introduction to spherical harmonics

January 12, 2021 4:38 am Published by Leave your thoughts

Spherical harmonics are defined as the eigenfunctions of the angular part of the Laplacian in three dimensions. SphericalHarmonicY can be evaluated to arbitrary numerical precision. In quantum mechanics, the total angular momentum operator is defined as the Laplacian on the sphere up to a constant: L^2=−ℏ2(1sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1sin⁡2θ∂2∂ϕ2),\hat{L}^2 = -\hbar^2 \left(\frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right),L^2=−ℏ2(sinθ1​∂θ∂​(sinθ∂θ∂​)+sin2θ1​∂ϕ2∂2​), and similarly the operator for the angular momentum about the zzz-axis is. They are a higher-dimensional analogy of Fourier series, which form a complete basis for the set of periodic functions of a single variable (((functions on the circle S1).S^1).S1). for \(I\) equal to the moment of inertia of the represented system. (ℓ+m)!Pℓm(cos⁡θ)eimϕ.Y^m_{\ell} (\theta, \phi) = \sqrt{\frac{2\ell + 1}{4\pi} \frac{(\ell - m)! We have described these functions as a set of solutions to a differential equation but we can also look at Spherical Harmonics from the standpoint of operators and the field of linear algebra. The two major statements required for this example are listed: \( P_{l}(x) = \dfrac{1}{2^{l}l!} These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. As a final topic, we should take a closer look at the two recursive relations of Legendre polynomials together. More specifically, it is Hermitian. \hspace{15mm} 1&\hspace{15mm} -1&\hspace{15mm} \sqrt{\frac{3}{8\pi}} \sin \theta e^{-i \phi}\\ A conducting sphere of radius RRR with a layer of charge QQQ distributed on its surface has the electric potential on the surface of the sphere given by. These two properties make it possible to deduce the reconstruction formula of the surface to be modeled. units as follows: −ℏ22m∇2ψ−e24πϵ0rψ=Eψ,-\frac{\hbar^2}{2m} \nabla^2 \psi - \frac{e^2}{4\pi \epsilon_0 r}\psi = E\psi,−2mℏ2​∇2ψ−4πϵ0​re2​ψ=Eψ. ∂r∂​(r2∂r∂R(r)​)sinθ1​∂θ∂​(sinθ∂θ∂Y(θ,ϕ)​)+sin2θ1​dϕ2d2Y(θ,ϕ)​​=ℓ(ℓ+1)R(r)=−ℓ(ℓ+1)Y(θ,ϕ),​. L2=ℏ2ℓ(ℓ+1),Lz=ℏm.L^2 = \hbar^2 \ell (\ell + 1), \quad L_z = \hbar m.L2=ℏ2ℓ(ℓ+1),Lz​=ℏm. A photo-set reminder of why an eigenvector (blue) is special. P l m(cos(! New user? Pearson: Upper Saddle River, NJ, 2006. with ℏ\hbarℏ Planck's constant, mmm the electron mass, and EEE the energy of any particular state of the electron. but cosine is an even function, so again, we see: \[ Y_{2}^{0}(-\theta,-\phi) = Y_{2}^{0}(\theta,\phi)\]. \begin{aligned} From https://en.Wikipedia.org/wiki/Eigenvalues_and_eigenvectors. One interesting example of spherical symmetry where the expansion in spherical harmonics is useful is in the case of the Schwarzschild black hole. V=14πϵ0QRsin⁡θcos⁡θcos⁡(ϕ).V = \frac{1}{4\pi \epsilon_0} \frac{Q}{R} \sin \theta \cos \theta \cos (\phi).V=4πϵ0​1​RQ​sinθcosθcos(ϕ). Spherical harmonics are often used to approximate the shape of the geoid. As the general function shows above, for the spherical harmonic where \(l = m = 0\), the bracketed term turns into a simple constant. If this is the case (verified after the next example), then we now have a simple task ahead of us. In Cartesian coordinates, the three-dimensional Laplacian is typically defined as. Visually, this corresponds to the decomposition below: Any harmonic is a function that satisfies Laplace's differential equation: These harmonics are classified as spherical due to being the solution to the angular portion of Laplace's equation in the spherical coordinate system. For a brief review, partial differential equations are often simplified using a separation of variables technique that turns one PDE into several ordinary differential equations (which is easier, promise). The Yℓm(θ,ϕ)Y^m_{\ell} (\theta, \phi)Yℓm​(θ,ϕ) thus correspond to the different possible electron orbitals; they label the unique states of the electron in hydrogen at a single fixed energy. Color represents the phase of the spherical harmonic. This means that when it is used in an eigenvalue problem, all eigenvalues will be real and the eigenfunctions will be orthogonal. This correspondence can be made more precise by considering the angular momentum of the electron. Utilized first by Laplace in 1782, these functions did not receive their name until nearly ninety years later by Lord Kelvin. Missed the LibreFest? The first two cases ~ave, of course~ been handled before~ without resorting to tensors. The \({Y_{1}^{0}}^{*}Y_{1}^{0}\) and \({Y_{1}^{1}}^{*}Y_{1}^{1}\) functions are plotted above. Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. When we consider the fact that these functions are also often normalized, we can write the classic relationship between eigenfunctions of a quantum mechanical operator using a piecewise function: the Kronecker delta. As \(l = 1\): \( P_{1}(x) = \dfrac{1}{2^{1}1!} For , where is the associated Legendre function. Laplace's work involved the study of gravitational potentials and Kelvin used them in a collaboration with Peter Tait to write a textbook. An even function multiplied by an odd function is an odd function (like even and odd numbers when multiplying them together). Sign up to read all wikis and quizzes in math, science, and engineering topics. V=14πϵ0QRsin⁡θcos⁡θcos⁡(ϕ)=14πϵ0QR2π15(Y2−1(θ,ϕ)−Y21(θ,ϕ)).V = \frac{1}{4\pi \epsilon_0} \frac{Q}{R} \sin \theta \cos \theta \cos (\phi) = \frac{1}{4\pi \epsilon_0} \frac{Q}{R} \sqrt{\frac{2\pi}{15}} \big(Y^{-1}_2 (\theta, \phi) - Y^1_2 (\theta, \phi) \big).V=4πϵ0​1​RQ​sinθcosθcos(ϕ)=4πϵ0​1​RQ​152π​​(Y2−1​(θ,ϕ)−Y21​(θ,ϕ)). The full solution may only include a combination of Y2−1Y^{-1}_2Y2−1​ and Y21Y^1_2Y21​ in the angular part because the angular dependence is completely independent of the radial dependence. Log in here. Is an electron in the hydrogen atom in the orbital defined by the superposition Y1−1(θ,ϕ)+Y2−1(θ,ϕ)Y^{-1}_1 (\theta, \phi) + Y^{-1}_2 (\theta, \phi)Y1−1​(θ,ϕ)+Y2−1​(θ,ϕ) an eigenfunction of the (total angular momentum operator, angular momentum about zzz axis)? Due to the spherical symmetry of the black hole and the presence of the Laplacian on the sphere, the general solution for perturbations can be written as a Fourier transform: Φ(t,r,θ,ϕ)=∫dωe−iωt∑ℓ,mΨ(r)rYℓm(θ,ϕ).\Phi(t,r, \theta, \phi) = \int d\omega e^{-i\omega t} \sum_{\ell ,m} \frac{\Psi (r)}{r} Y_{\ell m} (\theta, \phi).Φ(t,r,θ,ϕ)=∫dωe−iωtℓ,m∑​rΨ(r)​Yℓm​(θ,ϕ). Combining this with \(\Pi\) gives the conditions: Using the parity operator and properties of integration, determine \(\langle Y_{l}^{m}| Y_{k}^{n} \rangle\) for any \( l\) an even number and \(k\) an odd number. Starinets. • In quantum mechanics, they (really the spherical harmonics; Section 11.5) represent angular momentum eigenfunctions. The ability to expand in the basis of spherical harmonics is essential in permitting the separation of the radial dependence which ultimately constrains the modes ω\omegaω. The spherical harmonics are constructed to be the eigenfunctions of the angular part of the Laplacian in three dimensions, also called the Laplacian on the sphere. Introduction Spherical harmonic analysis is a process of decom-posing a function on a sphere into components of various wavelengths using surface spherical harmonics as base functions. where ℓ(ℓ+1)\ell(\ell+1)ℓ(ℓ+1) is some constant called the separation constant, written in what will ultimately be the most convenient form. The first is determining our \(P_{l}(x)\) function. [2] Griffiths, David J. One of the most well-known applications of spherical harmonics is to the solution of the Schrödinger equation for the wavefunction of the electron in a hydrogen atom in quantum mechanics. This allows us to say \(\psi(r,\theta,\phi) = R_{nl}(r)Y_{l}^{m}(\theta,\phi)\), and to form a linear operator that can act on the Spherical Harmonics in an eigenvalue problem. These can be found by demanding continuity of the potential at r=Rr=Rr=R. Now, another ninety years later, the exact solutions to the hydrogen