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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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: 1955

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: 13.07.2005

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$(\chi, \eta)$ scalar product of vectors      232 273
$(\nu, m)$ Hankels symbol      10
$(\phi, \psi)$ scalar product of functions      153 264
$(\|\chi\|)$ length of vector $\chi$      232
$A_{1}$(t), $A_{2}$(t) Airy integrals      29
$B_{n;\mu,\nu}(z) Gegenbauers polynomial      34
$C^{\lambda}_{n}(x)$ Gegenbauer polynomials      174
$D_{\nu}(z)$ Parabolic cylinder function      117
$E*(x)$ modified exponential integral      143
$e_ {\alpha} = \gamma(\omega_{\alpha})$      330
$E_{1}(x)$, exponential integral      143
$E_{mn}$ Appells polynomials      271
$E_{n}(x)$ function used in astrophysics and nuclear physics      134
$e_{n}(x)$ Truncated exponential series      136
$E_{\nu}(z)$ Webers function      35
$F_{mn}$ Appells polynomial      271
$g_{2}$, $g_{3}$ invariants of Weierstrass elliptic functions      299 305
$G_{m_{1}, ..., m_{n}}(x_{1}, ..., x_{n})$ Hermite polynomials of several variables      285
$hei_{\nu}(x)$, $hei_{\nu}(x)$ Kelvins functions, 6 H (x) error function      147
$H^{(1)}_{\nu}(z)$, $H^{(2)}_{\nu}(z)$ Bessel functions of the third kind      4
$H_{m_{1}, ..., m_{n}}(x_{1}, ..., x_{n})$ Hermite polynomial of several variables      285
$h_{m} = 1 + \frac{1}{2} + ... + \frac{1}{m}$      8
$H_{n}(x)$ Hermite polynomials      193
$H_{\nu}(z)$ Struves function      37
$i^{n}$ erfc x repeated integral of error function      149
$I_{\nu}(z) modified Bessel function of the first kind      5
$J(\tau)$ absolute invariant      375
$J_{\nu, m}(z)$ cut Bessel function of the first kind      21
$J_{\nu}(z)$ Angers function      35
$J_{\nu}(z)$ Bessel function of the first first kind      4
$k$ modulus of Jacobis elliptic functions and integrals      300 306
$kei_{\nu}x$, $kei_{\nu}x$ modified Kelvin functions      6
$K_{n}(x)$ function used in astrophysics and nuclear physics      134
$K_{\nu}(z)$ modified Bessel function of the third kind      5
$L^{\alpha}_{n}(x)$ Laguerre polynomial      188
$l_{n}{\alpha}$ polynomial      140
$L_{\nu}(z)$ modified Struve function      38
$mathfrak{G}^{u}_{x}$ Gauss transform      195
$O_{n}(z)$ Neumanns polynomial      32
$P^{\alpha,\beta}_{n}(x)$ Jacobi polynomial      168
$q = e^{i\pi\tau}$      345
$q^{(\alpha, \beta)}_{n}(x)$ polynomials associated with Jacobi polynomials      171
$Q^{(\alpha,\beta)}_{n}(x)$ Jacobi function of the second kind      170
$Q_{n}(x)$ Legendre function of the second kind      180
$Q_{n}(x)$ Legendre function of the second kind on the cut      181
$R_{m, \nu}(z)$ Lommels polynomials      34
$S_{n}(z)$ Schl$\ddot{a}$flis polynomial      34
$s_{\mu,\nu}(z)$ Lommels function      40
$S_{\mu\nu}(z)$ Lommels function      40
$T_{n}(x)$ Tchebichef polynomial      183
$U^{s}_{m_{1}, ..., m_{n}}(x_{1}, ..., x_{n})$ polynomials of Hermite and Didon      277
$U_{n}(x)$ Tchebichef polynomial      183
$U_{\nu}(w, z)$ Lommels function of two variables      42
$V^{s}_{m_{1}, ..., m_{n}}(x_{1}, ..., x_{n})$ polynomials of Hermite and Didon      274 ff.
