scholarly journals On the Reduction of Ferrers' Associated Legendre Function

1929 ◽  
Vol 1 (4) ◽  
pp. 241-243
Author(s):  
Hrishikesh Sircar
Keyword(s):  
Finite Number ◽  
Infinite Series ◽  
Infinite Number ◽  
Legendre Function ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Reduced Order ◽  
Zonal Harmonics ◽  
Integral Degree ◽  
Associated Legendre Function

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.

The Analysis of the Associated Legendre Functions with Non-Integral Degree

2011 ◽  
Vol 130-134 ◽  
pp. 3001-3005
Author(s):  
Jian Qiang Wang ◽  
Hao Yuan Chen ◽  
Yin Fu Chen
Keyword(s):  
Spherical Harmonic ◽  
Legendre Function ◽  
Scalar Potential ◽  
Regional Model ◽  
Legendre Functions ◽  
Spherical Cap ◽  
Associated Legendre Functions ◽  
Core Content ◽  
Local Gravity ◽  
Integral Degree

Spherical cap harmonic (SCH) theory has been widely used to format regional model of fields that can be expressed as the gradient of a scalar potential. The functions of this method consist of trigonometric functions and associated Legendre functions with integral-order but non-integral degree. Evidently, the constructing and computing of Legendre functions are the core content of the spherical cap functions. In this paper,the approximated calculation method of the normalized association Legendre functions with non-integral degree is introduced and an analysis of the entire order of associated non-Legendre function calculation is presented. Besides, we use the Muller method to search out for all intrinsic values. The results showed that the highest order of spherical harmonic function for constructing regional model of fields is limited, thus high-resolution spherical harmonic structure of local gravity field need to be improved.

scholarly journals Fast computation of sine/cosine series coefficients of associated Legendre function of arbitrary high degree and order

2018 ◽  
Vol 8 (1) ◽  
pp. 162-173
Author(s):  
T. Fukushima
Keyword(s):  
Legendre Function ◽  
Previous Method ◽  
Legendre Functions ◽  
Fast Computation ◽  
Fixed Degree ◽  
Associated Legendre Functions ◽  
Cosine Series ◽  
Recurrence Formulas ◽  
High Degree ◽  
Associated Legendre Function

Abstract In order to accelerate the spherical/spheroidal harmonic synthesis of any function, we developed a new recursive method to compute the sine/cosine series coefficient of the 4π fully- and Schmidt quasi-normalized associated Legendre functions. The key of the method is a set of increasing-degree/order mixed-wavenumber two to four-term recurrence formulas to compute the diagonal terms. They are used in preparing the seed values of the decreasing-order fixed-degree, and fixed-wavenumber two- and three-term recurrence formulas, which are obtained by modifying the classic relations. The new method is accurate and capable to deal with an arbitrary high degree/ order/wavenumber. Also, it runs significantly faster than the previous method of ours utilizing the Wigner d function, say around 20 times more when the maximum degree exceeds 1,000.

scholarly journals Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

2021 ◽  
Author(s):  
Howard S. Cohl ◽  
◽  
Justin Park ◽  
Hans Volkmer ◽  
◽  
...  
Keyword(s):  
Hypergeometric Function ◽  
Complex Plane ◽  
Fourier Expansion ◽  
Legendre Function ◽  
Legendre Functions ◽  
Detailed Review ◽  
Associated Legendre Function ◽  
Branch Cut

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at [1,infty). In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.

XXVII.—Expansions of Lamé Functions into Series of Legendre Functions

1948 ◽  
Vol 62 (3) ◽  
pp. 247-267
Author(s):  
A. Erdélyi
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Infinite Series ◽  
Recurrence Relations ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Jacobi Series ◽  
Characteristic Numbers ◽  
Transcendental Equations

Summary28. This paper contains the investigation of certain properties of periodic solutions of Lamé's differential equation by means of representation of these solutions by (in general infinite) series of associated Legendre functions. Terminating series of associated Legendre functions representing Lamé polynomials have been used by E. Heine and G. H. Darwin. The latter used them also for numerical computation of Lamé polynomials. It appears that infinite series of Legendre functions representing transcendental Lamé functions have not been discussed previously. In two respects these series seem to be superior to the generally used power-series and Fourier-Jacobi series, (i) They are convergent in some parts of the complex plane of the variable where both power-series and Fourier-Jacobi series diverge, (ii) They permit by simply replacing Legendre functions of first kind by those of second kind, to deal with Lamé functions of second kind as well as Lamé functions of first kind (§ 15).In §§ 2 and 8 of the present paper the series are heuristically deduced from the integral equations satisfied by periodic Lamé functions. Inserting the series found heuristically, with unknown coefficients, into Lamé's differential equation, recurrence relations for the coefficients are obtained (§§ 9–12). These recurrence relations yield the (in general transcendental) equations in form of (in general infinite) continued fractions for the determination of the characteristic numbers. The convergence of the series can be discussed completely.There are altogether forty-eight different series. Every one of the eight types of Lamé polynomials is represented by six different series (see table in § 13). There are interesting relations (§ 14) between series representing the same function.Next infinite series representing transcendental Lamé functions are discussed. It is seen that transcendental Lamé functions are only simply-periodic (§§ 18, 19). Lamé functions of real (§§ 20–22) and imaginary (§§ 23-24) period are represented by series of Legendre functions the variables of which are different in both cases.The paper concludes with a brief discussion of the most important limiting cases, and a short mention of other types of series of Legendre functions representing Lamé functions.

