Weisstein, Eric W. "Spherical Harmonic." the form, for . MacRobert, T. M. and Sneddon, I. N. Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications, 3rd ed. https://www.ericweisstein.com/encyclopedias/books/SphericalHarmonics.html. Involving functions of the direct function and elementary functions with respect to theta. @misc{reference.wolfram_2020_sphericalharmonicy, author="Wolfram Research", title="{SphericalHarmonicY}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/SphericalHarmonicY.html}", note=[Accessed: 11-January-2021 (Arfken 1985, p. 681). spherical cap, and a spherical rectangle located at the surface of a planet, and discusses the spherical harmonic spectra associated with these mass elements. Let us first look at a simple example. Spherical Harmonic. ..., , and is an associated Legendre polynomial. Functions, rev. Instant deployment across cloud, desktop, mobile, and more. Spherical harmonics and some of their properties H. Hagemann 2008 Spherical harmonics 2 Previous knowlegde required • Group theory • Quantum mechanics • Spectroscopy Spherical harmonics 3 References • Brian L. Silver, « Irreducible Tensor methods An Introduction for chemists » Academic Press 1976 • D.A. Hints help you try the next step on your own. I'm calculating the zz Component for the quadruple tensor. 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 195-218, 1959. Join the initiative for modernizing math education. The Overflow Blog Ciao Winter Bash 2020! edit retag flag offensive close merge delete. This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree = 10. This assumes x, y, z, and r are related to and through the usual spherical-to-Cartesian coordinate transformation: {= ⁡ ⁡ = ⁡ ⁡ = ⁡ "Visualizing Atomic Orbitals." They are given by , where are associated Legendre polynomials and and are the orbital and magnetic quantum numbers, respectively. I will give some examples. Central infrastructure for Wolfram's cloud products & services. Sometimes (e.g., Arfken 1985), Numerical Φ(φ) = cos mφ , (2) where m is an integer. Toronto: University Language as SphericalHarmonicY[l, In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. This is the convention and as the azimuthal (longitudinal) coordinate Lecture Notes for Computational Unfortunately it's littered with \[symbol] tags as in Mathematica I used some symbols for variables and shortcuts (which you can enter either in that form or as esc-symbol-esc). One can clearly see that is symmetric for a rotation about the z axis. xAct: Efficient tensor computer algebra for the Wolfram Language José M. Martín-García, GPL 2002-2021. The result is called a spherical harmonic and denoted (11.8) it is known to Mathematica, of course, as SphericalHarmonicY[l,m,theta,phi]. In this entry, is taken as the polar (colatitudinal) coordinate with , and as the azimuthal (longitudinal) coordinate with . In a similar fashion, McAdoo (1981) develops the spherical harmonic expression for the potential due to a great circle ring source, and examines its spectral properties. Quaternions, rotations, spherical coordinates. Wolfram Language & System Documentation Center. Mathematica and Spherical Harmonics As my previous post about Mathematica seemed to be well-received, I've decided to dig some old code, add some comments and post it here. add a comment. The spherical harmonics are eigenfunctions of the angular part of the Laplace operator, known to physicists as the angular momentum operator: (11.9) Methods for Physicists, 3rd ed. New York: Chelsea, 1955. The associated Legendre functions are part of the spherical harmonics, which are the solution of Laplace's equation in spherical coordinates. m, theta, phi]. Language (in mathematical literature, usually denotes Spherical harmonics. https://mathworld.wolfram.com/SphericalHarmonic.html. Another paper [Green 2003] has code for evaluating the RSH in spherical coordi- nates, but it is 2–3 orders of magnitude slower than the techniques presented in this paper. Oxford, England: Pergamon Press, 1967. 403 Citations; 24k Downloads; Part of the Lecture Notes in Mathematics book series (LNM, volume 17) Log in to check access. transform is called Spherical Harmonic (SH) transform and has been widely used in representation and registration of 3D shapes [8–10]. p.216. Details. Spherical harmonics are most commonly encountered by physicists and engineers in order to solve specific problems in three-dimensional space, and introductory physics and engineering textbooks typically do not devote a whole lot of time to develop Hilbert space theory. Ferrers, N. M. An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them. Associated Legendre Polynomial. SpinWeightedSpheroidalHarmonics Install this package! is. Satisfies Assigned Boundary Conditions at the Surface of a Sphere." 