$V_{\nu}(w, z)$ Lommels function of two variables      42
$W$ Wronskian      12
$Y^{m}_{n}(\theta,\phi)$ spherical surface harmonic      250
$Y_{\nu}(z)$ Bessel function of the second kind      4
$Z_{\nu}(z)$ Bessel function      2 48
$_{n,\nu}$(z) Gegenbauers polynomial      34
$\alpha(x)$ error function      147
$\Delta = g^{3}_{2} - 27g^{2}_{3}$ discriminant      332
$\Delta$ Laplaces operator      2 115 234
$\Delta(\phi, k)$      317
$\eta = \zeta(\omega)$, $\eta = \zeta(\omega)$      329
$\eta_{\alpha} = \zeta(\omega_{\alpha})$      330
$\gamma$ Euler Mascheroni constant      vol. I p.
$\gamma(z)$ Weierstrass elliptic function      323 328
$\Gamma(\alpha, x)$ complementary incomplete gamma function      133
$\gamma(\alpha, x)$ incomplete gamma function      133
$\gamma(\alpha, x)$ modified incomplete gamma function      140
$\gamma*(\alpha, x)$ modified incomplete gamma function      133
$\int^{(0+)}_{+\infty}$ loop integral      15
$\lambda($\tau)$ modular function      374
$\mathbf{B}$ complete elliptic integral      321
$\mathbf{C}$ complete elliptic integral      321
$\mathbf{D}$ complete elliptic integral      321
$\mathbf{E, E}$ complete elliptic integrals of the second kind      314 317
$\mathbf{E}$ Incomplete elliptic integral of the second kind      300 313
$\mathbf{K, K}$ complete elliptic integrals of the first kind      314 317
$\mathfrak{G}^{u}_{x}$ multi-dimensional Gauss transform      289 290
$\mathfrak{J}^{mn}_{mn}$ Appells polynomial      270
$\nabla_{\nu}$ Bessels differential operator      4
$\omega, \omega$ periods of Weierstrass elliptic functions      328
$\Omega_{n}(z)$ Neumanns polynomial      34
$\omega_{\alpha}$ periods of Weierstrass elliptic functions      330
$\Pi$ complete elliptic integral of the third kind      317
$\Pi$ incomplete elliptic integral of the third kind      301 313
$\sigma(z)$ Weierstrass sigma function      329
$\sigma_{\alpha}(z)$ sigma functions      330
$\tau = \omega/\omega$      328
$\theta_{1}(v), ...,\theta_{4}(v)$ Theta functions      355
$\Theta_{\mu\nu(v)$ Hermites theta function      360
$\zeta(z)$ Weierstrass zeta function      329
A(t) Airy function      200
Abelian integrals      295 ff.
Absolute invariant      375
Airys integrals      22
am u Jacobis function      322
Angers function $\mathbf{J}_{\nu(z)$      35 ff. 84 99 103
Appell series      280 ff.
Approximation of quadratically integrable functions      156
Associated polynomials      162 ff.
Automorphic functions      296 374
Barnes integral representations of Bessel functions      21 ff.
Bassets function      see Modified Bessel function of the third kind
bei $x$, $bei_{\nu}x$ Kelvins functions      6
ber x, $bei_{\nu}x$ Kelvins functions      6
Bessel coefficients      6
Bessel coefficients, generating function for      7
Bessel coefficients, integral representations for      13 ff. 81
Bessel function of the first kind      4
Bessel function of the first kind, derivative with respect to the order      7
Bessel function of the first kind, duplication formula for      45
Bessel function of the first kind, inequalities for      14 66
Bessel function of the first kind, series involving      63 ff.
Bessel function of the first kind, series involving, Bessel function of the first kind, zeros of      59 ff.
Bessel function of the second kind      4
Bessel function of the second kind, of integer order      7
Bessel function of the second kind, of order zero      8
Bessel function of the second kind, zeros of      61 ff.
Bessel function of the third kind      4
Bessel function of the third kind, zeros of      62
Bessel functions, addition theorems for      43 ff. 101
Bessel functions, addition theorems for, Bessel functions, analytic continuation of      12 80
Bessel functions, and wave motion      2 ff.
Bessel functions, as limits of Jacobi polynomials      173
Bessel functions, as limits of Laguerre polynomials      191
Bessel functions, asymptotic expansions for      22 ff. 85
Bessel functions, asymptotic expansions for, differential equations for      13
Bessel functions, differentiation formulas for      11
Bessel functions, integral representations for      14 ff. 57 81
Bessel functions, integrals involving      45 ff. 57 90
Bessel functions, notations for      3
Bessel functions, of imaginary order      87 ff.
