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.

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.

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.

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 The Legendre-tau technique for the determination of a source parameter in a semilinear parabolic equation

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
Abbas Saadatmandi ◽  
Mehdi Dehghan ◽  
Antonio Campo
Keyword(s):  
Control Parameter ◽  
Legendre Function ◽  
Numerical Procedure ◽  
Source Control ◽  
Operational Matrices ◽  
Semilinear Parabolic Equation ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Semilinear Parabolic

A numerical procedure for an inverse problem concerning diffusion equation with source control parameter is considered. The proposed method is based on shifted Legendre-tau technique. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution as a shifted Legendre function with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre functions. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

scholarly journals Multi-Integral Representations for Associated Legendre and Ferrers Functions

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1598
Author(s):  
Howard S. Cohl ◽  
Roberto S. Costas-Santos
Keyword(s):  
Integral Representation ◽  
Legendre Function ◽  
Order Parameters ◽  
Integral Representations ◽  
Special Values ◽  
Representation Formulas ◽  
Integration Formulas ◽  
Parameter Values ◽  
Associated Legendre Function

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.

Novel Technologıes and Nanotoxıcology of Medıcal Implants

2020 ◽  
Vol 16 ◽  
Author(s):  
Ramazan Akçan ◽  
Halit Canberk Aydogan ◽  
Mahmut Şerif Yıldırım ◽  
Burak Taştekin ◽  
Necdet Sağlam
Keyword(s):  
Human Health ◽  
Human Body ◽  
Diagnostic Procedures ◽  
Toxic Effects ◽  
Medical Implants ◽  
Healthcare Applications ◽  
Detailed Review ◽  
Novel Technologies ◽  
Technological Developments

Background/aim: Use of nanomaterials in the healthcare applications increases in parallel to technological developments. It is frequently utilized in diagnostic procedures, medications and in therapeutic implementations. Nanomaterials take place among key components of medical implants, which might be responsible for certain toxic effects on human health at nano-level. In this review, nanotoxicological effects, toxicity determination of nanobiomaterials used in human body and their effects on human health are discussed. Material and Method: A detailed review of related literature was performed and evaluated as per nanomaterials and medical implants. Results and Conclusion: The nanotoxic effects of the materials applied to human body and the determination of its toxicity are important. Determination of toxicity for each nanomaterial requires a detailed and multifactorial assessment considering the properties of these materials. There are limited studies in the literature regarding the toxic effects of nanomaterials used in medical implants. Although these implants are potentially biocompatible and biodegradable, it is highly important to discuss nanotoxicological characteristics of medical implant.

scholarly journals DETERMINATION OF RASA PANCHAKA (5 AYURVEDIC PRINCIPLES OF DRUG ACTION) OF A FOLK DRUG - Alstonia venenata R. Br.

2020 ◽  
Vol 8 (8) ◽  
pp. 4125-4132
Author(s):  
Anusha P R ◽  
Chandrakanth Bhat ◽  
Hariprasad Shetty ◽  
Sudhakar Bhat
Keyword(s):  
Stem Bark ◽  
Drug Action ◽  
Albino Rats ◽  
Test Drug ◽  
Herbal Drug ◽  
Small Tree ◽  
Detailed Review ◽  
Snake Bites ◽  
Trial Drug

Background and Objective: Systematic study of the folklore knowledge on herbal drug contributes to its conservation and preservation. Documentation of drugs in Ayurveda is based on the five principles called Rasa Panchaka (5 Ayurvedic principles of drug action). Alstonia venenata R. Br is a small tree belonging to Apocynaceae family. Its stem bark is used by tribes in fever, epilepsy and as anti-venom in snake bites. The aim of this study is to determine the Rasa Panchaka (5 Ayurvedic principles of drug action) of Al-stonia venenata R. Br. Methods: Detailed review of the trial drug was carried out. Rasa (taste) was determined by using direct per-ception method on 30 healthy volunteers. Veerya (potency) was determined by assessing the exothermic and endothermic reaction of the drug in water. Vipaka (taste after digestion), Guna (properties) and Prab-hava (specific action) were assessed by experimental study of the drug on 12 Wister Albino rats. Result and Conclusion: After the study Rasa panchaka (5 Ayurvedic principles of drug action) of the test drug was accessed as Tikta (bitter) Rasa (taste), Laghu (light) Rooksha (dry) Guna (property), Sheetha (cold) Veerya (potency)and Katu (pungent) Vipaka (taste after digestion).

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