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Virchenko N. — Generalized Associated Legendre Functions and Their Applications
Virchenko N. — Generalized Associated Legendre Functions and Their Applications

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Название: Generalized Associated Legendre Functions and Their Applications

Автор: Virchenko N.

Аннотация:

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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 195

Добавлена в каталог: 31.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\delta$-sequences method      160
$\Gamma$-function      44
b-function      179
B-function, incomplete      xiv xv 88 92
Barnes integral representation      179
Bessel function      88
Boundary value problems      xi 116
Buschman — Erdelyi's integral operator      147
Cauchy Cone functions      10
Cauchy problem      160
Cauchy theorem      12
Confluent hypergeometric function      148
Dual integral equations      116
E-function of Mac-Robert      xviii
Elliptic integrals      94
Elliptic type equation      164
Euler integral      179
Factorization of integral operators      152
Fourier cosine transform      134
Fourier inversion formula      134
Fourier sine transform      135
Fourier transform      134
Fractional integral operator      144
Fredholm integral equation      134
Fundamental function      161
Fundamental solution      162
G-function of Meijer      xviii
Gauss hypergeometric function      xii 175
Gauss hypergeometric function, analytic expansions      177
Gauss hypergeometric function, asymptotic expansion      181
Gauss hypergeometric function, integral representations      179
Goetze integral transform      103
Green function      98
H-function of Fox      xviii 116 172
Heaviside's unit function      117 174
Heine formula      xiv
Homogeneous integral equation      126
Hybrid dual integral equations      133
Hybrid generalized differential Legendre operator      168
Hypergeometric differential equation      xii 175
Hypergeometric generalized function      xviii
Inhomogeneous integral equations      131
Integral transforms      96
Jacobi functions      71
Jacobi functions, differential formula      72
Jacobi functions, generating functions      75
Jacobi functions, integral relation      73
Jacobi functions, orthogonality relation      72
Jacobi functions, recurrence formulas      71
Jordan double contour integral      xiii
Kummer series      176
Laplas integral transform      148
Laplas integral transform, modified      157
Laplas operator      xii
Legendre associated functions      3
Legendre associated functions, asymptotic expansions      9
Legendre associated functions, estimates      8
Legendre associated functions, integral representations      4
Legendre associated functions, recurrence formulas      6
Legendre differential equation      1
Legendre functions      1
Legendre functions, analytical expansions      1 2
Legendre functions, integral representations      2
Legendre generalized associated functions (of $1^{st}$ and $2^{nd}$ kind)      11
Legendre generalized associated functions generalized transform      104
Legendre generalized associated functions on the cross cut      14
Legendre generalized associated functions' type integral transform      98
Legendre generalized associated functions, asymptotic formulas      40 48
Legendre generalized associated functions, connections with Jacobi functions      74
Legendre generalized associated functions, derivatives      28
Legendre generalized associated functions, derivatives, logarithmic      50
Legendre generalized associated functions, integral representations      54 60
Legendre generalized associated functions, orthogonality      75
Legendre generalized associated functions, recurrence relations      32
Legendre generalized associated functions, series representations      17
Legendre generalized associated functions, zeros      67
Legendre generalized differential equation      11
Legendre polynomials      1
Legendre polynomials, analytical expansion      1
Legendre polynomials, generating function      3
Mehler — Fock integral transform      96
Mellin integral transform      157
Meromorphic function      4
Neumann integral      8
Neumann integral, generalization      8
Neumann's type integral      78
Nonsingular matrix      137
Operator $T^{m,n}_{k}$      115
Operator differential for $P^{m,n}_{k}(z)$, $Q^{m,n}_{k}(z)$      36
Operator equation      134
Operator Orthogonal polynomials      xi
Parseval equality      105
Pochhammer's notation      12
Pochhammer's symbol      xii
Potential theory      116
Prolate elliptic coordinates      xv
Regular function      3
Riemann equation      11 175
Riemann — Liouville fractional integral      144
Rodrigues formula for Jacobi function      73
Rodrigues formula for Legendre function      7
Rodrigues formula for Legendre function, extensions      7
Singular points      17
Slater theorem      152
Spherical functions      3
Spherical functions, associated      3
Titschmarsh method      106
Torus functions      10
Triple integral equations      126
Triple integral equations, systems      137
Vector equation      143
Vector equation, analogues      27
Weight function      123
Whipple formula for $P^{m,n}_{k}(z)$, $Q^{m,n}_{k}(z)$      74
Whipple formula for Legendre functions      6
Wright's generalized Bessel function      xviii
Wright's generalized functions of Kuipers and Meulenbeld      xii
Wright's generalized Legendre functions      xviii
Wright's hypergeometric function      xviii
Wronskians for $P^{m,n}_{k}(z)$, $Q^{m,n}_{k}(z)$      25
Wronskians for $P^{m,n}_{k}(z)$, $Q^{m,n}_{k}(z)$, recurrence formulas      29
Wronskians for $P^{\mu}_{v}(z)$, $Q^{\mu}_{v}(z)$      8
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