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Topic: Spherical harmonics

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Associated Legendre polynomials

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Spherical trigonometry

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	fUSION Anomaly. Harmonics
		Harmonics, series of subsidiary vibrations that accompany a primary, or fundamental, wave-motion vibration, most notably in musical instruments.
		Harmonics result when the vibrating body, for example a stretched string or an enclosed air column, vibrates simultaneously as a whole and in equal parts (halves, thirds, fourths, and so on), producing wave frequencies that are in simple ratios with the fundamental frequency.
		Galactic Harmonics is a grand discipline occupied with the examination of the underlying patterns and themes present in all levels of material existence.
	www.fusionanomaly.net /harmonics.html (909 words)

	Spherical Harmonics - LoveToKnow 1911
		SPHERICAL HARMONICS, in mathematics, certain functions of fundamental importance in the mathematical theories of gravitation, electricity, hydrodynamics, and in other branches of physics.
		The harmonic of positive degree n corresponding to that of degree -n - I in the expression (7) is 2.2n 2 I + 2.4.20 y4 II.4202-3'...
		ix.), of the poles of a spherical harmonic.
	www.1911encyclopedia.org /Spherical_Harmonics (8585 words)

	Spherical Harmonics
		The derivative of a solid harmonic is also a solid harmonic, since the operation D of differentiation commutes with the Laplacian operator.
		All the harmonics in between have both circles of latitude and meridians as zeros, and are called tesseral harmonics, since they seem to divide the spherical surface into tesserae.
		Although any spherical harmonics will have the same general behaviour and form, they may be defined with slightly different constants, which can make the comparison of references difficult.
	mysite.du.edu /~jcalvert/math/harmonic/harmonic.htm (2342 words)

	Spherical harmonics - DmWiki
		Mathematically, they can be used as a basis for all real- or complex-valued functions on the sphere; because of the way spherical harmonics are defined, it is easy to approximate other functions using a finite number of harmonics.
		Spherical harmonics are particularly important in precomputed radiance transfer.
		The incoming light is also represented using spherical harmonics, and the two can be multiplied using just their coefficients, yielding a spherical-harmonic representation of the light on the object's surface, which can then be evaluated by a shader.
	www.devmaster.net /wiki/Spherical_harmonics (166 words)

	SPHERICAL HARMONICS - Online Information article about SPHERICAL HARMONICS
		We thus see that the spherical harmonics of degree n are of the form n+m r"sin mip sin med—n+m(t~2-I)" where i denotes cos a; by giving m the values o, t, 2...n we thus have the 2n+1 functions required.
		Every ordinary harmonic of degree n is expressible as a linear function of the system of 2n-1-I zonal, tesseral and sectorial harmonics of degree n; thus the general form of the surface harmonic is n aoP,(A) +Z(am cos mcti +bm sin m0)P„0u).
		The harmonic of positive degree n corresponding to that of degree —n — in the expression (7) is I _ r2 2.211—1+2.4.2n y9II.42n—3—..
	encyclopedia.jrank.org /SOU_STE/SPHERICAL_HARMONICS.html (6907 words)

	Geodesy for the Layman
		Harmonic Expressions, formed by combining sines, cosines and arbitrary constants, are mathematical devices for curve-fitting and interpolation in either two or three dimensions.
		Each harmonic term has two parts: the trigonometric part which is harmonic and controls the frequency of the oscillations, and a constant multiplier which controls the amplitude.
		The coordinates normally associated with spherical harmonics are the spherical coordinates: radius r, co-latitude or polar distance (Theta), and longitude (Lambda).
	www.floridageomatics.com /publications/gfl/spherical-harmonics.htm (1855 words)

	Spherical Harmonic -- from Wolfram MathWorld
		Spherical harmonics satisfy the spherical harmonic differential equation, which is given by the angular part of Laplace's equation in spherical coordinates.
		is prepended to the definition of the spherical harmonics.
		Hobson, E. The Theory of Spherical and Ellipsoidal Harmonics.
	mathworld.wolfram.com /SphericalHarmonic.html (442 words)

	The Spherical Harmonics
		are the wavefunctions for any particle that is free to move in the spherical polar angles theta and phi (i.e., that has no dependence on these angles in the particle's potential energy function, as in the hydrogen atom).
		They are the building blocks for atomic wavefunctions in general, and their shapes and orientations in space are important to learn.
		the theta, phi spherical polar angular axes are in red
	www.dartmouth.edu /~chem81/thps/Ylm.html (216 words)

	Evaluation of the rotation matrices in the basis of real spherical harmonics. (Site not responding. Last check: )
		Rotation matrices (or Wigner D functions) are the matrix representations of the rotation operators in the basis of the spherical harmonics.
		Ordinary complex spherical harmonics are easier to manipulate theoretically, due to a number of useful relations that loose their simplicity when stated in terms of the RSH.
		, is characteristic of the spherical symmetry of atoms, being the eigenfunction of
	www.elsevier.com /homepage/saa/eccc3/paper48/eccc3.html (2838 words)

