Asymptotic Expansion of the Associated Legendre Function Over the Interval $0 \leq \theta \leq \pi$

2011 ◽  
Vol 59 (4) ◽  
pp. 1424-1427
Author(s):  
J. S. Gardner
Keyword(s):  
Asymptotic Expansion ◽  
Legendre Function ◽  
Associated Legendre Function

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 asymptotic expansion of legendre function of large degree and order

1957 ◽  
Vol 249 (971) ◽  
pp. 597-620 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Complex Variable ◽  
Bessel Functions ◽  
Legendre Function ◽  
Large Degree ◽  
Legendre Functions ◽  
Airy Functions ◽  
Exponential Functions ◽  
The Third ◽  
Shaped Domain

New expansions for the Legendre functions and are obtained; m and n are large positive numbers, is kept fixed as is an unrestricted complex variable. Three groups of expansions are obtained. The first is in terms of exponential functions. These expansions are uniformly valid as with respect to z for all z lying in except for the strips given by. The second set of expansions is in terms of Airy functions. These expansions are uniformly valid with respect to z throughout the whole z plane cut from except for a pear-shaped domain surrounding the point z = — 1 and a strip lying immediately below the real z axis for which . The third group of expansions is in terms of Bessel functions of order m . These expansions are valid uniformly with respect to z over the whole cut z plane except for the pear-shaped domain surrounding z = — 1. No expansions have been given before for the Legendre functions of large degree and order.

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.

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.

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