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.

scholarly journals The Asymptotic Expansions of the Spherical Harmonics

1922 ◽  
Vol 41 ◽  
pp. 82-93
Author(s):  
T. M. MacRobert
Keyword(s):  
Real Axis ◽  
Spherical Harmonics ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
The Real ◽  
Image Position

Associated Legendre Functions as Integrals involving Bessel Functions. Let,where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

scholarly journals Some Integrals involving Legendre Functions

1959 ◽  
Vol 11 (3) ◽  
pp. 161-165 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Bessel Functions ◽  
Legendre Functions ◽  
Dirichlet Integral ◽  
Associated Legendre Functions ◽  
Different Order

In this note generalisations of certain integrals involving Legendre functions including the Mehler-Dirichlet integral for Legendre functions of the first kind are given, these new results expressing associated Legendre functions of the first or second kinds as integrals involving corresponding functions of the same degree but different order. These integrals appear to be analogous to Sonine's integral in the theory of Bessel functions.

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.

scholarly journals An Expansion in Terms of Associated Legendre Functions

1952 ◽  
Vol 1 (1) ◽  
pp. 13-15
Author(s):  
T. M. Macrobert
Keyword(s):  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Right Hand ◽  
Image Position ◽  
The Right

If the right-hand side of the expansionis integrated m times from 1 to µ, it becomes

XIII.—Some Integrals, with respect to their Degrees, of Associated Legendre Functions

1935 ◽  
Vol 54 ◽  
pp. 135-144 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Type A ◽  
Fourier Integral ◽  
Contour Integration ◽  
Legendre Functions ◽  
Integral Type ◽  
Integral Theorem ◽  
Associated Legendre Functions ◽  
In Series ◽  
Special Case

Little is known regarding the integration of Legendre Functions with respect to their degrees. In this paper several such integrals are evaluated, three different methods being employed. In § 2 proofs are given of a number of formulae which are required later. In § 3 an example is given of the evaluation of an integral by contour integration. The following section contains the proof of a formula of the Fourier Integral type, a special case of which was given in a previous paper (Proc. Roy. Soc. Edin., vol. li, 1931, p. 123). In § 5 an integral is evaluated by employing Fourier's Integral Theorem; while in § 6 other integrals are evaluated by. means of expansions in series.

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.

Some series and integrals involving associated Legendre functions (II)

1931 ◽  
Vol 27 (3) ◽  
pp. 381-386 ◽  
Author(s):  
W. N. Bailey
Keyword(s):  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Image Position

1. In a paper published recently in these Proceedings, I have proved the formulaewhere R(m)>0;where R(m) > −½; andwhere R(m) >− 1. In each case n is unrestricted.

Some series and integrals involving associated Legendre functions

1931 ◽  
Vol 27 (2) ◽  
pp. 184-189 ◽  
Author(s):  
W. N. Bailey
Keyword(s):  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Definite Integrals ◽  
Image Position ◽  
Limiting Case

1. In a recent paper I have given some definite integrals involving Legendre functions which, as a limiting case, give known results involving Bessel functions. In another paper I have shown how some integrals involving Bessel functions can be obtained from Bateman's integraland the well-known expansion

scholarly journals On Neumann's Formula for the Legendre Functions

1952 ◽  
Vol 1 (1) ◽  
pp. 10-12 ◽  
Author(s):  
T. M. Macrobert
Keyword(s):  
Positive Integer ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Image Position

§1. Introductory. The formulawhere w is zero or a positive integer and | ζ | > 1, was given by F. E. Neumann “Crelle's Journal, XXXVII (1848), p. 24”. In § 2 of this paper some related formulae are given; the extension to the case when n is not integral is dealt with in § 3; while in § 4 the corresponding formulae for the Associated Legendre Functions when the sum of the degree and the order is a positive integer are established.

scholarly journals Integrals Involving E-Functions and Associated Legendre Functions

1955 ◽  
Vol 2 (3) ◽  
pp. 127-128 ◽  
Author(s):  
T. M. Macrobert
Keyword(s):  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Image Position

The formulae to be proved are as follows.If p ≧ q + 1, R(l + m) > 0, R(αr − l − m + n)> −1, R(αr − l − m − n)> 0, r = 1,2,…p,If p ≧ q + 1, R(l >0, R(l+m >0, R(αr − l − m + n)> −1,

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