Generalized associated Legendre functions and their applications

✍ Scribed by Virchenko N., Fedotova I.

• Publisher: WS
• Year: 2001
• Tongue: English
• Leaves: 217
• Category: Library

✦ Synopsis

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 "YFq", 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 Legendre functions of the first and the second kind, "Pm,nN(Z)" and "Qm,nN(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 Legendre functions as their series representations, asymptotic formulas in a neighbourhood 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,nN(Z)" and "Qm,nN(Z)", the classes of dual and triple integral equations associated with the function "Pm,n-1/2+/omega(chdelta)" etc.

✦ Table of Contents

Contents......Page 20
Foreword......Page 8
Preface......Page 12
1. A general information on Legendre functions\r\n......Page 22
2. The generalized associated Legendre functions\r\n......Page 32
3. The series representations of the generalized associated Legendre functions\r\n......Page 38
4. Relations between different solutions of the generalized Legendre equation. Wronskians of linearly independent solutions\r\n......Page 46
5. Relations between contiguous generalized associated Legendre functions\r\n......Page 53
6. Differential operators generated by the generalized associated Legendre equation\r\n......Page 56
7. Asymptotic formulas for the generalized associated Legendre functions in a neighborhood of singular points\r\n......Page 61
8. Asymptotic representations of the generalized associated Legendre functions as functions of parameters\r\n......Page 69
9. Integral representations of the generalized associated Legendre functions of the first kind\r\n......Page 75
10. Integral representations of the generalized associated Legendre functions of the second kind\r\n......Page 81
11. Zeros of the generalized associated Legendre functions\r\n......Page 88
12. Connection of the generalized associated Legendre functions with the Jacobi functions\r\n......Page 92
13. Integral relations and series with the generalized associated Legendre functions\r\n......Page 99
14. Relations between the generalized associated Legendre functions Bessel functions elliptic integrals and incomplete B-functions\r\n......Page 109
15. Integral transforms with the generalized associated Legendre functions\r\n......Page 117
16. Dual integral equations with the generalized Legendre function of the first kind ......Page 137
17. Dual integral equations with P and the weight function [1+G(x)]......Page 144
18. Triple integral equations with the generalized Legendre function of the first kind P......Page 147
19. Hybrid dual integral equations\r\n......Page 154
20. Systems of triple integral equations with the generalized associated Legendre functions of the first kind......Page 158
21. On a property of the fractional Riemann-Liouville integral of Pm,n k (z) ......Page 165
22. On some generalized integral operators of Buschman-Erdlyi's type\r\n......Page 168
23. Factorization of integral operators with the generalized associated Legendre functions and its application for solving integral equations\r\n......Page 173
24. Some generalizations of integral transforms of Mehler-Fock type and their applications\r\n......Page 181
25. Examples of the integrals involving the generalized associated Legendre functions\r\n......Page 192
Appendix. The hypergeometric function......Page 196
Bibliography......Page 204
Subject index......Page 214