Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. Index Abramowitz function for parabolic cylinder functions, computed by Clenshaw’s method, 378 74 for prolate spheroidal harmonics, absolute error, 356 364 Airy function for Scorer functions, 361 contour integral for, 166 for toroidal harmonics, 366 Airy functions of Remes, 290 algorithm, 359 analytic continuation of generalized asymptotic estimate of, 18 hypergeometric function, 27 asymptotic expansions, 81, 360 anomalous behavior of recursions, 118 Chebyshev expansions, 80, 85 a warning, 122 computing confluent hypergeometric complex arguments, 359 functions, 120 Gauss quadrature, 145 exponential integrals, 121 scaled functions, 359 first order inhomogeneous zeros, 224 equation, 121 connection formulas, 360, 361 modified Bessel functions, 118 contour integral for, 264 anti-Miller algorithm, 110, 112 differential equation, 249, 359 associated Legendre functions relation with hypergeometric computation for z>0, 363 function, 28 asymptotic expansion used in uniform asymptotic uniform, 237 expansion, 250 asymptotic expansions Airy-type asymptotic expansion alternative asymptotic for modified Bessel functions of representation for (z),49 purely imaginary order, 375 alternative expansion for parabolic cylinder functions, for (z),49 383 for (a, z),47 obtained from integrals, 249, 264 convergent asymptotic algorithm representation, 46 for Airy functions, 359 converging factor, 40 for computing zeros of Bessel exponentially improved, 39 functions, 385 for (a, z),39 for modified Bessel functions, 370 exponentially small remainders, 38 for oblate spheroidal harmonics, hyperasymptotics, 40 365 of exponential integral, 37, 38 405 From "Numerical Methods for Special Functions" by Amparo Gil, Javier Segura, and Nico Temme Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. 406 Index of incomplete gamma function order estimate, 336 (a, z),37 Bessel functions of modified Bessel function Kν(z), Airy-type expansions, 250 43 algorithms for computing, 369 of Poincaré type, 34 computing zeros, 197, 204, 385 of the exponential integral, 34 asymptotic expansions, 200, 233 Stokes phenomenon, 40 asymptotic expansions of Airy to compute zeros, 199, 200 type, 204 of Airy functions, 224 eigenvalue problems, 208, 212 of Bessel functions, 233 McMahon expansions, 200, 204 of Bessel functions with differential equation, 19, 24 McMahon expansions, 200 J0(x) computation of error functions, 229 Chebyshev expansion, 83 of orthogonal polynomials, 234 numerical inversion of Laplace of parabolic cylinder functions, transform, 349 233 the trapezoidal rule, 128 of Scorer functions, 227 Jν(z) as hypergeometric function, transforming into factorial series, 28 44 Neumann function Yν(z),25 uniform, 239 recurrence relations, 96 for the incomplete gamma recursion for J (z) and Y (z),87 functions, 240 ν ν series expansion for J (z),24 upper bound for remainder, 39 ν Wronskian, 255 for log (z),39 Bessel polynomials, 348 Wagner’s modification, 48 best approximation, 51 Watson’s lemma, 36 Jackson’s theorem, 63 asymptotic inversion polynomial, 290 of distribution functions, 317 versus Chebyshev series, 291 of incomplete beta functions, 318 of incomplete gamma functions, rational, 290 312 oscillations of the error curve, of the incomplete beta function 290 error function case, 322 binomial coefficient incomplete gamma function gamma functions, 27 case, 324 Pochhammer symbol, 27 symmetric case, 319 bisection method, 191, 193, 195 order of convergence, 194 backsubstitution in Olver’s method, 117 Bolzano’s theorem, 193 backward recurrence algorithm, see also Boole’s summation method, 336 Miller algorithm boundary value problem for computing continued fractions, for differential equations in the 181 complex plane backward sweep, 215 Taylor-series method, 293 base-2 floating-point arithmetic, 356 Bühring’s analytic continuation formula Bernoulli numbers and polynomials, for hypergeometric functions, 131, 331 31 From "Numerical Methods for Special Functions" by Amparo Gil, Javier Segura, and Nico Temme Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. Index 407 Carlson’s symmetric elliptic integrals, Clenshaw’s method 345 for evaluating a Chebyshev sum, Casorati determinant, 89 65, 75 its use in anti-Miller algorithm, 110 error analysis, 76 Cauchy’s form for the remainder of modification, 78 Taylor’s formula, 16 for solving differential equations, Cauchy’s inequality, 16 70 Cauchy–Riemann equations, 162 for the Abramowitz function, 74 chaotic behavior in the complex