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Topic: Associated Legendre polynomials

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	Legendre polynomials - Wikipedia, the free encyclopedia
		The left-hand side of the equation is the generating function for the Legendre polynomials.
		Legendre polynomials are symmetric or antisymmetric, that is
		Legendre polynomials of fractional order exist and follow from insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the gamma function) into the Rodrigues' formula.
	en.wikipedia.org /wiki/Legendre_polynomials (708 words)

	Legendre Polynomials
		The weight function w(x) of the Legendre polynomials is unity, and this is what distinguishes them from the others and determines them.
		The polynomials can also be found by solving the differential equation by determining the coefficients of a power series substituted in the equation.
		The more general problem requires the introduction of related functions called the associated Legendre functions that are actually built up from Jacobi polynomials, and can also be expressed in terms of derivatives of the Legendre polynomials.
	www.du.edu /~jcalvert/math/legendre.htm (1164 words)

	Associated Legendre polynomials - Wikipedia, the free encyclopedia
		They are a generalization of the "Legendre polynomials", but are not as general as the "Legendre functions".
		In many occasions in physics, associated Legendre polynomials in terms of angles occur where spherical symmetry is involved.
		As such, Legendre polynomials can be generalized to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces.
	en.wikipedia.org /wiki/Associated_Legendre_polynomials (856 words)

	Manual do Maxima: 63. orthopoly
		The exceptions are the half-integer Bessel functions and the associated Legendre function of the second kind.
		The Bessel functions are evaluated using an explicit representation, while the associated Legendre function of the second kind is evaluated using recursion.
		Numerically, the half-integer Bessel functions are evaluated using the SLATEC code, and the associated Legendre functions of the second kind is numerically evaluated using the same algorithm as its symbolic evaluation uses.
	maxima.sourceforge.net /docs/manual/pt/maxima_63.html (2302 words)

	Legendre Polynomial -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-04)
		The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p.
		The "shifted" Legendre polynomials are a set of functions analogous to the Legendre polynomials, but defined on the interval (0, 1).
		The associated Legendre functions are part of the spherical harmonics, which are the solution of Laplace's equation in spherical coordinates.
	mathworld.wolfram.com.cob-web.org:8888 /LegendrePolynomial.html (663 words)

	GNU Scientific Library -- Reference Manual: Associated Legendre Polynomials and Spherical Harmonics
		These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, x
		These routines compute the normalized associated Legendre polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics.
		This function computes an array of normalized associated Legendre functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ for m >= 0, l = m,..., lmax, x
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_121.html (220 words)

	GNU Scientific Library -- Reference Manual - Associated Legendre Polynomials and Spherical Harmonics (Site not responding. Last check: 2007-11-04)
		This function computes an array of Legendre polynomials @math{P_l^m(x)} for @c{$m \ge 0$} @math{m >= 0}, @c{$l = m, \dots, lmax$} @math{l = m,..., lmax}, @c{$x
		These routines compute the normalized associated Legendre polynomial @math{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$} suitable for use in spherical harmonics.
		This function computes an array of normalized associated Legendre functions @math{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$} for @c{$m \ge 0$} @math{m >= 0}, @c{$l = m, \dots, lmax$} @math{l = m,..., lmax}, @c{$x
	www.math.utah.edu:8080 /software/gsl/gsl-ref_121.html (259 words)

	Legendre Polynomials (Site not responding. Last check: 2007-11-04)
		One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Legendre polynomials.
		From the Legendre polynomials can be generated another important class of functions for physical problems, the associated Legendre functions.
		The associated Legendre functions can be used to construct another important set of functions, the spherical harmonics.
	hyperphysics.phy-astr.gsu.edu /hbase/math/legend.html (255 words)

	Orthogonal Polynomials
		are obtained from derivatives of the Legendre polynomials according to
		Section 3.2.10 discusses the generalization of Legendre polynomials to Legendre functions, which can have non-integer degrees.
		Legendre, Gegenbauer and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials.
	documents.wolfram.com /v4/MainBook/3.2.9.html (240 words)

	Legendre polynomials (Site not responding. Last check: 2007-11-04)
		Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates.
		Associated polynomials have to be used when the solutions have an azimuthal component
		and m, and even then it may be only one of the two linearly independent solutions which is a polynomial, but the full implications of this situation will only become fully apparent when the spectral density is discussed.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node58.html (287 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		LEGENDRE FUNCTIONS: Legendre Functions of the First Kind and Legendre Polynomials.
		Routines in the package calculate the associated Legendre functions of the first kind, indefinite (multiple) integrals involving the Legendre polynomials, derivatives of the Legendre polynomials, spherical harmonic functions, and an expansion of an arbitrary (single valued) function.
		LEGENDRE FUNCTIONS contains routines to expand functions in Legendre polynomials, routines for the real and imaginary parts of the spherical harmonics in spherical polar coordinates, derivatives of the Legendre polynomials and multiple integrals of Legendre polynomials P
	www-rsicc.ornl.gov /codes/psr/psr1/psr-108.html (162 words)

