单词 | Legendre Polynomial | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
释义 | Legendre Polynomial![]() The Legendre Functions of the First Kind are solutions to theLegendre Differential Equation. If The Rodrigues Formula provides the Generating Function
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The Legendre polynomials are orthogonal over
![]() A Complex Generating Function is
Additional integrals (Byerly 1959, p. 172) include
The first few Legendre polynomials are ![]() The first few Powers in terms of Legendre polynomials are ![]() For Legendre polynomials and Powers up to exponent 12, see Abramowitz and Stegun (1972, p. 798). The Legendre Polynomials can also be generated using Gram-Schmidt Orthonormalization in theOpen Interval
Normalizing so that ![]() The ``shifted'' Legendre polynomials are a set of functions analogous to the Legendre polynomials, but defined on theinterval (0, 1). They obey the Orthogonality relationship
![]() The associated Legendre polynomials
where ![]() ![]() ![]()
Associated polynomials are sometimes called Ferrers' Functions(Sansone 1991, p. 246). If
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![]() ![]() ![]() Written in terms ![]() ![]() The derivative about the origin is
References Abramowitz, M. and Stegun, C. A. (Eds.). ``Legendre Functions'' and ``Orthogonal Polynomials.'' Ch. 22 in Chs. 8 and 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 331-339 and 771-802, 1972. Arfken, G. ``Legendre Functions.'' Ch. 12 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 637-711, 1985. Binney, J. and Tremaine, S. ``Associated Legendre Functions.'' Appendix 5 in Galactic Dynamics. Princeton, NJ: Princeton University Press, pp. 654-655, 1987. Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959. Iyanaga, S. and Kawada, Y. (Eds.). ``Legendre Function'' and ``Associated Legendre Function.'' Appendix A, Tables 18.II and 18.III in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1462-1468, 1980. Legendre, A. M. ``Sur l'attraction des Sphéroides.'' Mém. Math. et Phys. présentés à l'Ac. r. des. sc. par divers savants 10, 1785. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 593-597, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 252, 1992. Sansone, G. ``Expansions in Series of Legendre Polynomials and Spherical Harmonics.'' Ch. 3 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 169-294, 1991. Snow, C. Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. Washington, DC: U. S. Government Printing Office, 1952. Spanier, J. and Oldham, K. B. ``The Legendre Polynomials Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. |
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