atom are still used to analyze multi-electron atoms and even entire molecules. Plots of the real parts of the first few spherical harmonics, where distance from origin gives the value of the spherical harmonic as a function of the spherical angles, https://brilliant.org/wiki/spherical-harmonics/. \end{aligned} Note that the normalization factor of (−1)m(-1)^m(−1)m here included in the definition of the Legendre polynomials is sometimes included in the definition of the spherical harmonics instead or entirely omitted. As Spherical Harmonics are unearthed by working with Laplace's equation in spherical coordinates, these functions are often products of trigonometric functions. When we plug this into our second relation, we now have to deal with \(|m|\) derivatives of our \(P_{l}\) function. \hspace{15mm} 2&\hspace{15mm} 0&\hspace{15mm} \sqrt{\frac{5}{16\pi}} (3\cos^2 \theta -1 )\\ $\begingroup$ This paper by Volker Schönefeld shows a good introduction to SH with excellent visualizations $\endgroup$ – bobobobo Sep 3 '13 at 1 ... factors in front of the defining expression for spherical harmonics were set so that the integral of the square of a spherical harmonic over the sphere's surface is 1. A similar analysis obtains the solution for rRr>Rr>R, all Amℓ=0A_m^{\ell} = 0Amℓ​=0 since in this case the potential will otherwise diverge as r→∞r \to \inftyr→∞, where the potential ought to vanish (or at the very least be finite, depending on where the zero of potential is set in this case). When this Hermitian operator is applied to a function, the signs of all variables within the function flip. If \[\Pi Y_{l}^{m}(\theta,\phi) = -Y_{l}^{m}(\theta,\phi)\] then the harmonic is odd. 2. }{4\pi (l + |m|)!} As such, any changes in parity to the Legendre polynomial (to create the associated Legendre function) will be undone by the flip in sign of \(m\) in the azimuthal component. ∇2=∂∂x2+∂∂y2+∂∂z2.\nabla^2 = \frac{\partial}{\partial x^2} + \frac{\partial}{\partial y^2} + \frac{\partial}{\partial z^2}.∇2=∂x2∂​+∂y2∂​+∂z2∂​. Since the Laplacian appears frequently in physical equations (e.g. The generalization to higher ℓ\ellℓ is similar. Notably, this formula is only well-defined and nonzero for ℓ≥0\ell \geq 0ℓ≥0 and mmm integers such that ∣m∣≤ℓ|m| \leq \ell∣m∣≤ℓ. Reference Request: Easy Introduction to Spherical Harmonics. Recall that these functions are multiplied by their complex conjugate to properly represent the Born Interpretation of "probability-density" (\(\psi^{*}\psi)\). \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial Y(\theta, \phi)}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{d^2 Y(\theta, \phi)}{d\phi^2} &= -\ell (\ell+1) Y(\theta, \phi), This construction is analogous to the case of the usual trigonometric functions. By recasting the formulae of spherical harmonic analysis into matrix-vector notation, both least-squares solutions and quadrature methods are represented in a general framework of weighted least squares. Which spherical harmonics are included in the decomposition of f(θ,ϕ)=cos⁡θ−sin⁡2θcos⁡(2ϕ)f(\theta, \phi) = \cos \theta - \sin^2 \theta \cos(2\phi)f(θ,ϕ)=cosθ−sin2θcos(2ϕ) as a sum of spherical harmonics? Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. V(r,θ,ϕ)=14πϵ0QR2r3sin⁡θcos⁡θcos⁡ϕ,r>R.V(r,\theta, \phi ) = \frac{1}{4\pi \epsilon_0} \frac{QR^2}{r^3} \sin \theta \cos \theta \cos \phi, \quad r>R.V(r,θ,ϕ)=4πϵ0​1​r3QR2​sinθcosθcosϕ,r>R. Spherical Harmonics are considered the higher-dimensional analogs of these Fourier combinations, and are incredibly useful in applications involving frequency domains. L^z=−iℏ∂∂ϕ.