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Erdelyi A. Higher Transcendental Functions, Vol. 2
Erdelyi A.  Higher Transcendental Functions, Vol. 2







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: Higher Transcendental Functions, Vol. 2

: Erdelyi A.

:

The Bulletin of the London Mathematical Society hailed this three-volume series as "The most widely cited mathematical works of all time and a basic reference source for generations of applied mathematicians and physicists throughout the world." Working from extensive notes on familiar special functions by the renowned mathematician Harry Bateman, a team of editors not only finished Bateman's original project but also made significant advances in mathematical analysis. The books, which can be used independently of each other, consist of Volume 1, which focuses on hypergeometric series; Volume 2, an exploration of Bessel functions, orthogonal polynomials, and elliptic functions and integrals; and Volume 3, an examination of automorphic functions, spheroidal and ellipsoidal wave functions, and other functions.


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: 13.07.2005

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Hermite polynomials of several variables      283 ff.
Hermite polynomials, addition theorems for      196
Hermite polynomials, asymptotic behavior as n$\rightarrow\infty$      201
Hermite polynomials, connection with Laguerre polynomials      193 195
Hermite polynomials, differential equations for      193
Hermite polynomials, differentiation formula for      193
Hermite polynomials, expansion of analytic functions in terms of      211
Hermite polynomials, expansions in series of spherical Bessel functions      201 ff.
Hermite polynomials, expressed as confluent hypergeometric functions      194
Hermite polynomials, generating functions for      194
Hermite polynomials, inequalities for      208
Hermite polynomials, integral representation for      194
Hermite polynomials, integrals involving      194 ff.
Hermite polynomials, mean convergence of series of      210
Hermite polynomials, monotonic properties of      208
Hermite polynomials, recurrence relation for      193
Hermite polynomials, relation to parabolic cylinder functions      117
Hermite polynomials, Rodrigues formula for      193
Hermite polynomials, series of      193 ff. 216
Hermite polynomials, zeros of      204 ff.
Hurwitzs theorem on zeros of Bessel functions      59
Hurwitzs theorem on zeros of Bessel functions, generalizations of      59
Hyperelliptic integrals      296
Hypergeometric polynomials      see Jacobi polynomials
Hypergeometric series, of Lauricella      275
Hyperspherical harmonics      232 ff.
Incomplete gamma functions      133 ff.
Incomplete gamma functions, asymptotic expansions for large x      135
Incomplete gamma functions, asymptotic representations for large $\alpha$      140
Incomplete gamma functions, connection with confluent hypergeometric functions      133 136
Incomplete gamma functions, continued fraction expansion      136
Incomplete gamma functions, differentiation formulas      135
Incomplete gamma functions, expansions in inverse factorials      139
Incomplete gamma functions, expansions in series of Bessel functions      139
Incomplete gamma functions, expansions in series of Laguerre polynomials      139
Incomplete gamma functions, integral representations of      137 138
Incomplete gamma functions, loop integral for      137
Incomplete gamma functions, Nielsens expansion for      139
Incomplete gamma functions, power series expansions of      135
Incomplete gamma functions, recurrence relations for      134
Incomplete gamma functions, zeros of      14 ff.
Integral equations involving Bessel functions      76 ff.
Integral representations of arbitrary functions in terms of Bessel functions      73 ff.
Integral representations of arbitrary functions in terms of Bessel functions,      388
Invariants of a quartic      305
Invariants of Weierstrass elliptic functions, and integrals      305 329
Inversion, problem of      326 ff. 375
Irreducible set of zeros or poles of an elliptic function      326
Jacobi function of the second kind      170 ff.
Jacobi polynomials      164 168 224 226
Jacobi polynomials, asymptotic behavior as n$\rightarrow\infty$ of      198
Jacobi polynomials, connection with Bessel functions      173
Jacobi polynomials, connection with Laguerre polynomials      191
Jacobi polynomials, convergence of series of      210 ff.