Bessel functions, of order $n +^{1}/_{ 2}$      see Spherical Bessel functions
Bessel functions, of order $\pm^{1}/_{ 2}$      10 79
Bessel functions, power series for products of      10 ff.
Bessel functions, recurrence relations for      12
Bessel functions, relations with Legendre functions      55 ff.
Bessel functions, series involving      58 63 98
Bessel functions, series involving, Bessel functions, Wronskians of      12 79
Bessel functions, series involving, Bessel functions, zeros of      58 ff.
Bessel polynomials      10
Bessels differential equation      3 ff.
Bessels differential equation, Bessels inequality      157
Bilinear forms      284
Biorthogonal system      265
Birational invariant      295
Birational transformation      295
C (x) Fresnel integral      149
C (x, $\alpha$) generalized Fresnel integral      149
cd u Glaishers function      322
cell      325
Chi x modified cosine integral      146
Christoffel numbers      161
Christoffel Darboux formula      159 269
Ci x cosine integral      145
Classical orthogonal polynomials      163 ff. (see also Legendre Gegenbauer Hermite Laguerre
Classical orthogonal polynomials, characterization of      164
Classical orthogonal polynomials, differential equation for      166 ff.
Classical orthogonal polynomials, differential equation for, properties of      164 166
Classical orthogonal polynomials, differentiation formula for      167
cn u Jacobis elliptic function      322
Complete elliptic integrals      see Elliptic integrals
Confluent hypergeometric functions, expansion in terms of parabolic cylinder functions      124
Conformal mappings, involving elliptic functions and integrals      376 ff.
Convergence in mean of generalized Fourier expansions      157
Convolution      45
Cornus spiral      151
Cosine integral      145 ff.
cs u Glaishers function      322
Cut Bessel functions      22
dc u Glaishers function      322
Didon series      280 ff.
Didon series, Dini series of Bessel functions      70 ff.
dn u Jacobis elliptic function      322
Doubly-periodic functions      323 ff. (see also Elliptic functions)
ds u Giaishers function      322
E(u) Jacobis function      343
Ei (x) exponential integral      143
Elliptic functions      294 ff. 322 325
Elliptic functions, addition theorems satisfied by      328
Elliptic functions, differential equations satisfied by      327
Elliptic functions, expression of, in terms of $\gamma(z)$, $\gamma(z)$      335
Elliptic functions, expression of, in terms of sigma functions      337
Elliptic functions, expression of, in terms of zeta functions      336 ff.
Elliptic functions, general properties of      325 ff.
Elliptic functions, integrals of, expressed in terms of Weierstrass functions      337
Elliptic functions, Jacobian      322 ff. 340
Elliptic functions, Jacobian, addition theorems for      344
Elliptic functions, Jacobian, degenerate cases of      354
Elliptic functions, Jacobian, expressed in terms of theta functions      362
Elliptic functions, Jacobian, expressed in terms of Weierstrass functions      340 ff.
Elliptic functions, Jacobian, Landens transformation of      372
Elliptic functions, Jacobian, linear transformation of      367 ff.
Elliptic functions, Jacobian, periods, zeros, poles and residues of      341
Elliptic functions, Jacobian, quadratic transformations of      372 ff.
Elliptic functions, Jacobian, special values of      346 ff.
Elliptic functions, Jacobian, with $0 < k < 1$      349 ff.
Elliptic functions, Nevilles notation for      294 342
Elliptic functions, order of      326
Elliptic functions, residues of      326
Elliptic functions, transformations of      365 ff.
Elliptic functions, Weierstrass      323 328
Elliptic functions, Weierstrass duplication formula for      333
Elliptic functions, Weierstrass expressed in terms of theta functions      360 ff.
Elliptic functions, Weierstrass, addition theorem for      332 ff.
Elliptic functions, Weierstrass, degenerate cases of      339 ff.
Elliptic functions, Weierstrass, differential equation for      331 ff.
Elliptic functions, Weierstrass, Landens transformation of      371 ff.
Elliptic functions, Weierstrass, linear transformations of      367 ff.
Elliptic functions, Weierstrass, quadratic transformations of      371 ff.
Elliptic functions, Weierstrass, with real invariants      338 ff.
Elliptic integrals      294 ff.
