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Luke Y.L. — Special Functions and Their Approximations. Volume II

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Название: Special Functions and Their Approximations. Volume II

Автор: Luke Y.L.

Аннотация:

These volumes are designed to provide scientific workers with a self-contained and unified development for many of the mathematical functions which arise in applied problems, as well as the attendant mathematical theory for their approximations. These functions are often called the special functions of mathematical physics or more simply the special functions.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1969

Количество страниц: 506

Добавлена в каталог: 17.11.2013

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Предметный указатель

Pade approximation, convergence of      II 80
Pade approximation, table      II 66 75—77
Painleve's transcendent      II 89
Parabolic cylinder functions      I 135 212
Pi ( $\pi$ ), asymptotic expansion for      I 35 36
Pi ( $\pi$ ), value of      II 292
Polynomial approximation      see "Approximation"
Polynomials      see "Bessel orthogonal etc."
Power series and ratio of power series, Pade approximation for      II 76
Powers of z, expansions for      see "Expansion"
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, approximation      see "Approximation"
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, asymptotic expansion      I 33
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, definition and elementary properties      I 12 13
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, expansion      see "Expansion"
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, integral representations      I 13—15 28
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, power series expansion      I 26
R      see "Recurrence formula"
Rational approximation      see "Approximation"
Recurrence formula for $G^{m,k}_{p+1,q+1}(z|^{a_{p+1}}_{b_{q},n})$ and family of functions which satisfy the same differential equation as does the above function      II 156
Recurrence formula for $G^{m,k}_{p+1,q+2}(z|^{a_{p+1}}_{b_{q},n,-n-\lambda})$ and family of functions which satisfy the same differential equation as does the above function      II 153—159
Recurrence formula for $G^{m,k}_{p+1,q+2}(z|^{a_{p+1}}_{b_{q},n,-n-\lambda})$ and family of functions which satisfy the same differential equation as does the above function, a special case      II 26
Recurrence formula for $u_{n} _{p+1}F_{q}(a_{p+1};b_{q};z)$ with all parameters as in the preceding example and family of functions which satisfy the same differential equation as does the above function      II 156 157
Recurrence formula for $[u_{n}/(n+\lambda)_{n}] _{p+1}F_{q+1}(a_{p+1};b_{q},\lambda+1+2n;z)$ , $a_{i}=\alpha_{i}+n$ , i=1,2,...,p+1, $b_{j}=p_{j}+n$ , j=1,2,...,q, $u_{n}=(\alpha_{p+1})_{n}z^{n}/(p_{q})_{n}n!$ and family of functions which satisfy the same differential equation as does the above function      II 157 158
Recurrence formula for $[u_{n}/(n+\lambda)_{n}] _{p+1}F_{q+1}(a_{p+1};b_{q},\lambda+1+2n;z)$ , $a_{i}=\alpha_{i}+n$ , i=1,2,...,p+1, $b_{j}=p_{j}+n$ , j=1,2,...,q, $u_{n}=(\alpha_{p+1})_{n}z^{n}/(p_{q})_{n}n!$ and family of functions which satisfy the same differential equation as does the above function, special case p=0, q=1      II 35 37
Recurrence formula for $[u_{n}/(n+\lambda)_{n}] _{p+1}F_{q+1}(a_{p+1};b_{q},\lambda+1+2n;z)$ , $a_{i}=\alpha_{i}+n$ , i=1,2,...,p+1, $b_{j}=p_{j}+n$ , j=1,2,...,q, $u_{n}=(\alpha_{p+1})_{n}z^{n}/(p_{q})_{n}n!$ and family of functions which satisfy the same differential equation as does the above function, special case p=2, q=1      II 34 48 159
Recurrence formula for $[u_{n}/(n+\lambda)_{n}] _{p+1}F_{q+1}(a_{p+1};b_{q},\lambda+1+2n;z)$ , $a_{i}=\alpha_{i}+n$ , i=1,2,...,p+1, $b_{j}=p_{j}+n$ , j=1,2,...,q, $u_{n}=(\alpha_{p+1})_{n}z^{n}/(p_{q})_{n}n!$ and family of functions which satisfy the same differential equation as does the above function, special case p=3, q=2      II 158
Recurrence formula for $[u_{n}/(n+\lambda)_{n}] _{p+1}F_{q+1}(a_{p+1};b_{q},\lambda+1+2n;z)$ , $a_{i}=\alpha_{i}+n$ , i=1,2,...,p+1, $b_{j}=p_{j}+n$ , j=1,2,...,q, $u_{n}=(\alpha_{p+1})_{n}z^{n}/(p_{q})_{n}n!$ and family of functions which satisfy the same differential equation as does the above function, special case p=q=1      II 35
Recurrence formula for $_{1}F_{1}(-n; -2n+a-\nu; z)$       II 190 191
Recurrence formula for $_{1}F_{1}(-n; -2n+a-\nu; z)$ , $c_{n} = _{2}F_{3}(b, c; d, e, f; x^{2})$ ; b=a-n, $c=\frac{1}{2}-n$ , $d=\nu + 2$ , e=a-2n, $f=a-\nu-1-2n$ , a=0 or 1      II 82 193
Recurrence formula for $_{1}F_{1}(-n; -2n+a-\nu; z)$ , $c_{n}^{2}$ , $d_{n}^{2}$ , $c_{n}d_{n}$       II 193
Recurrence formula for $_{1}F_{1}(-n; -2n+a-\nu; z)$ , $d_{n} =_{2}F_{3}$ as above with b = -n, $c=a-\frac{1}{2}-n$ , $d = \nu + 1$ , e=a-1-2n, $f=a-\nu-1-2n$ , a=0 or 1      II 82 193
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=-n, v=-n+1/x, w=1-a+1/x, z=x      II 181
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=-n, v=-n+a- $\sigma$ , w=-2n+a- $\rho$ , z=-z      II 177
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=-n, v=-n- $\frac{1}{2}$ , w= $\frac{1}{2}$ , z=z      II 185
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=-n, v=x-n, w=x+1-a, z=x      II 180
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=1, v=m+ $\frac{1}{2}$ +b, w=m+ $\frac{3}{2}$ +b, z=-x      II 276
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=1-n, v=-n+ $\frac{1}{2}$ , w= $\frac{3}{2}$ , z=z      II 185
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=a-n, v=a-n-1/x, w=1+a-1/x, z=x      II 181
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=a-n, v=a-x-n, w=1+a-x, z=x      II 180
Recurrence formula for $_{2}F_{1}(u,v;w;z)$ where u, v, w, and z are given in the following, a=0 or 1, b=0 or 1/2, u=n+ $\omega$ , v= $\omega$ , w=n+1, z=z      II 36
Recurrence formula for $_{4}F_{1}(a,b,c,d;e;x)$ , a=-m, b=m+1+ $\epsilon$ , c=m+ $\nu$ + $\epsilon$ , d=1-m- $\nu$ , e= $\frac{1}{2}$ + $\epsilon$       II 232
Recurrence formula for $_{r+3}F_{s}(^{-n+a+k,n+\lambda+a+k,a_{r}+k,1}_{\ \ \ \ \ \ \ \ \ \ b_{s}+k}|1/\gamma)$ as a function of k      II 152
Recurrence formula for Bernoulli polynomials, generalized      I 20 22 35
Recurrence formula for coefficients in the expansion of $_{1}F_{1}$ in series of the Bessel functions $I_{n+\lambda}(z/2)$       II 48
Recurrence formula for coefficients in the expansions of the functions named below in series of Chebyshev polynomials of the first kind, $\int^{\infty}_{x} t^{\mu} K_{\nu} dt$       II 241
Recurrence formula for coefficients in the expansions of the functions named below in series of Chebyshev polynomials of the first kind, $\psi$ -function or confluent hypergeometric function      II 26 239
Recurrence formula for coefficients in the expansions of the functions named below in series of Chebyshev polynomials of the first kind, $_{0}F_{1}$       II 35 235—238
Recurrence formula for coefficients in the expansions of the functions named below in series of Chebyshev polynomials of the first kind, $_{1}F_{1}$       II 35
Recurrence formula for coefficients in the expansions of the functions named below in series of Chebyshev polynomials of the first kind, $_{2}F_{1}$       II 34

Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial      II 134—148 156
Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=0, q=1      II 170—172 177 179
Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=0, q=2      II 229 230
Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=0, q=3      II 449
Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=1, q=2      II 48 147
Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=1, q=3      II 146
Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=2, q=3      II 145 146
Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=q=0      II 190—192
Recurrence formula for Jacobi polynomial, extended, $_{p+2}F_{q}(-n,n+\lambda,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=q=1      II 197 198
Recurrence formula for Laguerre polynomial, extended, $_{p+1}F_{q}(-n,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial      II 143—145.
Recurrence formula for Laguerre polynomial, extended, $_{p+1}F_{q}(-n,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=0, q=1      II 198—201
Recurrence formula for Laguerre polynomial, extended, $_{p+1}F_{q}(-n,\alpha_{p};\rho_{q};z)$ and family of functions which satisfy same differential equation as does the above polynomial, special case p=1, q=2      II 147
Recurrence formula for Pade approximations      II 78 88 133 see
Recurrence formula for polynomials in polynomial and rational approximations to generalized hypergeometric functions      II 134—166 see
Recurrence formulas, use of in computation      I 317—319 325—329 II 26—28 159—166 282—285
Representation formulas for polynomials      II 139 140 142 144
RF      see "Rational function"
RFP      see "Rational function which is also a Pade approximation"
Riccati differential equations      II 77—91
Saalschuetz's formula      I 103
Selected points, method of      II 69
Sine and cosine      I 23 39 210 see "Expansion"
Sine and cosine integrals      I 135 221—223 227 II 27 see "Expansion"
Stokes phenomenon      I 128 199
Struve functions      see also "Approximation" "Expansion"
Struve functions, asymptotic expansion      I 219
Struve functions, definition      I 217
Struve functions, difference-differential properties      I 218
Struve functions, expressed as a G-function      I 227 233
Struve functions, integrals involving      I 165 220 II 40
Tables (of) Bernoulli polynomials and numbers      I 19 20 34
Tables (of) coefficients for expansion of functions in series of Chebyshev polynomials of the first kind and for approximations of functions in the form of polynomials and rational functions      II 282—452 see "Expansion"
Tables (of) mathematical constants      II 292 293
Tangent and cotangent      I 23 see "Expansion"
Tangent and cotangent, inverse of      I 38 40 210
tr      see "Trapezoidal or modified trapezoidal rule integration formula"
Trapezoidal-type integration formulas      II 214—226
Ultraspherical polynomial      I 273 279
Vandermonde's theorem      I 99
Watson's formula      I 104
Watson's lemma      I 4—7
Weber function      see "Anger — Weber"
Whipple's formula      II 104
Whittaker functions      see also "Hypergeometric function confluent"
Whittaker functions, definition      I 134
Whittaker functions, expressed as G-function      I 225 226 228 231 233 234
Wronskians      I 84 85 124 II 101
Y-transform      I 165
Zeta-function (Riemann)      I 27 II 287 293 306

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