Note on the generalized Mehler transform

1964 ◽  
Vol 60 (1) ◽  
pp. 57-59 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Boundary Value Problems ◽  
Inversion Formula ◽  
Integral Transform ◽  
Legendre Function ◽  
Boundary Value ◽  
Diffusion Equations ◽  
Legendre Functions ◽  
Image Position ◽  
And Diffusion ◽  
Associated Legendre Function

1. The integral transform. One result of recent studies of boundary-value problems of the wave and diffusion equations involving wedge- or conically-shaped boundaries has been the interest shown in integrals in which the variable of integration appears as the order of Bessel or Legendre functions. An integral of this type occurs as the inversion formula for the generalized Mehler transform which is defined bywhere ψ(μ, k) = Γ(½ − k + iμ)Γ(½ − k − iμ) and is the associated Legendre function of the first kind. Oberhettinger and Higgins (4) have given a table of transform pairs corresponding to the above transform.

VIII.—Some Series and Integrals involving Associated Legendre Functions, regarded as Functions of their Degrees

1936 ◽  
Vol 55 ◽  
pp. 85-90 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Infinite Series ◽  
Contour Integration ◽  
Legendre Functions ◽  
Dirichlet Integral ◽  
Associated Legendre Functions ◽  
Image Position ◽  
Dirichlet Integrals

§ 1. Introductory—In a former paper (Proc. Roy. Soc. Edin., vol. liv, 1934, pp. 135–144) the author discussed the evaluation of a number of integrals of Associated Legendre Functions, regarded as functions of their degrees. The methods employed depended mainly on contour integration, and most of the integrals were evaluated in terms of infinite series of Associated Legendre Functions. In the present paper the methods employed are of a more elementary character, depending mainly on the use of Dirichlet Integrals; the results obtained are more general; and the integrals and the corresponding series are evaluated in simpler forms. The same notation is employed as in the previous paper. The Mehler-Dirichlet Integralwhere o < θ < π, μ > – ½, is used throughout.

On the expansion in a series of the attraction of a spheroid

1833 ◽  
Vol 2 ◽  
pp. 162-163
Keyword(s):  
Finite Number ◽  
Infinite Series ◽  
Infinite Number ◽  
Joint Effect ◽  
The Other ◽  
Original Function ◽  
Simple Mode ◽  
Rectangular Coordinates ◽  
Rotatory Motion

Mr. Ivory’s principal object in this paper appears to be the removal of some difficulties in the demonstration of the method of developing the attractions of spheroids in an infinite series, as employed by Laplace in the Mécanique Céleste . It is natural to think, he observes, that the theory of the figure of the planets would be placed on a firmer basis if it were deduced directly from the general principles of the case, than when it is made to depend on a nice and somewhat uncertain point of analysis; and he conjectures that the theory will probably be found to hinge on this proposition,—that a spheroid, whether homogeneous or heterogeneous, cannot be in equilibrium by means of a rotatory motion about an axis, and the joint effect of the attraction of its own particles and of the other bodies of the system, unless its radius be a function of three rectangular coordinates; for if this proposition were clearly and rigorously demonstrated, the analysis of Laplace, on changing the ground on which it is built, would require little or no alteration in other respects. Without, however, attempting to demonstrate this proposition in all its extent, the author has substituted a more direct and simple mode of argument than that of Laplace, which is perfectly conclusive with respect to all the cases to which the theorem in question can possibly require to be applied. He has shown that by immediately transforming a given expression into a function of three rectangular coordinates, we obtain the same development as is deduced in the Mécanique Céleste , by a more general and complicated mode of reasoning, which seems to be so far objectionable, as it tends to introduce a variety of quantities into the series which do not alter its total value, since they destroy each other, but which may possibly interfere with the accuracy of its application to particular cases, in which it may be employed as a symbolical representation: for example, when any finite number of terms is assumed as affording an approximate value; since, if the expression developed has not been reduced to the form of a function of three rectangular coordinates, the development may contain an infinite number of terms, which are introduced by the operation without being essential to its final result. He takes for the example of such a case the equation of a spheroid, prominent between the equator and the poles, somewhat resembling the figure which was once attributed to Saturn; and he shows that its development in the form required will contain an infinite number of quantities arising from the expansion of a radical, which are not to be found in the original function.

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