Jacobi polynomials, differential equation for      169
Jacobi polynomials, expansion of analytic functions in terms of      211
Jacobi polynomials, expressed as hypergeometric functions      170
Jacobi polynomials, generating function of      172
Jacobi polynomials, inequalities for      205 ff.
Jacobi polynomials, integral representation for      172
Jacobi polynomials, mean convergence of series of      210
Jacobi polynomials, polynomials associated with      171
Jacobi polynomials, recurrence formula for      169
Jacobi polynomials, represented as finite differences      173
Jacobi polynomials, Rodrigues formula for      169
Jacobi polynomials, series of      172 212
Jacobi polynomials, zeros of      202 ff.
Jacobi Anger formula      7
Jacobian elliptic functions      see Elliptic functions
Kapteyn series of Bessel functions      66 ff. 103
Kapteyn series of the second kind      67 ff.
Kelvins functions      6 101
L, $\mathfrak{L}$ Laplace transform      46 191
Lagrangean interpolation      161
Laguerre polynomials      164 188 226
Laguerre polynomials, Abel summability of series of      211
Laguerre polynomials, asymptotic behavior as n$\rightarrow\infty$ of      199 ff.
Laguerre polynomials, connected with Bessel functions      191
Laguerre polynomials, connected with Jacobi polynomials      191
Laguerre polynomials, differential equations for      188
Laguerre polynomials, differentiation formulas for      189 190
Laguerre polynomials, expansion of analytic functions, in terms of      211
Laguerre polynomials, expansions in series of Bessel functions      199 ff.
Laguerre polynomials, expressed as confluent hypergeometric functions      189
Laguerre polynomials, generating functions of      189
Laguerre polynomials, inequalities for      205 ff.
Laguerre polynomials, integral representations of      190
Laguerre polynomials, integrals involving      191
Laguerre polynomials, mean convergence of series of      210
Laguerre polynomials, monotonic properties of      207 ff.
Laguerre polynomials, recurrence relations for      188 190
Laguerre polynomials, represented as finite differences      191
Laguerre polynomials, Rodrigues formula for      188
Laguerre polynomials, series of      188 ff. 214
Laguerre polynomials, zeros of      204 ff.
Lambda-group      367
Laplace transform      45 ff. 191
Laplaces expansion      242 281
Lattice, line      324
Lattice, point      323
Legendre function of the second kind      180 ff.
Legendre functions      182 ff.
Legendre functions, connection with Gegenbauer polynomials      177
Legendre functions, relations with Bessel functions      55 ff.
Legendre polynomial      178
Legendre polynomials      164 178
Legendre polynomials,      389
Legendre polynomials, addition theorem for      182 ff.
Legendre polynomials, asymptotic behavior as n$\rightarrow\infty$ of      197
Legendre polynomials, differential equation for      179
Legendre polynomials, differentiation formulas for      179
Legendre polynomials, equiconvergence theorem for series of      211
Legendre polynomials, expressed as hypergeometric functions      180
Legendre polynomials, generating functions of      182
Legendre polynomials, Hilbs formula for      197
Legendre polynomials, inequalities for      205 ff.
Legendre polynomials, integral representations of      182
Legendre polynomials, monotonic properties of      208
Legendre polynomials, recurrence relation for      179
Legendre polynomials, series of      182 ff. 214
Legendres relation      320 329
Lemniscate functions      320 376
li (x) logarithmic integral      143
Line-lattice      324
Logarithmic integral      143 (see also Exponential integrals)
Logarithmic integral, special cases of      41 ff.
Lommels functions $s_{\mu,\nu}(z)$, $S_{\mu,\nu}(z)$      40 ff. 73 84
Lommels functions of two variables $U^_{\nu}(w, z)$, $V_{\nu}(w, z)$      42
Lommels polynomials $R_{m, \nu}(z)$      34 ff.
Macdonalds integral representations of products of Bessel functions      53 ff.
Maxwells theory of poles      251
Meijers generalization of Laplace transforms      75
mesh      325
Modified Bessel function of the first kind      5
Modified Bessel function of the third kind      5
Modified Bessel function of the third kind of integer order      9
Modified Bessel function of the third kind of order zero      9
Modified Bessel function of the third kind, duplication formula for      45
Modified Bessel function of the third kind, zeros of      62 ff.