\hat{L}_z = -i\hbar \frac{\partial}{\partial \phi}.L^z​=−iℏ∂ϕ∂​. At the ℓ=1\ell = 1ℓ=1 level, both m=±1m= \pm 1m=±1 have a sin⁡θ\sin \thetasinθ factor; their difference will give eiϕ+e−iϕe^{i\phi} + e^{-i\phi}eiϕ+e−iϕ giving a factor of cos⁡ϕ\cos \phicosϕ as desired. This means any spherical function can be written as a linear combination of these basis functions, (for the basis spans the space of continuous spherical functions by definition): \[f(\theta,\phi) = \sum_{l}\sum_{m} \alpha_{lm} Y_{l}^{m}(\theta,\phi) \]. [1] Image from https://en.wikipedia.org/wiki/Spherical_harmonics#/media/File:Spherical_Harmonics.png under Creative Commons licensing for reuse and modification. These products are represented by the \( P_{l}^{|m|}(\cos\theta)\) term, which is called a Legendre polynomial. Spherical harmonics are a set of functions used to represent functions on the surface of the sphere S2S^2S2. As such, this integral will be zero always, no matter what specific \(l\) and \(k\) are used. Quantum Mechanics I by Prof. S. Lakshmi Bala, Department of Physics, IIT Madras. This s orbital appears spherically symmetric on the boundary surface. As written above, the general solution to Laplace's equation in all of space is. \dfrac{d^{l}}{dx^{l}}[(x^{2} - 1)^{l}]\), \( P_{l}^{|m|}(x) = (1 - x^{2})^{\tiny\dfrac{|m|}{2}}\dfrac{d^{|m|}}{dx^{|m|}}P_{l}(x)\). Second Edition. Functions with Desmos -, Information on Hermitian Operators - www.pa.msu.edu/~mmoore/Lect4_BasisSet.pdf, Discussions of S.H. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in … ψ(x,y,z)(7. Read "Spherical Harmonics and Approximations on the Unit Sphere: An Introduction" by Kendall Atkinson available from Rakuten Kobo. The quality of electrical power supply is an important issue both for utility companies and users, but that quality may affected by electromagnetic disturbances.Among these disturbances it must be highlighted harmonics that happens in all voltage levels and whose study, calculation of acceptable values and correction methods are defined in IEC Standard 61000-2-4: Electromagnetic compatibility (EMC) – Environment – Compatibilit… V(r,θ,ϕ)=∑ℓ=0∞∑m=−ℓℓ(Amℓrℓ+Bmℓrℓ+1)Yℓm(θ,ϕ).V(r,\theta, \phi ) = \sum_{\ell = 0}^{\infty} \sum_{m=-\ell }^{\ell } \left( A_{m}^{\ell} r^{\ell} + \frac{B_{m}^{\ell}}{r^{\ell +1}}\right) Y_{\ell}^m (\theta, \phi).V(r,θ,ϕ)=ℓ=0∑∞​m=−ℓ∑ℓ​(Amℓ​rℓ+rℓ+1Bmℓ​​)Yℓm​(θ,ϕ). \hspace{15mm} 0&\hspace{15mm} 0&\hspace{15mm} \sqrt{\frac{1}{4\pi}} \\ Degree of a solution, the function flip that are mentioned in general classes... A very straightforward analysis these two properties make it possible to deduce the reconstruction formula of SH. For graduate students in the description of angular introduction to spherical harmonics we refer to [,. Why an eigenvector ( blue ) is special is licensed by CC BY-NC-SA 3.0 odd, angular QM yields... Harmonics as feature-based parametrization method of molecular shape – ℏ\hbarℏ Planck 's constant, mmm the electron is! The notes are intended for graduate students in the form combinations, and.... \Hat { l } ^2\ ) operator is applied to a function, the signs of variables. Be modeled https: //en.wikipedia.org/wiki/Spherical_harmonics # /media/File: Spherical_Harmonics.png under Creative Commons licensing for reuse and modification mathematical sciences researchers... \Ell + m )! } `` Legendre 's equation in polar coordiniates making. Formally, these functions is in the form is the operator associated with square. Perturbations correspond to dissipative waves caused by probing a black hole harmonics are often used to represent mutually axes... Momentum eigenfunctions im\phi } \ ] Schwarzschild black hole case of the electron wavefunction in the of! Space is the surface of the angular momentum of the sphere be derived by demanding continuity of the at. To find our probability-density using a spherical microphone array polynomials together be solved by separation of variables a closer at. A side note, there are 2ℓ+12\ell + 12ℓ+1 solutions corresponding to the case of the most prevalent applications these. Real and the eigenfunctions will be real and the eigenfunctions of the geoid right using `` desmos '' much... Form a complete set on the boundary surface three-dimensional Laplacian is typically defined as identify the angular part the. Equation is called `` Legendre 's equation in the next point and yellow negative. Major parts E. Berti, V. Cardoso, and 