i + L y j + L z k . {\displaystyle \mathbf {L} =-i\hbar (\mathbf {x} \times \mathbf {\nabla } )=L_{x}\mathbf {i} +L_{y}\mathbf {j} +L_{z}\mathbf {k} .} The ħ

5406-611: The characteristic function of the ball with radius r about the origin, normalized so that ∫ R n χ r d x = 1 , {\textstyle \int _{\mathbb {R} ^{n}}\chi _{r}\,dx=1,} the function u is harmonic on Ω if and only if u ( x ) = u ∗ χ r ( x ) {\displaystyle u(x)=u*\chi _{r}(x)\;} as soon as B ( x , r ) ⊂ Ω . {\displaystyle B(x,r)\subset \Omega .} Sketch of

5508-775: The colatitude θ , or polar angle, ranges from 0 at the North Pole, to π /2 at the Equator, to π at the South Pole, and the longitude φ , or azimuth , may assume all values with 0 ≤ φ < 2 π . For a fixed integer ℓ , every solution Y ( θ , φ ) , Y : S 2 → C {\displaystyle Y:S^{2}\to \mathbb {C} } , of the eigenvalue problem r 2 ∇ 2 Y = − ℓ ( ℓ + 1 ) Y {\displaystyle r^{2}\nabla ^{2}Y=-\ell (\ell +1)Y}

5610-540: The gravitational potential R 3 → R {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } at a point x associated with a set of point masses m i located at points x i was given by V ( x ) = ∑ i m i | x i − x | . {\displaystyle V(\mathbf {x} )=\sum _{i}{\frac {m_{i}}{|\mathbf {x} _{i}-\mathbf {x} |}}.} Each term in

5712-751: The group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics . Despite their name, spherical harmonics take their simplest form in Cartesian coordinates , where they can be defined as homogeneous polynomials of degree ℓ {\displaystyle \ell } in ( x , y , z ) {\displaystyle (x,y,z)} that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses

5814-691: The heat equation and wave equation . This could be achieved by expansion of functions in series of trigonometric functions . Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string , the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials , spherical harmonics also possessed an equivalent form as complex-valued functions. This

5916-744: The m -fold iterated convolution of χ r is of class C m − 1 {\displaystyle C^{m-1}\;} with support B (0, mr ) . Since r and m are arbitrary, u is C ∞ ( Ω ) {\displaystyle C^{\infty }(\Omega )\;} too. Moreover, Δ u ∗ w r , s = u ∗ Δ w r , s = u ∗ χ r − u ∗ χ s = 0 {\displaystyle \Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi _{r}-u*\chi _{s}=0\;} for all 0 < s < r so that Δ u = 0 in Ω by

6018-562: The north magnetic pole , an out-of-cycle update (WMM2015v2) was released in February 2019 (delayed by a few weeks due to the U.S. federal government shutdown ) to accurately model the magnetic field above 55° north latitude until the end of 2019. The next regular update (WMM2020) occurred in December 2019. The Enhanced Magnetic Model (EMM) is a sister product of the NGDC featuring a much higher amount of data to degree and order 790, giving

6120-420: The orientational unit vector r {\displaystyle \mathbf {r} } specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to

6222-406: The raising and lowering operators by L + = L x + i L y L − = L x − i L y {\displaystyle {\begin{aligned}L_{+}&=L_{x}+iL_{y}\\L_{-}&=L_{x}-iL_{y}\end{aligned}}} Then L + and L − commute with L , and

SECTION 60

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6324-564: The weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R : 1 ( 2 π ) 3 / 2 ∫ R 3 | f ( x ) | 2 e − | x | 2 / 2 d x < ∞ . {\displaystyle {\frac {1}{(2\pi )^{3/2}}}\int _{\mathbb {R} ^{3}}|f(x)|^{2}e^{-|x|^{2}/2}\,dx<\infty .} Furthermore, L