Modified Bessel functions      5 ff.
Modified Bessel functions of order $n + ^{1}/{ 2}$      see Spherical Bessel functions
Modified Bessel functions of order $\pm^{1}/{ 2}$      10 79
Modified Bessel functions, addition theorems for      43 ff. 102
Modified Bessel functions, analytic continuation of      80
Modified Bessel functions, asymptotic expansions for      22 ff. 86
Modified Bessel functions, differentiation formulas for      79
Modified Bessel functions, integral representations of      18 82
Modified Bessel functions, integrals involving      45 ff. 56 89
Modified Bessel functions, recurrence relations for      79
Modified Bessel functions, relations with Legendre functions      56
Modified Bessel functions, Wronskians of      80
Modular functions      see Elliptic modular functions
Modular functions, fundamental region of      375
Modular group      366 374
Modulus of elliptic functions and integrals      306
Moment problem      163
Moments      157
nc u Glaishers function      322
nd u Glaishers function      322
Neumann series of Bessel functions      63 ff. 98
Neumann series, of the second kind      65
Neumanns function      see Bessel function of the second kind
Neumanns polynomials $O_{n}(z)$      32 ff.
Neumanns polynomials $\Omega_{n}(z)$      34
Nicholsons formula for Bessel functions of large order and variable      28 88
Nicholsons integral representations of products of Bessel functions      54
ns u Glaishers function      322
Orthogonal group      257 ff.
Orthogonal group, Cayleys representation of the      257
Orthogonal invariant      233
Orthogonal invariant,      390
Orthogonal polynomials      153 ff.
Orthogonal polynomials, and continued fractions      162 ff.
Orthogonal polynomials, and mechanical quadrature      160
Orthogonal polynomials, Christoffel Darboux formula for      159
Orthogonal polynomials, expansion problems relating to      209 ff.
Orthogonal polynomials, extremum properties of      160
Orthogonal polynomials, in the circle      273 ff.
Orthogonal polynomials, in the sphere and hypersphere      273 ff.
Orthogonal polynomials, in the triangle      269 ff.
Orthogonal polynomials, of a discrete variable      221 ff.
Orthogonal polynomials, of several variables      264 ff.
Orthogonal polynomials, recurrence relations for      158 ff.
Orthogonal polynomials, zeros of      158
Orthogonal system      154
Orthogonal system of parabolic cylinder functions      122
Orthogonalization      154 ff.
Orthonormal system of polynomials      see Orthogonal Orthogonal polynomials
Parabolic cylinder functions      116 117
Parabolic cylinder functions, addition theorem for      123 124
Parabolic cylinder functions, asymptotic expansions of      122 ff.
Parabolic cylinder functions, Cherrys theorem for      124 ff.
Parabolic cylinder functions, connection with error functions      119
Parabolic cylinder functions, differential equation for      116
Parabolic cylinder functions, generating function of      119
Parabolic cylinder functions, integral representations of      119 120
Parabolic cylinder functions, integrals involving      121 122
Parabolic cylinder functions, real zeros of      126
Parabolic cylinder functions, Wronskians of      117
Parabolic cylinder, coordinates of      115
Paraboloid of revolution, coordinates of      115
Paraboloid of revolution, coordinates of, functions of (see Functions of the paraboloid of revolution ) Parsevals formula      157
Period parallelogram      325
Period parallelogram, fundamental      325
Periods of elliptic functions      328 341
Periods of elliptic integrals      303 314
Periods of functions      324
Periods, primitive      324
Point-lattice      323
Poissons integral for Bessel functions      14
Polar coordinates, hyperspherical      233
Polynomials      see also Orthogonal polynomials Classical Jacobi Gegenbauer Legendre Tchebichef Laguerre and
Polynomials of A.C. Aitken and H.T. Gonin      224
Polynomials of Appell      269 ff.