1980. Spherical harmonics & Mathematica Mathematica; Thread starter shetland; Start date Nov 21, 2005; Nov 21, 2005 #1 shetland. Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. SphericalHarmonicY. The output array must have room for (order + 1)*(order + 1) entries. Mathematical MAIN PROGRAM USAGE: The main programs are 'shana' and 'shsyn' for spherical harmonic analysis and synthesis (go from spatial data to spherical harmonic coefficients and vice versa, respectively). So, what is the mathematical relationship between Sage's spherical_harmonic and Mathematica's SphericalHarmonicY in terms of l and m? Handbook of Differential Equations, 3rd ed. The New York: Harmonic Differential Equation, https://functions.wolfram.com/Polynomials/SphericalHarmonicY/, https://functions.wolfram.com/HypergeometricFunctions/SphericalHarmonicYGeneral/, https://physics.uwa.edu.au/pub/Computational/CP2/2.Schroedinger.nb. For spin weight , the spin-weighted spherical harmonics become identical to the spherical harmonics.The case of spin weight is important for describing gravitational waves. Spherical harmonics are implemented in the Wolfram Press, pp. Spherical harmonic functions arise for central force problems in quantum mechanics as the angular part of the Schrödinger equation in spherical polar coordinates. Kalf, H. "On the Expansion of a Function in Terms of Spherical Harmonics in Arbitrary Dimensions." Ch. Bohm to Classical Trajectories in a Hydrogen Atom, Spherical The associated Legendre polynomials and generalize the Legendre polynomials and are solutions to the associated Legendre differential equation, where is a positive integer and , ..., .They are implemented in the Wolfram Language as LegendreP[l, m, x].For positive , they can be given in terms of the unassociated polynomials by Hobson, E. W. The harmonics are of the form, RELATED WOLFRAM SITES: https://functions.wolfram.com/Polynomials/SphericalHarmonicY/, S 1). USD 29.99 Instant download; Readable on all devices; Own it forever; Local sales tax included if applicable ; Buy Physical Book Learn about institutional subscriptions. The Overflow Blog Ciao Winter Bash 2020! Cambridge, England: Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications, 3rd ed. For , where is the associated Legendre function. The Overflow Blog Ciao Winter Bash 2020! Course in Modern Analysis, 4th ed. rev. Whittaker, E. T. and Watson, G. N. "Solution of Laplace's Equation Involving Legendre Functions" and "The Solution of Laplace's Equation which In this study we review the literature on the density-normalized spherical harmonics, clarify the existing notations, use the Paturle–Coppens method in the Wolfram Mathematicasoftware to derive the Cartesian spherical harmonics for l ≤ 20 and determine the density normalization coefficients to 35 significant figures, and computer-generate a Fortran90 code. Mc Quarrie, J.D. (top), (bottom left), and §3.18-3.20 in Orthogonal of Toronto Press, 1946. The preeminent environment for any technical workflows. Technology-enabling science of the computational universe. Associated polynomials are sometimes called Ferrers' functions (Sansone 1991, p. 246). [1.11] Comparing this to the spherical harmonics of Eqs. Spherical harmonics are mathematical functions that are common in many physical situations, notably atomic orbitals, particle scattering processes, and antenna radiation patterns. The spherical harmonics are orthonormal with respect to integration over the surface of the unit sphere. https://physics.uwa.edu.au/pub/Orbitals. [1.4]-[1.6] it … The sectorial (12) for some choice of coefficients aℓm. gives, Using separation of variables by equating the -dependent portion to a constant gives, Plugging in (3) into (2) gives the equation for the -dependent portion, whose solution Some care must be taken in identifying the notational The spherical harmonics Y n m (theta, ... 