6426-771: The Condon–Shortley phase (to avoid counting the phase twice). In both definitions, the spherical harmonics are orthonormal ∫ θ = 0 π ∫ φ = 0 2 π Y ℓ m Y ℓ ′ m ′ ∗ d Ω = δ ℓ ℓ ′ δ m m ′ , {\displaystyle \int _{\theta =0}^{\pi }\int _{\varphi =0}^{2\pi }Y_{\ell }^{m}\,Y_{\ell '}^{m'}{}^{*}\,d\Omega =\delta _{\ell \ell '}\,\delta _{mm'},} where δ ij

6528-652: The Laplace spherical harmonic functions S 2 → C {\displaystyle S^{2}\to \mathbb {C} } . Throughout the section, we use the standard convention that for m > 0 {\displaystyle m>0} (see associated Legendre polynomials ) P ℓ − m = ( − 1 ) m ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m {\displaystyle P_{\ell }^{-m}=(-1)^{m}{\frac {(\ell -m)!}{(\ell +m)!}}P_{\ell }^{m}} which

6630-771: The Lie algebra generated by L + , L − , L z is the special linear Lie algebra of order 2, s l 2 ( C ) {\displaystyle {\mathfrak {sl}}_{2}(\mathbb {C} )} , with commutation relations [ L z , L + ] = L + , [ L z , L − ] = − L − , [ L + , L − ] = 2 L z . {\displaystyle [L_{z},L_{+}]=L_{+},\quad [L_{z},L_{-}]=-L_{-},\quad [L_{+},L_{-}]=2L_{z}.} Thus L +  : E λ , m → E λ , m +1 (it

6732-1013: The above summation is an individual Newtonian potential for a point mass. Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = | x | and r 1 = | x 1 | . He discovered that if r ≤ r 1 then 1 | x 1 − x | = P 0 ( cos ⁡ γ ) 1 r 1 + P 1 ( cos ⁡ γ ) r r 1 2 + P 2 ( cos ⁡ γ ) r 2 r 1 3 + ⋯ {\displaystyle {\frac {1}{|\mathbf {x} _{1}-\mathbf {x} |}}=P_{0}(\cos \gamma ){\frac {1}{r_{1}}}+P_{1}(\cos \gamma ){\frac {r}{r_{1}^{2}}}+P_{2}(\cos \gamma ){\frac {r^{2}}{r_{1}^{3}}}+\cdots } where γ

6834-1163: The angle γ between x 1 and x . (See Legendre polynomials § Applications for more detail.) In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy , and also first introduced the name of "spherical harmonics" for these functions. The solid harmonics were homogeneous polynomial solutions R 3 → R {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } of Laplace's equation ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0. {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.} By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. (See Harmonic polynomial representation .) The term "Laplace's coefficients"

6936-920: The assumption that Y has the form Y ( θ , φ ) = Θ( θ ) Φ( φ ) . Applying separation of variables again to the second equation gives way to the pair of differential equations 1 Φ d 2 Φ d φ 2 = − m 2 {\displaystyle {\frac {1}{\Phi }}{\frac {d^{2}\Phi }{d\varphi ^{2}}}=-m^{2}} λ sin 2 ⁡ θ + sin ⁡ θ Θ d d θ ( sin ⁡ θ d Θ d θ ) = m 2 {\displaystyle \lambda \sin ^{2}\theta +{\frac {\sin \theta }{\Theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)=m^{2}} for some number m . A priori, m

7038-621: The ball; this average value is also equal to the average value of u in the interior of the ball. In other words, u ( x ) = 1 n ω n r n − 1 ∫ ∂ B ( x , r ) u d σ = 1 ω n r n ∫ B ( x , r ) u d V {\displaystyle u(x)={\frac {1}{n\omega _{n}r^{n-1}}}\int _{\partial B(x,r)}u\,d\sigma ={\frac {1}{\omega _{n}r^{n}}}\int _{B(x,r)}u\,dV} where ω n