	News \| Gainesville.com \| The Gainesville Sun \| Gainesville, Fla. (Site not responding. Last check: )
		Spherical harmonics are important in many theoretical and practical applications, particularly in the computation of atomic electron configurations, the representation of the gravitational field, geoid, and magnetic field of planetary bodies, as well as characterization of the cosmic microwave background radiation.
		In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre functions, or to append it to the definition of the spherical harmonic functions.
		The analog of the spherical harmonics for the Lorentz group are given by the hypergeometric series; indeed, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) is a subgroup of PSL(2,C).
	www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=spherical_harmonics (1299 words)

	Springer Online Reference Works (Site not responding. Last check: )
		The simplest spherical harmonics are the zonal spherical harmonics.
		Expansions in spherical harmonics are largely analogous to expansions in Fourier series, of which they are essentially a generalization.
		spherical harmonics are sometimes also called surface spherical harmonics.
	eom.springer.de /s/s086690.htm (192 words)

	Environment Spherical Harmonics — Illuminate Labs
		This demo scene shows Environment Lighting using Spherical Harmonics generated by Turtle 4.
		With Turtle 4 you can precompute all light cast from the environment (direct and indirect) as Spherical Harmonics.
		This can be used for extremely efficient relighting of geometry moving through the scene.
	www.illuminatelabs.se /turtle/environment-spherical-harmonics (99 words)

	Spherical Harmonics
		The derivative of a solid harmonic is also a solid harmonic, since the operation D of differentiation commutes with the Laplacian operator.
		All the harmonics in between have both circles of latitude and meridians as zeros, and are called tesseral harmonics, since they seem to divide the spherical surface into tesserae.
		Although any spherical harmonics will have the same general behaviour and form, they may be defined with slightly different constants, which can make the comparison of references difficult.
	www.du.edu /%7Ejcalvert/math/harmonic/harmonic.htm (2342 words)

	Spherical Harmonics (Site not responding. Last check: )
		One of the varieties of special functions which are encountered in the solution of physical problems, is the class of functions called spherical harmonics.
		The functions in this table are placed in the form appropriate for the solution of the Schrodinger equation for the spherical potential well, but occur in other physical problems as well.
		in spherical polar coordinates is a modified form of the associated Legendre functions.
	hyperphysics.phy-astr.gsu.edu /hbase/math/sphhar.html (68 words)

	mangle
		The spherical harmonics of a mask is the sum of the spherical harmonics of its polygons, each weighted according to its weight.
		The spherical harmonics are computed analytically, in the manner described in the Appendix of Hamilton (1993), MNRAS, ApJ 417, 19-35.
		The spherical harmonics of an angular mask are given by integrals of spherical harmonics over the area of the mask.
	casa.colorado.edu /~ajsh/mangle (12121 words)

	SHTOOLS - Tools for working with spherical harmonics (Site not responding. Last check: )
		SHTOOLS is an archive of fortran 95 based software that can be used to perform (among others) spherical harmonic transforms and reconstructions, rotations of spherical harmonic coefficients, and multitaper spectral analyses on the sphere.
		Spherical harmonic transforms can be calculated by exact quadrature rules using either (1) the sampling theorem of Driscoll and Heally (1994) where data are equally sampled (or spaced) in latitude and longitude, or (2) Gauss-Legendre quadrature.
		The spherical harmonic transforms are proven to be accurate to approximately degree 2800, corresponding to a spatial resolution of better than 4 arc minutes.
	www.ipgp.jussieu.fr /~wieczor/SHTOOLS/SHTOOLS.html (301 words)

	cgCharacter: Shareware
		Spherical Harmonics are a group of mathematical functions that encode information about around a 3d polar coordinate system to encode complex information into very small parameter set.
		Once higher order spherical harmonics are used we will be able to use them for encding the specular high lights.
		THis shader is now superseeded by the ability of a Spherical Harmonics map to convert a map to a diffuse light source.
	www.cgcharacter.com /shareware (496 words)

	gnuplot / spherical_harmonics / spharm (2E) (Site not responding. Last check: )
		In the quantum mechanics, l and m values of the spherical harmonics Y[lm] are the angular momentum and magnetic momentum numbers.
		When m is odd, the spherical harmonics contains an imaginary part which comes from exp(-im phi) term.
		The complex conjugate function of the spherical harmonics can be given by the relation,
	t16web.lanl.gov /Kawano/gnuplot/spherical_harmonics/spharm2-e.html (256 words)

	40 (Site not responding. Last check: )
		We discussed the role of the spherical harmonics in solving Laplace's equation in 3 dimensions, as well as other PDEs in 3-dimensions for spherically symmetric systems.
		Also noted why spherical harmonics are so central in 3-d physics: why, for any physical problem with spherical symmetry, spherical harmonics remain the angular eigenfunctions, with only the radial eigenfunctions altered by differing radial potentials, say, in Schrodinger's equation.
		Spherical harmonics are the eigenstates of the laplacian in spherical coordinates; physically, they are the eigenstates of angular momentum in quantum mechanics, the energy eigenstates of the hydrogen atom, and of any other rotationally invariant system.
	www.physics.emory.edu /faculty/benson/320/notes/40/40.html (621 words)