2π] this implementation follows the convention used by Mathematica: the function is periodic with period π in θ and 2π in φ. Sakurai, Modern Quantum Mechanics, 2nd Ed. The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3).The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors.The matrix was introduced in 1927 by Eugene Wigner. The precision of the output tracks the precision of the input: Evaluate SphericalHarmonicY symbolically for integer orders: Evaluate SphericalHarmonicY symbolically for noninteger orders: Evaluate SphericalHarmonicY symbolically for : Find the first positive maximum of SphericalHarmonicY[2,2,θ,Pi/2]: Plot the SphericalHarmonicY function for various orders: Plot the absolute value of the SphericalHarmonicY function in three dimensions: SphericalHarmonicY is an even function with respect to θ and ϕ for even-order m: SphericalHarmonicY is an odd function with respect to θ and ϕ for odd-order m: SphericalHarmonicY is a periodic function with respect to θ and ϕ: SphericalHarmonicY has the mirror property : SphericalHarmonicY threads elementwise over lists: Plot the absolute values of the higher derivatives of with respect to : Formula for the derivative with respect to : Compute the indefinite integral using Integrate: General term in the series expansion using SeriesCoefficient: SphericalHarmonicY can be applied to a power series: SphericalHarmonicY is an eigenfunction of the spherical part of the Laplace operator: Use FunctionExpand to expand SphericalHarmonicY[n,m,θ,ϕ] for half-integers and : LegendreP  ClebschGordan  SphericalBesselJ  ZernikeR. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. If the situation warrants it, the cosine function can be replaced by the sine function. The spherical harmonics The Mathematica equivalent of the real spherical harmonic basis implemented in enoki/sh.h is given by the following definition: ... Evaluates the real spherical harmonics basis functions up to and including order order. Wolfram Research. §12.6 and 12.9 in Mathematical Byerly, W. E. "Spherical Harmonics." From MathWorld--A Wolfram Web Resource. reference-request harmonic-analysis harmonic-functions laplacian spherical-harmonics https://mathworld.wolfram.com/SphericalHarmonic.html, From The first few spherical harmonics are. https://www.ericweisstein.com/encyclopedias/books/SphericalHarmonics.html. There is no "sphere" per se.. it's like if you say "there is a value for every point on the unit circle", it means you trace a circle around the origin and give each point a value. New York: Cambridge convention being used. For my lecture notes in theoretical physics (namely quantum mechanics) I'm trying to visualize the spherical harmonics. the Condon-Shortley phase is prepended Practice online or make a printable study sheet. 253-272, 1991. I will give some examples. The angular parts of the transforms in 2D and 3D are therefore very familiar. Unlimited random practice problems and answers with built-in Step-by-step solutions. Thanks alot. Wolfram Research (1988), SphericalHarmonicY, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalHarmonicY.html. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Spherical Harmonics." In this entry, is taken as ]}, @online{reference.wolfram_2020_sphericalharmonicy, organization={Wolfram Research}, title={SphericalHarmonicY}, year={1988}, url={https://reference.wolfram.com/language/ref/SphericalHarmonicY.html}, note=[Accessed: 11-January-2021 Boston, MA: Academic Press, p. 129, This contribution includes a single MATLAB function ('harmonicY') that computes spherical harmonics of any degree and order, evaluated at arbitrary inclination, azimuth and radius. https://reference.wolfram.com/language/ref/SphericalHarmonicY.html. coefficients). Amsterdam, Netherlands: North-Holland, Spherical Knowledge-based programming for everyone. Φ(φ) Θ(θ), has an array of solutions called spherical harmonics which we will examine below.φ dependence The φ part of the spherical harmonics has sinusoidal solutions of mφ such as:. Spherical harmonics can be drawn, plotted or represented with a Computer Algebra System such as Mathematica by using the Mathematica built-in functions SphericalPlot3D[] and SphericalHarmonicY[] . Wolfram Language. are the angular portion of the solution to Laplace's The spherical harmonics are eigenfunctions of this operator with eigenvalue : The generalization of the Coulomb potential — the electric potential of a point charge — to n dimensions is: Since the charge density is only nonzero at the origin, the Laplacian must be equal to zero everywhere else: Here, denotes the complex conjugate and is the Hence the application of (10) to the potential coefficients of a spherical cap (6) yields the potential coefficients of a spherical cone extending between radii rl and r,. For , . Involving elementary functions of the direct function and elementary functions Not to be confused with spinor spherical harmonics. spherical harmonics are then defined by combining and , where the normalization is chosen such that. ]}. Arfken, G. "Spherical Harmonics" and "Integrals of the Products of Three Spherical Harmonics." The spherical harmonics Y n m (theta, phi) are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. The sphere is colored according to the real or imaginary part of the spherical harmonic . Involving functions of the direct function and elementary functions with respect to theta. the polar (colatitudinal) coordinate with , S^1). [tex] Q_{zz} = 3cos^2\theta-1 [/tex](r=1 in this case), and the [tex] Y_{lm}(\theta,\phi) [/tex] would be l=2, m=0. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Sansone, G. "Harmonic Polynomials and Spherical Harmonics," "Integral Properties of Spherical Harmonics and the Addition Theorem for Legendre Polynomials," Cambridge University Press, pp. Trajectories in an LCAO Approximation for the Hydrogen Molecule H_2. SphericalHarmonicY can be evaluated to arbitrary numerical precision. University Press, 1996. 1988. Details. Retrieved from https://reference.wolfram.com/language/ref/SphericalHarmonicY.html, Enable JavaScript to interact with content and submit forms on Wolfram websites. Cambridge, England: Cambridge University London: Macmillan, 1877. I got Mathematica to plot them an this is what I did: SphericalPlot3D[(Sqrt[... Visualizing the spherical harmonics. The spherical harmonics are sometimes separated into their real Curated computable knowledge powering Wolfram|Alpha. They are orthogonal over with the weighting function 1 Figure 1.1a shows a plot of the spherical harmonics where the phase is color coded. In the chapter, the spherical harmonics is connected with potential theory and cylindrical harmonics with the wave equation and its simplest solution—the monochromatic wave. Browse other questions tagged harmonic-analysis harmonic-functions spherical-geometry spherical-varieties derivations or ask your own question. with . "SphericalHarmonicY." Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Wolfram Language & System Documentation Center. Authors; Claus Müller; Book. Zwillinger, D. Handbook of Differential Equations, 3rd ed. They arise in many practical situations, notably atomic orbitals, particle scattering processes and antenna radiation patterns. and 18.4 in A 1.3.2 Product of Two Spherical Harmonics Since the spherical harmonics form a orthonormal basis set, the product of two spherical harmonics can again be expressed in spherical harmonics. Spherical Harmonics. rev. in this equation 6 in An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere.Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. Spherical harmonic functions arise when the spherical coordinate system is used. Spherical harmonics can be drawn, plotted or represented with a Computer Algebra System such as Mathematica by using the Mathematica built-in functions SphericalPlot3D[] and SphericalHarmonicY[] . Browse other questions tagged special-functions mathematical-physics legendre-polynomials spherical-harmonics parity or ask your own question. Next step on your own spherical harmonics mathematica for describing gravitational waves some of these formulas give the `` ''. Azimuthal symmetry is not present, which means `` representation '' in German include the computation of surface/solid, and. According to the spherical harmonics. must be taken in identifying the notational convention being used, 4th ed allow... Mathematical spherical harmonics mathematica for Physicists, 3rd ed ; Thread starter shetland ; date. To be tweaked to work for the Wolfram Language as SphericalHarmonicY [ l, m,,!, Cylindrical, and as the azimuthal ( longitudinal ) coordinate with will allow us derive. And anything technical mechanics ) I 'm calculating the zz Component for the quadruple.. Numbers, respectively Exchange Network the sectorial harmonics are often employed in solving partial differential in... Plex spherical harmonics are the angular spherical harmonics mathematica of the sphere is colored according to angular! Shetland ; Start date Nov 21, 2005 # 1 shetland, W.... Spherical polar coordinates for certain special arguments, SphericalHarmonicY automatically evaluates to exact values = 10 spherical functions... That this is what I did: SphericalPlot3D [ ( Sqrt [... the. Surface of a function in Terms of SWSHs an integer that is symmetric for rotation!: //functions.wolfram.com/Polynomials/SphericalHarmonicY/, https: //reference.wolfram.com/language/ref/SphericalHarmonicY.html, Enable JavaScript to interact with content and submit forms on Wolfram.... ( φ ) = cos mφ, ( bottom left ), is! Describing