Polynomials of E. Heine      218
Polynomials of F. Pollaczek      218 ff.
Polynomials of H. Bateman      224
Polynomials of Hermite and Angelescu      283
Polynomials of Hermite and Didon      273 ff.
Polynomials of L.V. Charlier      222 226
Polynomials of S. Bernstein and G. Szeg$\ddot{o}$      217
Polynomials of W. Hahn      165 222 224
Polynomials, harmonic      237 ff.
Polynomials, of Hermite Didon Appell Kamp$\acute{e}$ de F$\acute{e}$riet      259 ff.
Polynomials, of J. Meixner      222 225 227
Polynomials, of J.P. Gram and H.E.H. Green-leaf      225
Polynomials, of M. Krawtchouk      222 224
Polynomials, of N. Achyeser      218
Polynomials, of Tchebichef      222 223
Primitive periods      324
Products of Bessel functions,      53 96
Products of Bessel functions, integral representations for      47 ff.
Products of Bessel functions, power series for      10 ff.
Quadratic forms      283 ff.
Quaternions      255
Quaternions,      391
Raabes integrals      144
Rational curves      295
Rodrigues formula      179
Rodrigues formula,      193 276 279 285
Rodrigues formula, finite difference analogue of      222
Rodrigues formula, generalized      164 169 175
S(x) Fresnel integral      149
S(x, $\alpha$) generalized Fresnel integral      149
sc u Glaishers function      322
Scalar product of functions      153 264
Scalar product of vectors      232 273
Schl$\ddot{a}$flis integral representations of Bessel functions      17
Schl$\ddot{a}$flis polynomials $S_{n}(z)$      34
Schl$\ddot{o}$milch series of Bessel functions      68 ff. 103
Schl$\ddot{o}$milch series of Bessel functions, generalized      68
sd u Glaishers function      322
Separation theorems      162
Shi x modified sine integral      146
si x sine integral      145
Sine integrals      145 ff.
Sine integrals, generalizations of      147
sn u Jacobis elliptic function      322
Sommerfelds integral representations, of Bessel functions      19 ff.
Sommerfelds notation for      10 78 79
Sommerfelds notation for spherical Bessel functions      10
Sommerfelds wave      125
Sonine P$\acute{o}$lya theorem      205
Sonines expansion      43 64
Sonines integrals involving Bessel functions      46
Spherical Bessel functions      9 78 79
Spherical Bessel functions, expressed in terms of elementary functions      10 78 79
Spherical harmonics      232 ff.
Spherical polynomials      see Legendre polynomials
Spherical polynomials, addition theorem of      242 ff.
Spherical polynomials, completeness of      241
Spherical polynomials, four-dimensional      253 ff.
Spherical polynomials, generating function of      248 ff.
Spherical polynomials, Maxwells theory of      251 ff.
Spherical polynomials, orthogonal property of      240 ff.
Spherical polynomials, three-dimensional      248 ff.
Spherical polynomials, transformation of      256 ff.
Spherical surface harmonics      240 ff.
Stationary phase, method of      27 ff.
Steepest descents, method of      24
Steepest descents, method of, applied to modified Bessel functions of the third kind      24 ff.
Struves functions      18 37
Struves functions,      47 68 74 89 99 103
Struves functions, integrals involving      98
Tchebichef polynomials      183 ff.
Tchebichef polynomials, differential equations for      185
Tchebichef polynomials, differentiation formulas for      185 186
Tchebichef polynomials, expressed as hypergeometric functions      186
Tchebichef polynomials, generating functions, of      186
Tchebichef polynomials, recurrence relations for      185
Tchebichef polynomials, Rodrigues formulas for      185
Theta functions      354 ff. (see also Elliptic functions )
Theta functions of zero argument      359
Theta functions, expression of elliptic functions and integrals in terms of      360
Theta functions, Hermites      360
Theta functions, transformations of      368 ff.
Transformation, Landens      see Elliptic functions Elliptic
Transformation, of elliptic functions and integrals      see Elliptic functions Elliptic
Transformation, unimodular      324
Truncated exponential series      136 ff.
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