Elliptic integrals, addition theorems for      315
Elliptic integrals, complete, expressed in terms of hypergeometric series      318
Elliptic integrals, complete, integration formulas for      322
Elliptic integrals, complete, Legendres relation for      320
Elliptic integrals, complete, of the first, second, and third kind      314 317
Elliptic integrals, complete, particular cases of      320
Elliptic integrals, complete, transformations of      318 ff.
Elliptic integrals, differentiation formulas for      317 321
Elliptic integrals, expressed in terms of theta functions      363ff.
Elliptic integrals, expressed in terms of Weierstrass functions      337 ff.
Elliptic integrals, interchange theorem for      303 ff. 315
Elliptic integrals, inversion of      322 ff.
Elliptic integrals, Landens transformation of      317
Elliptic integrals, Legendres form of      300 ff. 314
Elliptic integrals, Legendres, evaluation of      308 ff.
Elliptic integrals, linear transformations of      315 ff.
Elliptic integrals, Lows form of      301
Elliptic integrals, moduli of periodicity of      303
Elliptic integrals, of the first, second, and third Elliptic integrals, kinds      299 ff. 313
Elliptic integrals, periods of      303 314
Elliptic integrals, reduction of      296 ff. 304
Elliptic integrals, reduction to Legendres normal form      305 ff.
Elliptic integrals, reduction to Weierstrass normal form      304 ff.
Elliptic integrals, singularities of      303 314
Elliptic integrals, Weierstrass form of      299 ff.
Elliptic modular functions      374 ff.
Equianharmonic elliptic functions and integrals      306 320
Erf x error function      147
Erfc x complementary error function      147
Erfi x modified error function      147
Error functions      147 ff.
Error functions, connection with parabolic Error functions, cylinder functions      119
Error functions, expansions in terms of Bessel Error functions, functions      148
Error functions, power series expansions of      147
Error functions, repeated integrals of      149
Exponential integrals      143 ff.
Exponential integrals, expressed in terms of confluent Exponential integrals, hypergeometric functions      143
Exponential integrals, expressed in terms of incomplete Exponential integrals, gamma functions      143
Exponential integrals, generalizations of      145
F Incomplete elliptic integral of the first kind      300 313
Field of elliptic functions      327
Field, differential      327
Fourier coefficients (generalized)      156
Fourier series (generalized)      156
Fourier Bessel series      70 ff. 104
Fresnel integrals      149 ff.
Fresnel integrals, connection with error functions      149
Fresnel integrals, connection with incomplete gamma functions      149
Functions of the parabolic cylinder      see Parabolic cylinder functions
Functions of the paraboloid of revolution      126 ff.
Fundamental period-parallelogram,      325
Fundamental region of the $\lambda$-group      375
Fundamental region of the modular group      375
Funk Hecke theorem      247
Gauss transforms      194
Gauss transforms, multi-dimensional      289 ff.
Gegenbauer polynomials      164 174 235
Gegenbauer polynomials, asymptotic behavior as n$\rightarrow$$\infty$      198
Gegenbauer polynomials, connection with Legendre functions      177
Gegenbauer polynomials, expressed as hypergeometric functions      175 ff.
Gegenbauer polynomials, generating functions of      177
Gegenbauer polynomials, inequalities for      206 ff.
Gegenbauer polynomials, integral representations for      177
Gegenbauer polynomials, monotonic properties of      208
Gegenbauer polynomials, recurrence formula for      175
Gegenbauer polynomials, Rodrigues formula for      175
Gegenbauer polynomials, series of      177 ff. 213
Gegenbauer polynomials, zeros of      203 ff. 387
Gegenbauers addition theorem for Bessel functions      43 ff.
Gegenbauers polynomials $A_{n,\nu}(z), B_{n;\mu,\nu}(z)$      34
Generalized Dirichlet series      72 ff.
Genus of algebraic curves      295
Glaishers notation,of Jacobian elliptic functions      322
Grafs addition theorem for Bessel functions      44 ff.
Grams determinant      155
Gublers integral representations, of Bessel functions      17 ff.
Hankels function      see Bessel function of the third kind
Hankels infinite integral, involving Bessel functions      49
Hankels integral representations of Bessel functions      15 ff.
Hankels inversion theorem      73
Hardys generalization of      73 ff.
Hardys generalization of, Hankels symbol ($\nu$, m)      10
Harmonic polynomials      237 ff.
Harmonic polynomials, complete set of      239 ff.
Hermite polynomials      164 192
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