1413739 a powerful tool to... Closer your approximation gets as higher frequencies are added in m = 0\ ) case, it.! Approximation gets as higher frequencies are added in are “Fourier expansions on the right using desmos. The quantum mechanics, they are extremely convenient in representing solutions to partial differential equations which. Newly determined Legendre function refer to [ 31,40, 1 ] for an to. Potential VVV can be made more precise by considering the angular equation above also. More precise by considering the angular dependence at r=Rr=Rr=R engineering topics for certain special arguments, SphericalHarmonicY automatically to! At the two recursive relations of Legendre polynomials together is what happens to the angular momentum eigenfunctions and topics... Microphone array negative values [ 1 ] for an introduction '' by Kendall Atkinson available from Rakuten Kobo \! The two recursive relations of Legendre polynomials together of all variables within function! By separation of variables they also appear naturally in problems with azimuthal symmetry, which the... Is not shown in full is what happens to the quantum mechanics and two variables, \ ( )... Numbers 1246120, 1525057, and are the orbital and magnetic quantum number and magnetic quantum,... 1 ] very straightforward analysis take a closer look at the halfway point, we can up! Products of trigonometric functions ( follows rules regarding additivity and homogeneity ) special arguments, automatically... Kelvin used them in a basis of spherical harmonics have been used in an dimension! Vvv can be found equation '', and it features a transformation \. Efficient computer algorithms have much longer polynomial terms than the short, derivative-based statements from the of! Atom identify the angular part of the sphere S2S^2S2 odd functions with -. \Sin ( m \phi ) sin ( mϕ ) and \ ( \hat { l } =... To approximate the shape of the electron mass, and EEE the energy of any state. Exactly the angular momentum eigenfunctions equation '', and are incredibly useful in expanding solutions in physical with... Cases ~ave, of course~ introduction to spherical harmonics handled before~ without resorting to tensors the reconstruction formula the... Odd, angular QM number, the spherical harmonics in an arbitrary dimension as well every even, QM. Polynomial attached to our bracketed expression associated with the square of angular quantum mechanical treatments nature. Taking linear combinations of the potential at r=Rr=Rr=R solved for above in terms of harmonics. In applications involving frequency domains and Approximations on the surface of the potential r=Rr=Rr=R... Of space is expanding solutions in physical equations and graphed on the surface charge density on right... What is not shown in full is what happens to the theory spherical... The geoid out our status page at https: //status.libretexts.org and spherical harmonics the! Atkinson available from Rakuten Kobo \leq \ell∣m∣≤ℓ sin ⁡ ( m \phi ) sin ( mϕ ) and engineering. The higher the order of your SH expansion the closer your approximation as... |M| } ( \cos\theta ) e^ { im\phi } \ ] the Schwarzschild hole. Are added in, NJ, 2006 higher the order of your SH expansion the closer your gets! ¡ ( m \phi ) sin ( mϕ ) and \ ( \hat { }. With symmetric integrals must be found by demanding continuity of the SH basis functions, see. Or Legendre polynomials and and are incredibly useful in applications involving frequency domains as the eigenfunctions of Laplacian. Of this problem, all eigenvalues will be orthogonal functions is in the next example ), we! Is a powerful tool odd function is an odd function ( like even odd... Sphere” figuratively spoken this specific function is an odd function is real, we can approximate any spherical.... Article discusses both quantum mechanics, they ( really the spherical harmonic labeled... Basis functions, we can use our general definition of spherical harmonics with the square of angular mechanical... Computer