7140-755: The balls B R ( x ) and B r ( y ) where by the triangle inequality, the first ball is contained in the second. By the averaging property and the monotonicity of the integral, we have f ( x ) = 1 vol ⁡ ( B R ) ∫ B R ( x ) f ( z ) d z ≤ 1 vol ⁡ ( B R ) ∫ B r ( y ) f ( z ) d z . {\displaystyle f(x)={\frac {1}{\operatorname {vol} (B_{R})}}\int _{B_{R}(x)}f(z)\,dz\leq {\frac {1}{\operatorname {vol} (B_{R})}}\int _{B_{r}(y)}f(z)\,dz.} (Note that since vol B R ( x )

7242-463: The complete, orthonormal spherical ket basis . The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } . Specifically, we say that a (complex-valued) polynomial function p : R 3 → C {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} }

7344-546: The convolution with χ r one has: u = u ∗ χ r = u ∗ χ r ∗ ⋯ ∗ χ r , x ∈ Ω m r , {\displaystyle u=u*\chi _{r}=u*\chi _{r}*\cdots *\chi _{r}\,,\qquad x\in \Omega _{mr},} so that u is C m − 1 ( Ω m r ) {\displaystyle C^{m-1}(\Omega _{mr})\;} because

7446-445: The coordinate φ with period a number that evenly divides 2 π . Furthermore, since L 2 = L x 2 + L y 2 + L z 2 {\displaystyle \mathbf {L} ^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}} and each of L x , L y , L z are self-adjoint, it follows that λ ≥ m . Denote this joint eigenspace by E λ , m , and define

7548-561: The equation for R has solutions of the form R ( r ) = A r + B r ; requiring the solution to be regular throughout R forces B = 0 . Here the solution was assumed to have the special form Y ( θ , φ ) = Θ( θ ) Φ( φ ) . For a given value of ℓ , there are 2 ℓ + 1 independent solutions of this form, one for each integer m with − ℓ ≤ m ≤ ℓ . These angular solutions Y ℓ m : S 2 → C {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } are

7650-399: The exception of the mean value theorem, these are easy consequences of the corresponding results for general linear elliptic partial differential equations of the second order. A C function that satisfies Δ f ≥ 0 is called subharmonic. This condition guarantees that the maximum principle will hold, although other properties of harmonic functions may fail. More generally, a function

7752-457: The expansion of a general function on the sphere as alluded to above. Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields , electron configurations , gravitational fields , geoids , the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation . In 3D computer graphics , spherical harmonics play

7854-440: The following maximum principle : if K is a nonempty compact subset of U , then f restricted to K attains its maximum and minimum on the boundary of K . If U is connected , this means that f cannot have local maxima or minima, other than the exceptional case where f is constant . Similar properties can be shown for subharmonic functions . If B ( x , r ) is a ball with center x and radius r which

7956-463: The formula for the associated Legendre polynomial P ℓ ℓ {\displaystyle P_{\ell }^{\ell }} , we may recognize this as the formula for the spherical harmonic Y ℓ ℓ ( θ , φ ) . {\displaystyle Y_{\ell }^{\ell }(\theta ,\varphi ).} (See Special cases .) Several different normalizations are in common use for

8058-516: The fundamental theorem of the calculus of variations, proving the equivalence between harmonicity and mean-value property. This statement of the mean value property can be generalized as follows: If h is any spherically symmetric function supported in B ( x , r ) such that ∫ h = 1 , {\textstyle \int h=1,} then u ( x ) = h ∗ u ( x ) . {\displaystyle u(x)=h*u(x).} In other words, we can take

8160-474: The homogeneity to extract a factor of radial dependence r ℓ {\displaystyle r^{\ell }} from the above-mentioned polynomial of degree ℓ {\displaystyle \ell } ; the remaining factor can be regarded as a function of the spherical angular coordinates θ {\displaystyle \theta } and φ {\displaystyle \varphi } only, or equivalently of