plane, for the J-Bessel function, 72 197 Clenshaw–Curtis quadratures, 62, 296, characteristic equation, 92 297 Chebyshev equioscillation theorem, 63 compact operator in a Hilbert space, 209 Chebyshev expansion complementary error function computing coefficients, 69 as normal distribution function, 242 convergence properties, 68 computed by numerical inversion analytic functions, 68 of Laplace transform, 350 for Airy functions, 80, 85 contour integral, 350 for error function, 83 in uniform asymptotic for J-Bessel functions, 83 approximations, 242 for Kummer U-function, 84 complex Gauss quadrature formula, 348 of a function, 66 nodes and weights, 349 Chebyshev interpolation, 62 complex orthogonal polynomials, 348 computing the polynomial, 64 compound trapezoidal rule, 126 of the second kind, 65 condition of TTRRs, 88 Chebyshev polynomial, 56, 140 confluent hypergeometric functions Chebyshev polynomials anomalous behavior of recursion, as particular case of Jacobi 120 polynomials, 62 Chebyshev expansion for discrete orthogonality relation, 59 U−function, 84 economization of power series, 80 differential equation, 19 equidistant zeros and extrema, 61 integral representation for expansion of a function, 66 U(a, c, z),43 minimax approximation, 58 M in terms of hypergeometric of the first kind, 56 function, 28 of the second, third, and fourth recurrence relations, 96, 99 kinds, 60 conical functions orthogonality relation, 59 computing zeros, 211, 223 polynomial representation, 59 recurrence relation, 103, 211 ∗ shifted polynomial Tn (x),60 conjugate harmonic functions, 162 Chebyshev sum continued fraction, 173 evaluated by Clenshaw’s method, computing, 181 75 backward recurrence algorithm, Christoffel numbers for Gauss 181 quadrature, 136 forward recurrence algorithm, classical orthogonal polynomials, 140 181 From "Numerical Methods for Special Functions" by Amparo Gil, Javier Segura, and Nico Temme Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. 408 Index forward series recurrence degree of exactness, 124, 132 algorithm, 181 difference equation modified Lentz algorithm, 183 first order inhomogeneous, 112 Steed’s algorithm, 181 second order homogeneous, 87 contractions, 175 differential equation convergence, 175, 179 Frobenius method, 22 equivalence transformations, 175 fundamental system of solutions, even and odd part, 175 21 for incomplete beta function, 189 homogeneous linear second order, for incomplete gamma function, 292 176 in the complex plane for incomplete gamma function Taylor-series method, 292 (a, z), 186 Taylor-series method for for ratios of Gauss hypergeometric boundary value problem, 293 function, 187 Taylor-series method for initial for special functions, 185 value problem, 292 Jacobi fraction, J-fraction, 179 inhomogeneous linear second linear transformations, 174 order, 292 nth convergent, nth approximant, of Airy functions, 359 174 of Bessel functions, 19 numerical evaluation, 181 of confluent hypergeometric of Gauss, 188 functions, 19 of Gauss hypergeometric functions, recursion for convergents, 174 18 relation with of Hermite polynomials, 19 ascending power series, 178, 179 of Legendre functions, 19, 363 Padé approximant, 278 of modified Bessel functions, 370 Padé approximants, 179 of purely imaginary order, 372 three-term recurrence relation, of parabolic cylinder functions, 19, 95 377 Stieltjes fraction, S-fraction, 178 of Whittaker functions, 19 theorems on convergence, 180 singular point, 19 value of the, 174 irregular, 19 contour integrals in the complex plane regular, 19 quadrature for, 157 Taylor expansion method, 291 convergence properties Dini–Lipschitz continuity, 64 Chebyshev expansion, 68 discrete cosine transform, 66 analytic functions, 68 dominant solution of a recurrence continued fraction, 175 relation, 90 convergent power series, 15 double factorial, 364 converging factor for asymptotic dual algorithm for computing toroidal expansion, 40 harmonics, 369 Coulomb wave functions recurrence relations, 98 economization of power series, 80 cylinder functions, 233, see also Bessel eigenvalue problem functions for Bessel functions, 208, 212, 213 From "Numerical Methods for Special Functions" by Amparo Gil, Javier Segura, and Nico Temme Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. Index 409 for compact infinite matrix, 210 for exponential integral, 45 for conical functions, 211 for incomplete gamma function for minimal solutions of three-term (a, z),45 recurrence relations, 207 Fadeeva function, 229 for orthogonal polynomials,