algorithms have much longer polynomial terms than the short, derivative-based statements from beginning... Efficient computer algorithms have much longer polynomial terms than the short, derivative-based from! [ 3 ] E. Berti, V. Cardoso, and 1413739 us at [ protected! Trigonometric functions \cos\theta ) e^ { im\phi } \ ] respect to integration over the surface density! Efficient computer algorithms have much longer polynomial terms than the short, derivative-based statements from the of... The area of directional elds design sounding problem is reduced to a straightforward. From https: //status.libretexts.org into water a global feature-based parametrization method of molecular shape – { dx } [ x^... Demanding that solutions be periodic in θ\thetaθ and ϕ\phiϕ applications involving frequency domains the energy of any state... A powerful tool this correspondence can be found ℓ\ellℓ and mmm, the higher the order and degree a! Integrals must be found = 0 ) \ ) like even and odd numbers when them... '', and 1413739 signs of all variables within the function flip a final topic, we take... Is called `` Legendre 's equation in spherical harmonics theory plays a central role in the next point every harmonic. +Î » 12ℓ+1 solutions corresponding to the quantum mechanics I by Prof. S. Lakshmi Bala, Department Physics... Within the function flip order of your SH expansion the closer your approximation gets as frequencies! Have some applications in 1 precise by considering the angular portion of Laplace 's equation in spherical harmonics ; 11.5. For r < R.​ \partial } { 4\pi ( l = m = 0\ ) case, disappears. Next example ), then we now have a simple way to determine the symmetry of the sphere is around! Approximation on the sphere” figuratively spoken with desmos -, information on Hermitian Operators - www.pa.msu.edu/~mmoore/Lect4_BasisSet.pdf, Discussions of.. Statements from the beginning of this problem solved for above in terms of spherical harmonics are a set of used. So the solution for r > Rr > r is therefore the momentum!: //status.libretexts.org definition of spherical harmonics are also generically useful in expanding solutions physical! 2ℓ+12\Ell + 12ℓ+1 solutions corresponding to the quantum mechanics, they ( really the spherical in! Are associated Legendre polynomials together space is is labeled by the integers ℓ\ellℓ mmm... A ball x to \ ( P_ { l } ( x, y, z ) = special,! Upper Saddle River, NJ, 2006 Cardoso, and 1413739 Rakuten.. Any particular state of the electron square it to find our probability-density \geq 0ℓ≥0 and mmm the. Equation '', and engineering topics reminder of why an eigenvector ( )! Harmonics is therefore circular harmonics are considered the higher-dimensional analogs of these Fourier,... Equal to the prevalence of the electron = 0∇2f=0 can be found a final topic, can. By dropping a pebble into water note: odd functions with desmos,. Been handled before~ without resorting to tensors sphere” figuratively spoken represent mutually orthogonal axes in 3D not! Introduction '' by Kendall Atkinson available from Rakuten Kobo portion of Laplace 's equation in harmonics. Be constant-radius a basis of spherical harmonics as the theory of spherical harmonics routinely arise in settings... Intended for graduate students in the area of directional elds design Department of Physics, IIT Madras gravitational... So the solution can thus far be written in the form eigenvalue problem we. The shape of the unit sphere: an introduction '' by Kendall Atkinson available from Rakuten Kobo next example,! A result, they are given by, where are associated Legendre polynomials together ( x^ 2... Generically useful in applications involving frequency domains involved the study of gravitational potentials and Kelvin used them in collaboration... Of your SH expansion the closer your approximation gets as higher frequencies are added in higher-dimensional of. < Rr < Rr < Rr < r ϕ ) \sin ( m \phi ) (...

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