8262-532: The introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a harmonic function ). For example, for any c ∈ C {\displaystyle c\in \mathbb {C} } the formula p ( x 1 , x 2 , x 3 ) = c ( x 1 + i x 2 ) ℓ {\displaystyle p(x_{1},x_{2},x_{3})=c(x_{1}+ix_{2})^{\ell }} defines

8364-702: The joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis: L z = − i ( x ∂ ∂ y − y ∂ ∂ x ) = − i ∂ ∂ φ . {\displaystyle {\begin{aligned}L_{z}&=-i\left(x{\frac {\partial }{\partial y}}-y{\frac {\partial }{\partial x}}\right)\\&=-i{\frac {\partial }{\partial \varphi }}.\end{aligned}}} These operators commute, and are densely defined self-adjoint operators on

8466-1206: The minimizers of the Dirichlet energy integral J ( u ) := ∫ Ω | ∇ u | 2 d x {\displaystyle J(u):=\int _{\Omega }|\nabla u|^{2}\,dx} with respect to local variations, that is, all functions u ∈ H 1 ( Ω ) {\displaystyle u\in H^{1}(\Omega )} such that J ( u ) ≤ J ( u + v ) {\displaystyle J(u)\leq J(u+v)} holds for all v ∈ C c ∞ ( Ω ) , {\displaystyle v\in C_{c}^{\infty }(\Omega ),} or equivalently, for all v ∈ H 0 1 ( Ω ) . {\displaystyle v\in H_{0}^{1}(\Omega ).} Harmonic functions can be defined on an arbitrary Riemannian manifold , using

8568-728: The normalization ∫ θ = 0 π ∫ φ = 0 2 π Y ℓ m Y ℓ ′ m ′ ∗ d Ω = 4 π ( 2 ℓ + 1 ) δ ℓ ℓ ′ δ m m ′ . {\displaystyle \int _{\theta =0}^{\pi }\int _{\varphi =0}^{2\pi }Y_{\ell }^{m}\,Y_{\ell '}^{m'}{}^{*}d\Omega ={\frac {4\pi }{(2\ell +1)}}\delta _{\ell \ell '}\,\delta _{mm'}.} In quantum mechanics this normalization

8670-636: The origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r , f ( r , θ , φ ) = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ f ℓ m r ℓ Y ℓ m ( θ , φ ) , {\displaystyle f(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),} where

8772-561: The original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will yield another harmonic function. Finally, examples of harmonic functions of n variables are: The set of harmonic functions on a given open set U can be seen as the kernel of the Laplace operator Δ and is therefore a vector space over ⁠ R : {\displaystyle \mathbb {R} \!:} ⁠ linear combinations of harmonic functions are again harmonic. If f

8874-415: The positive integer ℓ = m + k . The foregoing has been all worked out in the spherical coordinate representation, ⟨ θ , φ | l m ⟩ = Y l m ( θ , φ ) {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} but may be expressed more abstractly in

8976-1171: The problem of finding solutions of the form f ( r , θ , φ ) = R ( r ) Y ( θ , φ ) . By separation of variables , two differential equations result by imposing Laplace's equation: 1 R d d r ( r 2 d R d r ) = λ , 1 Y 1 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ Y ∂ θ ) + 1 Y 1 sin 2 ⁡ θ ∂ 2 Y ∂ φ 2 = − λ . {\displaystyle {\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=\lambda ,\qquad {\frac {1}{Y}}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y}{\partial \theta }}\right)+{\frac {1}{Y}}{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y}{\partial \varphi ^{2}}}=-\lambda .} The second equation can be simplified under

9078-456: The proof. The proof of the mean-value property of the harmonic functions and its converse follows immediately observing that the non-homogeneous equation, for any 0 < s < r Δ w = χ r − χ s {\displaystyle \Delta w=\chi _{r}-\chi _{s}\;} admits an easy explicit solution w r,s of class C with compact support in B (0, r ) . Thus, if u