	Alibris: Polynomials
		Polynomials pervade mathematics, virtually every branch of mathematics from algebraic number theory and algebraic geometry to applied analysis and computer science, has a corpus of theory arising from polynomials.
		The analysis of orthogonal polynomials associated with general weights has been a major theme in classical analysis this century.
		The zeros of compositions of polynomials are also investigated along with their growth, and some of these considerations lead to the...
	www.alibris.com /search/books/subject/Polynomials (955 words)

	I
		Because the associated Legendre polynomials satisfy the geometrically determined boundary conditions at the poles and because they satisfy Laplace's equation governing tidal motions on the sphere they are a natural choice for representing latitudinal variations.
		Due to the presence of fast gravity waves in the atmosphere, this constraint can result in an extremely small time step that is prohibitive in terms of computational speed, so some means of overcoming this limitation needs to be employed.
		In particular, the pressure gradient term in the momentum equation (or its analog in the vorticity and divergence equations) and the divergence term in the surface pressure tendency equation are stepped forward in time implicitly while the other terms are treated explicitly (e.g., leap frog scheme).
	grads.iges.org /agcm/agcm_hydro.html (2250 words)

	FredericPetit - (Site not responding. Last check: 2007-11-04)
		associated Legendre polynomials are well defined in SciPy, but you can't pass to them array arguments (and you need this to display the spatial distribution as 3D array).
		Note that SciPy has a builtin spherical Bessel function, but it has the same drawback as its associated Legendre polynomial; so you have to define your own spherical Bessel function, using scipy.special.jv, the Bessel function of real order v.
		eigenmodes depends on associated Legendre polynomials derivatives vs theta and phi angles; so you have to calculate them analytically (and simplify as possible).
	www.scipy.org /FredericPetit (701 words)

	GRIB
		The association of the numbers in octet PL (and following) with the particular row follows the scanning mode specification in Table 8.
		In this figure, the ordinate, n, is the zonal wave number, the abscissa, m, is the total meridional wave number, the vertical line at m = M is the zonal truncation, and the diagonal passing through (0,0) is the line n = m.
		On the n-axis, the horizontal line at n = K indicates the upper limit to n values, and the diagonal that intersects the n-axis at n = J indicates the upper limit of the area in which the Polynomials are defined.
	www.nco.ncep.noaa.gov /pmb/docs/on388/section2.html (527 words)

	Legendre Polynomial
		In the Sturm-Liouville Boundary Value Problem, there is an important special case called Legendre's Differential Equation which arises in numerous problems, especially in those exhibiting spherical symmetry.
		Since Legendre's differential equation is a second order ordinary differential equation, two sets of functions are needed to form the general solution.
		Recurrence Relation: A Legendre Polynomial at one point can be expressed by neighboring Legendre Polynomials at the same point.
	www.efunda.com /math/Legendre/index.cfm (210 words)

	GNU Scientific Library -- Reference Manual
		An iterative polynomial solver is also available for finding the roots of general polynomials with real coefficients (of any order).
		This function converts the divided-difference representation of a polynomial to a Taylor expansion.
		The roots of polynomial equations cannot be found analytically beyond the special cases of the quadratic, cubic and quartic equation.
	www.math.umn.edu /systems_guide/gsl-1.3/gsl-ref.html (9456 words)

	SLATEC5 (REBAK through ZBIRY) UCID-19631,19632,19633
		See J.M. Smith, F.W.J. Olver, and D.W. Lozier, Extended-Range Arithmetic and Normalized Legendre Polynomials, ACM Transactions on Mathematical Soft- ware, 93-105, March 1981, for a complete description of the algorithm and special arithmetic.
		The normalized Legendre polynomials are multiples of the associated Legendre polynomials of the first kind where the normalizing coefficients are chosen so as to make the integral from -1 to 1 of the square of each function equal to 1.
		0 the corre- sponding value of the normalized Legendre polynomial cannot be represented in single-precision because of overflow or under- flow.
	www.llnl.gov /LCdocs/slatec5/index.jsp?show=s2.215 (786 words)