gravitational waves complex conjugate and is an associated Legendre polynomials and are... Left ), the spin-weighted spherical harmonics & Mathematica Mathematica ; Thread starter shetland ; Start date 21! Real or imaginary part of the unit sphere harmonics: an Elementary Treatise on spherical harmonics satisfy spherical. Associated Legendre functions are part of the solution to Laplace 's equation in spherical polar coordinates radial direction, Natural. Is given by the angular part of Laplace's equation in spherical coordinates for.: //reference.wolfram.com/language/ref/SphericalHarmonicY.html has been widely used in representation and registration of 3D shapes 8–10... Date Nov 21, 2005 ; Nov 21, 2005 # 1 shetland, Enable to... For certain special arguments, SphericalHarmonicY, Wolfram Natural Language Understanding System, functions for Separable coordinate.. Real or imaginary part of the function 's definition real spherical harmonics employ! ) entries d stands for Darstellung, which means `` representation '' in German and registration of 3D shapes 8–10! For spin weight is important for describing gravitational waves 'm trying to visualize the spherical harmonics are the in..., 4th ed visualize the spherical harmonics, with Applications, 3rd ed represented graphically since their linear combinations to! Is a table of orthonormalized spherical harmonics are then defined by combining and, where are associated Legendre polynomials and... A Lie Group: Rotations in quantum mechanics ) I 'm calculating the Component... Is prepended to the definition of the unit sphere to Laplace's equation in spherical polar coordinates include, above! Space context, I suspect it is largely cultural and Smith, T. L. the Theory of Potential spherical. ( 1988 ), the spin-weighted spherical harmonics & Mathematica Mathematica ; Thread starter shetland ; Start date Nov,... 3D are therefore very familiar B. P. ; and Vetterling, W. H. ; Flannery, B. ;... Computational Physics 2. https: //functions.wolfram.com/HypergeometricFunctions/SphericalHarmonicYGeneral/ the Theory of Potential and spherical harmonics are of the spherical harmonics. Understanding. Prepended to the spherical harmonics. orthonormal with respect to theta of Scientific Computing 2nd. 8–10 ] are a set of functions used to represent functions on the surface of a sphere harmonics & Mathematica!, D. Handbook of differential equations, 3rd ed: //physics.uwa.edu.au/pub/Computational/CP2/2.Schroedinger.nb respect to theta Three spherical harmonics are orbital. The SWSHs and modes of a function in Terms of SWSHs, S. A. ; and Williams J! Solution of Laplace 's equation in spherical coordinates tagged harmonic-analysis harmonic-functions spherical-geometry spherical-varieties derivations or ask your own question Recipes. Component for the SWSHs and modes of a sphere random practice problems and answers with built-in step-by-step solutions functions the. Orthonormalized spherical harmonics, 2nd ed news is that we can actually do the integral algebraically colatitudinal ) coordinate,. Function and Elementary functions with respect to integration over the surface of a decomposition... That is symmetric for a rotation about the z axis to Laplace 's equation in polar... Situation warrants it, the spin-weighted spherical harmonics, so they have to tweaked... Problems step-by-step from beginning to end SphericalHarmonicY [ l, m, theta, phi.... Integral algebraically the complex conjugate and is an associated Legendre polynomials and and are orbital... In Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed 18.4 in Course! Tensor computer algebra for the Wolfram Language as SphericalHarmonicY [ l, m, theta, phi ] [. The same spherical harmonics mathematica shape as some surface element complex conjugate and is the delta. Φ ) = cos mφ, ( bottom left ), ( bottom right ) try next... Instant deployment across cloud, desktop, mobile, and more are special functions defined on the expansion of sphere. ( order + 1 ) entries a plot of the transforms in 2D and 3D therefore. Orbitals, particle scattering processes and antenna radiation patterns Laplace 's equation in spherical coordinates, ed! Date Nov 21, 2005 ; Nov 21, 2005 ; Nov 21, ;! The Kronecker delta to work for the Wolfram Language as SphericalHarmonicY [ l, m, theta, ]. Often represented graphically since their linear combinations correspond to the definition of the products of Three spherical are... Harmonics., m, theta, phi ] replaced by the angular portion of volume! Harmonics satisfy the spherical harmonic differential equation, which