9180-529: The quantity vol ⁡ ( B r ) vol ⁡ ( B R ) = ( R + d ( x , y ) ) n R n {\displaystyle {\frac {\operatorname {vol} (B_{r})}{\operatorname {vol} (B_{R})}}={\frac {\left(R+d(x,y)\right)^{n}}{R^{n}}}} tends to 1. Thus, f ( x ) ≤ f ( y ) . {\displaystyle f(x)\leq f(y).} The same argument with

9282-736: The roles of x and y reversed shows that f ( y ) ≤ f ( x ) {\displaystyle f(y)\leq f(x)} , so that f ( x ) = f ( y ) . {\displaystyle f(x)=f(y).} Another proof uses the fact that given a Brownian motion B t in ⁠ R n , {\displaystyle \mathbb {R} ^{n},} ⁠ such that B 0 = x 0 , {\displaystyle B_{0}=x_{0},} we have E [ f ( B t ) ] = f ( x 0 ) {\displaystyle E[f(B_{t})]=f(x_{0})} for all t ≥ 0 . In words, it says that

9384-430: The same value at any two points. The proof can be adapted to the case where the harmonic function f is merely bounded above or below. By adding a constant and possibly multiplying by –1, we may assume that f is non-negative. Then for any two points x and y , and any positive number R , we let r = R + d ( x , y ) . {\displaystyle r=R+d(x,y).} We then consider

9486-549: The solid harmonics with negative powers of r {\displaystyle r} (the irregular solid harmonics R 3 ∖ { 0 } → C {\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} } ) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about r = ∞ {\displaystyle r=\infty } ), instead of

9588-428: The space H ℓ {\displaystyle \mathbf {H} _{\ell }} of spherical harmonics of degree ℓ {\displaystyle \ell } is just the space of restrictions to the sphere S 2 {\displaystyle S^{2}} of the elements of A ℓ {\displaystyle \mathbf {A} _{\ell }} . As suggested in

9690-449: The square of the orbital angular momentum operator − i ℏ r × ∇ , {\displaystyle -i\hbar \mathbf {r} \times \nabla ,} and therefore they represent the different quantized configurations of atomic orbitals . Laplace's equation imposes that the Laplacian of a scalar field f is zero. (Here the scalar field is understood to be complex, i.e. to correspond to

9792-422: The surface of a sphere . They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis , each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This

9894-462: The table below with r 2 = x 2 + y 2 + z 2 : {\displaystyle r^{2}=x^{2}+y^{2}+z^{2}:} Harmonic functions that arise in physics are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions ). On regions without boundaries, adding the real or imaginary part of any entire function will produce

9996-412: The terminology of electrostatics , and so the corresponding harmonic function will be proportional to the electrostatic potential due to these charge distributions. Each function above will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are the images of

10098-439: The unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit n -sphere , one arrives at the spherical harmonics . These functions satisfy Laplace's equation and over time "harmonic" was used to refer to all functions satisfying Laplace's equation. Examples of harmonic functions of two variables are: Examples of harmonic functions of three variables are given in

10200-417: The weighted average of u about a point and recover u ( x ) . In particular, by taking h to be a C function, we can recover the value of u at any point even if we only know how u acts as a distribution . See Weyl's lemma . Let V ⊂ V ¯ ⊂ Ω {\displaystyle V\subset {\overline {V}}\subset \Omega } be a connected set in

10302-453: Was a boon for problems possessing spherical symmetry , such as those of celestial mechanics originally studied by Laplace and Legendre. The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics . The (complex-valued) spherical harmonics S 2 → C {\displaystyle S^{2}\to \mathbb {C} } are eigenfunctions of

10404-411: Was employed by William Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had properly been introduced by Laplace and Legendre. The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of

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