	Appendix A: List of associated Legendre functions. Three-dimensional mathematical analysis of particle shape using ...
		The associated Legendre functions with m = -M < 0 are simply given in terms of the equivalent functions with M > 0 according to
		Table 3: List of associated Legendre polynomials from n = 0 to n = 5.
		Table 4: List of associated Legendre polynomials from n = 6 to n = 8.
	ciks.cbt.nist.gov /~garbocz/paper134/node10.html (97 words)

	FORTRAN Routines for Computation of Special Functions
		mclqmn.for (CLQMN) Evaluate a sequence of associated Legendre functions of the second kind and their derivatives with complex arguments.
		mlpmns.for (LPMNS) Evaluate a sequence of the associated Legendre functions of the first kind and their derivatives with real arguments for a given order.
		mlqmns.for (LQMNS) Evaluate a sequence of the associated Legendre functions of the second kind and their derivatives with real arguments for a given order.
	jin.ece.uiuc.edu /routines/routines.html (2278 words)

	spherical harmonics (Site not responding. Last check: 2007-11-04)
		Further discussion of the Legendre polynomials and spherical harmonics
		While Maple or Mathematica can take care of doing calculations with special functions like the Legendre polynomials, it is helpful to remember where they came from, in order to be able to set calculations up properly.
		If we rewrite (11.9) in terms of the associated Legendre polynomials and the variable z - notice that sin(theta) d theta = - dz - we find
	www.mathphysics.com /pde/spharm.html (308 words)

	Gegenbauer biography
		Gegenbauer had many mathematical interests such as number theory, function theory, and the theory of integration, but he was chiefly an algebraist.
		He is remembered for the Gegenbauer polynomials, a class of orthogonal polynomials.
		The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Gegenbauer.html (281 words)

	Factorization Method for Solving a Partial Differential Equation: Spherical Harmonics
		This method differs from a power series or a numerical approach in that one solves a calculus problem without the use of calculus: one obtains the linear algebraic aspects of the problem (eigenvalues, all normalized eigenvectors, their properties, etc.) in one fell swoop without ever having to determine explicitly the detailed functional form (i.e.
		To be sure, one can readily determine and exhibit these solutions in explicit form in terms of Legendre and associated Legendre polynomials, and we shall do so.
		Its form is chosen so as to simplify the to-be-derived formula for the Legendre polynomials, Eq.(5.94).
	www.math.ohio-state.edu /~gerlach/math/BVtypset/node135.html (1349 words)

	GNU Scientific Library -- Reference Manual - Special Functions
		The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultraspherical polynomials.
		The Laguerre polynomials are defined in terms of confluent hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x).
		These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3.
	www.math.umn.edu /systems_guide/gsl-1.3/gsl-ref_7.html (6072 words)

	Manpage of backwardSHT
		where P(cos(theta)) = P_{l,m}(cos(theta)) are the associated Legendre polynomials, and i is the imaginary number.
		For more information about the calculation of the associated Legendre polynomials see the man page for calculateQlm().
		Instead most of the coding required to do the calculation of the associated Legendre polynomials is done in the functions calculateSmallQlm() and lmCalculations() which the transform does call.
	crd.lbl.gov /~cmc/ccSHTlib/doc/manHtml/backwardSHT.html (676 words)

	XI. Great Balls of PDEs
		While Mathematica can take care of doing calculations with special functions like the Legendre polynomials, you may find it helpful to read some further discussion of the Legendre polynomials when doing the exercises.
		Exercises XI Noticing that Legendre's equation is unchanged by changing z to - z, we can conclude that a fundamental pair of solutions can be found, one of which is even and the other odd.
		Verify that the first few Legendre polynomials satisfy the Legendre equation with these values of u.
	www.mathphysics.com /pde/ch11wr.html (2637 words)

	Astron. Astrophys. 326, 1235-1240 (1997) (Site not responding. Last check: 2007-11-04)
		In Appendix A of SCDT a relatively straightforward technique is given for constructing the orthogonal fitting polynomials
		The integrals are easily evaluated from the normalization of associated Legendre polynomials.
		Integrals of products of three associated Legendre polynomials also occur regularly in quantum mechanics when adding angular momenta (cf.
	aa.springer.de /papers/7326003/2301235/sc5.htm (178 words)

	Associated Legendre Polynomials and Spherical Harmonics - GNU Scientific Library -- Reference Manual
		Associated Legendre Polynomials and Spherical Harmonics - GNU Scientific Library -- Reference Manual
		Next: Conical Functions, Previous: Legendre Polynomials, Up: Legendre Functions and Spherical Harmonics
		These functions compute an array of Legendre polynomials P_l^m(x), and optionally their derivatives dP_l^m(x)/dx, for
	www.gnu.org /software/gsl/manual/html_node/Associated-Legendre-Polynomials-and-Spherical-Harmonics.html (246 words)

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