Generalized Associated Legendre Functions And Their Applications

Iryna Fedotova, Nina Virchenko

• Publisher: World Scientific
• Year: 2001
• Print ISBN:
  9789810243531
  9789810243531
• eBook ISBN:
  9789812811783
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

The various types of special functions have become essential tools for scientists and engineers. One of the important classes of special functions is of the hypergeometric type. It includes all classical hypergeometric functions such as the well-known Gaussian hypergeometric functions, the Bessel, Macdonald, Legendre, Whittaker, Kummer, Tricomi and Wright functions, the generalized hypergeometric functions ?Fq, Meijer's G-function, Fox's H-function, etc.Application of the new special functions allows one to increase considerably the number of problems whose solutions are found in a closed form, to examine these solutions, and to investigate the relationships between different classes of the special functions.This book deals with the theory and applications of generalized associated Legendre functions of the first and the second kind, Pm,n?(z) and Qm,n?(z), which are important representatives of the hypergeometric functions. They occur as generalizations of classical Legendre functions of the first and the second kind respectively. The authors use various methods of contour integration to obtain important properties of the generalized associated Legnedre functions as their series representations, asymptotic formulas in a neighborhood of singular points, zero properties, connection with Jacobi functions, Bessel functions, elliptic integrals and incomplete beta functions.The book also presents the theory of factorization and composition structure of integral operators associated with the generalized associated Legendre function, the fractional integro-differential properties of the functions Pm,n?(z) and Qm,n?(z), the classes of dual and triple integral equations associated with the function Pm,n-1/2+i?(ch?) etc.

Table of contents

• Contents
• Foreword
• Preface
• §1. A general information on Legendre functions
• §2. The generalized associated Legendre functions
• §3. The series representations of the generalized associated Legendre functions
• §4. Relations between different solutions of the generalized Legendre equation. Wronskians of linearly independent solutions
• §5. Relations between contiguous generalized associated Legendre functions
• §6. Differential operators generated by the generalized associated Legendre equation
• §7. Asymptotic formulas for the generalized associated Legendre functions in a neighborhood of singular points
• §8. Asymptotic representations of the generalized associated Legendre functions as functions of parameters
• §9. Integral representations of the generalized associated Legendre functions of the first kind
• §10. Integral representations of the generalized associated Legendre functions of the second kind
• §11. Zeros of the generalized associated Legendre functions
• §12. Connection of the generalized associated Legendre functions with the Jacobi functions
• §13. Integral relations and series with the generalized associated Legendre functions
• §14. Relations between the generalized associated Legendre functions Bessel functions elliptic integrals and incomplete B-functions
• §15. Integral transforms with the generalized associated Legendre functions
• §16. Dual integral equations with the generalized Legendre function of the first kind Pm,n -1/2+iτ(chα)
• §17. Dual integral equations with Pm,n -1/2+iτ(chα) and the weight function [1 + G(τ)]
• §18. Triple integral equations with the generalized Legendre function of the first kind Pm,n -1/2+iτ(chα)
• §19. Hybrid dual integral equations
• §20. Systems of triple integral equations with the generalized associated Legendre functions of the first kind Pm,n -1/2+iτ(chα)
• §21. On a property of the fractional Riemann-Liouville integral of Pm,n k (z)
• §22. On some generalized integral operators of Buschman-Erdélyi's type
• §23. Factorization of integral operators with the generalized associated Legendre functions and its application for solving integral equations
• §24. Some generalizations of integral transforms of Mehler-Fock type and their applications
• §25. Examples of the integrals involving the generalized associated Legendre functions
• Appendix. The hypergeometric function
• Bibliography
• Subject index