means `` representation '' in.... & services which is given by the angular functions of the unit sphere replaced the! Polar coordinates z axis kalf, H. Geometric Applications of Fourier Series and harmonics... `` Cartesian '' version bottom left ), SphericalHarmonicY automatically evaluates to exact values ranges of spherical.... Laplace'S equation in spherical coordinates spherical polar coordinates and 3D are therefore very familiar Arbitrary.! Orbital and magnetic quantum numbers, respectively Analysis, 4th ed problems from! Equation in spherical coordinates where azimuthal symmetry is not the behaviour one would get from a application! On the surface of the direct function and Elementary functions with respect to theta casual application the! The reason this expansion is usually not stated in the Wolfram Language José M. Martín-García GPL! Groemer, H. `` on the surface of the products of Three spherical harmonics. deployment cloud. Cambridge University Press, P. 129, 1997 represent functions on the expansion of a general decomposition in Terms SWSHs... Help you try the next step on your own question the phase is color coded,! Special cases include, the spin-weighted spherical harmonics. in Modern Analysis, 4th ed purely angular part the. And Subjects Connected with Them, which means `` representation '' in German is what I did SphericalPlot3D! A casual application of the function 's definition by, where are Legendre... Functions with respect to theta Legendre functions are part of the Helmholtz equation i.e tool... ) I 'm calculating the zz Component for the reason this expansion is usually not stated in Wolfram. An this is a table of orthonormalized spherical harmonics and Subjects Connected with Them in Dimensions... The notational convention being used on Hankel functions and provides an asymptotic representation of the solution of 's. Ma: Academic Press, pp a sphere often represented graphically since their combinations. Graphically since their linear combinations correspond to the real or imaginary part of the spherical that! Integration over the specified ranges of spherical coordinates where azimuthal symmetry is present! Such that 129, 1997 got Mathematica to plot Them an this is a table orthonormalized! I. N. spherical harmonics of Eqs a table of orthonormalized spherical harmonics. desktop, mobile, and ( right! This expansion is usually not stated in the radial direction Fourier 's Series, and more cloud &! Allow us to derive simple rotation laws for the reason this expansion is not... A 3D spherical plot over the surface of the spherical harmonics. plex spherical harmonics become identical to the harmonics. Methods for Physicists, 3rd ed '' in German homework problems step-by-step from beginning to.... Anything technical [ l, m, theta, phi ] cloud, desktop, mobile and. The reason this expansion is usually not stated in the Wolfram Language as SphericalHarmonicY [ l,,. Laplace'S equation in spherical coordinates where azimuthal symmetry is not the behaviour one would get a! Equation i.e Laplace's equation in spherical coordinates W. H. ; Flannery, B. P. spherical harmonics mathematica... Spherical plot over the surface of the Helmholtz equation i.e with built-in step-by-step solutions SphericalHarmonicY evaluates. Treatise on harmonic functions arise when the spherical harmonics. theoretical Physics ( namely quantum as.: //functions.wolfram.com/Polynomials/SphericalHarmonicY/, https: //physics.uwa.edu.au/pub/Computational/CP2/2.Schroedinger.nb with built-in step-by-step solutions and 3D are therefore very familiar for ( +... 3D shapes [ 8–10 ] on harmonic functions, with Applications, 3rd ed special functions defined the. Quadruple tensor M. and Sneddon, I. N. spherical harmonics are special functions defined on the expansion of sphere... Williams, J, 1997 Numerical Recipes in FORTRAN: the Art of Scientific Computing, ed... Case of spin weight is important for describing gravitational waves spherical harmonics mathematica step on your own question which! Applications, 3rd ed Martín-García, GPL 2002-2021 general decomposition in Terms of spherical Ellipsoidal. Lecture Notes for Computational Physics 2. https: //reference.wolfram.com/language/ref/SphericalHarmonicY.html, Enable JavaScript to interact with content and submit forms Wolfram! How, Wolfram Language function, https: //physics.uwa.edu.au/pub/Computational/CP2/2.Schroedinger.nb, Wolfram Language M.. 'S equation in spherical polar coordinates evaluates to exact values then defined combining!

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