	Fast Spherical Harmonic Transforms: SpharmonicKit
		This implies that for the spherical transforms in the Kit, a function of bandwidth B is sampled on the equiangular 2Bx2B grid on the sphere.
		Depending on the size of spherical transform one would like to do, it may be more efficient to use the routines provided in Spherepack.
		Though the spherical transforms in Spherepack are computed using the direct method, the amount of overhead involved in using any non-direct method may make the direct method faster, at least at small enough problem sizes.
	www.cs.dartmouth.edu /~geelong/sphere (1370 words)

	Surface Spherical Harmonics
		The first of each pair gives the view of from a 30 degree angle in respect to the equatorial plane, the second one from a 90 degree angle (the top).
		More on surface spherical harmonics can be found in books on mathematical physics, for example Mathematical methods for physicists by George Arfken.
		Spherical Harmonics Gallery Page for a different representation.
	stephan.sugarmotor.org /harmonics (182 words)

	Evaluation of the rotation matrices in the basis of real spherical harmonics. (Site not responding. Last check: )
		The aim of this contribution is to obtain a general algorithm to compute the representation matrix of any point-group symmetry operation in the basis of the real spherical harmonics, paying attention to the use of recurrence relations that allow the treatment of functions with high angular momenta.
		Ordinary complex spherical harmonics are easier to manipulate theoretically, due to a number of useful relations that loose their simplicity when stated in terms of the RSH.
		, is characteristic of the spherical symmetry of atoms, being the eigenfunction of
	www1.elsevier.com /homepage/saa/eccc3/paper48/eccc3.html (2838 words)

	MATH 308 Project: Spherical Harmonics
		A spherical harmonic is analogous to the sinusoidal wave from particle-on-a-line example.
		A spherical harmonic can be though of as a 3D-path that a particle can travel without “destroying” itself energetically.
		As such, the Schrodinger wave equation is decomposed into two separate parts: the radial (r) and angular elements (Φ,Θ) which arise from spherical harmonics.
	www.math.ubc.ca /%7Ecass/courses/m308-02b/projects/alo/SphericalHarmonics.html (279 words)

	The Shift Operators and Translations of Spherical Harmonics (ResearchIndex)
		Abstract: Solid and surface spherical harmonics functions have very simple transformation properties with respect to the gradient and angular momentum operators.
		These properties can be utilized for the derivation of translation relations of the spherical harmonic functions.
		21 The theory of spherical and ellipsoidal harmonics (context) - Hobson - 1955
	citeseer.ist.psu.edu /vangelderen98shift.html (326 words)

	laplace (Site not responding. Last check: )
		Spherical Harmonic Analysis consists of determining values for (and significance of) constants
		This is the differential equation for the Simple Harmonic Oscillator (SHO), or a mass on a spring:
		The undetermined coefficients A and B are determined by initial conditions (think of them as boundary conditions in time), namely the position, x, and velocity v, of the mass, when t = 0.
	solid_earth.ou.edu /notes/harmonic/harmonic.html (489 words)

	Angular Modes
		harmonics are related to the ordinary spherical harmonics as
		By virtue of their relation to the rotation matrices, the spin harmonics satisfy: the compatibility relation with spherical harmonics,
		The parity equation (6) tells us that the spin flips under a parity transformation so that unlike the s=0 spherical harmonics, the higher spin harmonics are not parity eigenstates.
	background.uchicago.edu /~whu/tamm/webversion/node5.html (374 words)

	[astro-ph/0508514] Fast spin +-2 spherical harmonics transforms and application in cosmology
		Second, we discuss the a priori O(L^4) asymptotic complexity of the spin +-2 spherical harmonics transforms, where 2L stands for the square-root of the number of sampling points on the sphere, also setting a band limit L for the spin +-2 functions considered.
		We derive an explicit expression for the spin +-2 spherical harmonics +-2_Y_lm (of integer l and m, with l<=2, m<=l) as linear combinations of the standard scalar spherical harmonics Y_lm and Y_(l-1)m.
		An exact algorithm is developed for the spin +-2 spherical harmonics transforms on equi-angular grids, based on the Driscoll and Healy fast scalar spherical harmonics transform.
	arxiv.org /abs/astro-ph/0508514 (374 words)

	Spherical Harmonics -- from Eric Weisstein's Encyclopedia of Scientific Books
		An Elementary Treatise on Spherical Harmonics and Subjects Connected with them.
		Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications, 3rd ed.
		The Theory of Potential and Spherical Harmonics, 2nd ed.
	www.ericweisstein.com /encyclopedias/books/SphericalHarmonics.html (42 words)

	gnuplot / spherical_harmonics / spharm (1E)
		The spherical harmonics in the polar coordinate Y[lm](theta,phi) is given by:
		The simplest spherical harmonics can be obtained by setting l=0 and m=0.
		The spherical harmonics becomes real when m is even.
	t16web.lanl.gov /Kawano/gnuplot/spherical_harmonics/spharm1-e.html (300 words)

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