Abramowitz Function Computed by Clenshaw's Method, 74 Absolute

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

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

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

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, 205 fast cosine transforms, 67 elliptic integral fast Fourier transform, 69, 298 other forms, 347 Fejér quadrature, 296 elliptic integrals first rule, 297 Carlson’s symmetric forms, 345 second rule, 297 incomplete of the first kind, 344 Filon’s method for oscillatory integrals, incomplete of the second kind, 344 303 of Legendre, 344 first order linear inhomogeneous of the third kind, 345 difference equations, 87 epsilon algorithm of Wynn, 278 fixed point method, 192, 193, 196 equidistant interpolation based on global strategies, 213 Runge phenomenon, 54 error bound, 194 equioscillation property, 67 Newton–Raphson method, 195 error order of convergence, 194 absolute, relative, 356 fixed point theorem, 193 bound for fixed point method, 194 floating-point error functions IEEE formats, 356 Chebyshev expansion, 83 IEEE-754 standard for base-2 computing zeros by using arithmetic, 356 asymptotic expansions, 229 numbers, 356 inversion, 330 forward differences, 53 Euler–Maclaurin formula, 131 forward elimination in Olver’s method, relation with the trapezoidal rule, 117 130 forward recurrence algorithm Euler’s summation formula, 331 for computing continued fractions, limitations, 336 181 exponential function forward series recurrence algorithm Padé approximants, 280 for computing continued fractions, exponential integral 181 anomalous behavior of recursion, fractal, 197 121 Frobenius method, 22 as solution of an inhomogeneous fundamental Lagrange interpolation linear first order difference polynomial, 52 equations, 115 fundamental system of solutions, 21 asymptotic expansions, 37, 38 expansion as factorial series, 45 sequence transformations, 288 gamma function exponentially improved asymptotic alternative asymptotic expansions, 39 representation, 49 asymptotic expansion, 243 factorial series, 44 numerical algorithm based on condition for convergence, 44 recursion, 246

410 Index

Gauss hypergeometric functions, 28 recursion for orthogonal Bühring’s analytic continuation polynomials, 143 formula, 31 Stieltjes procedure, 139 convergence domains of power Gegenbauer polynomial, 140 series, 31 and Gauss–Kronrod quadrature, deriving the continued fraction for 300 a ratio of, 187 generalized differential equation, 18 hypergeometric function, 27 Norlünd’s continued fraction, 105 analytic continuation, 27 other power series, 30 terminating series, 27 Padé approximants, 283 Laguerre polynomial, 140 recurrence relations in all global fixed point methods, 213 directions, 104 global strategies for finding zeros, 213 recursion for power series Golub–Welsch algorithm for Gauss coefficients, 29 quadrature, 133, 141, 145 removable singularities, 33 Gram–Schmidt orthogonalization, 134 special cases, 29 value at z = 1, 29 Hadamard-type expansions Gauss quadrature, 132, 135, 191 for modified Bessel function I (z), Christoffel numbers, 136 ν 41 computing zeros and weights, 133, Hankel transforms, 303 141, 145 Hermite example for Legendre polynomials, interpolation, 53, 54, 136 145 Hermite polynomial for computing Airy functions, 360 and Gauss–Kronrod quadrature, for computing the Airy function in 300 the complex plane, 145 Hermite polynomials Gauss–Kronrod, 299 differential equation, 19 Gauss–Lobatto, 299 Gauss–Radau, 299 Gauss quadrature, 145 generalized Hermite polynomials, generalized for Gauss quadrature, 141 141 Golub–Welsch algorithm, 133, 141, special case of parabolic cylinder 145 functions, 102 Hermite polynomials, 145 zeros, 102 Jacobi matrix hyperasymptotics, 40 nonorthonormal case, 144 for the gamma function, 40 orthonormal case, 142 hypergeometric functions, see Gauss Jacobi polynomials, 145 hypergeometric functions Kronrod nodes, 300 hypergeometric series, 26 Laguerre polynomials, 145 Meixner–Pollaczek polynomials, IEEE floating-point, see floating-point 141 ill-conditioned problem, 357 orthonormal polynomials, 134 incomplete beta function other rules, 298 asymptotic inversion, 318 Patterson, 301 error function case, 322

Index 411

incomplete gamma function initial value problem case, 324 for differential equations in the symmetric case, 319 complex plane deriving the continued fraction, 189 Taylor-series method, 292 in terms of the Gauss inner product of polynomials, 133 hypergeometric function, 189 interpolation incomplete gamma functions by orthogonal polynomials, 65 as solution of inhomogeneous Chebyshev, 62 linear first order difference of the second kind, 65 equation, 114 Hermite, 53, 54, 136 asymptotic expansions, 237 Lagrange, 52, 54 alternative representation, 47 Runge phenomenon, 54 for (a, z), 37, 238 interpolation polynomial simpler uniform expansions, 247 fundamental Lagrange, 52 uniform, 242 inversion of complementary error function, asymptotic inversion, 312, 329 309 continued fraction, 176 of error function, 330 computing (a, z), 177, 181, 182 of incomplete beta functions, 318 for (a, z), 186 of incomplete gamma functions, expansion as factorial series, 45 312, 329 normalized functions P(a, z) and irregular singular point of a differential Q(a, z), 241 equation, 19 numerical algorithm based on uniform expansion, 245 Jackson’s theorem, 63 Padé approximant, 284 Jacobi continued fraction, 179 indicial equation, 22 Jacobi matrix, 205 inhomogeneous Gauss quadrature Airy functions, 359 nonorthonormal case, 144 linear difference equations, 112 used in Gauss quadrature, 142 linear first order difference equation Jacobi polynomial, 60, 140 condition of the recursion, 113 as hypergeometric series, 60 for exponential integrals, 115 Gauss quadrature, 145 for incomplete gamma function, zeros, 191 114 Julia set, 197 minimal and dominant solutions, 113 Kummer functions, see confluent linear first order difference hypergeometric functions equations, 112 second order difference equations, Lagrange 115 interpolation, 52 example, 115 formula for the error, 125 Olver’s method, 116 fundamental polynomials, 124 subdominant solution, 115 polynomial, 54 superminimal solution, 115 remainder of Taylor’s formula, 16

412 Index

Laguerre polynomial, 140 Maclaurin series, 16 computing zeros, 221 mathematical libraries for computing Laguerre polynomials special functions, 355 Gauss quadrature, 145 McMahon expansions for zeros of Lambert’s W-function, 312 Bessel functions, 200, 204 Laplace transform Meixner–Pollaczek polynomials, 141 inversion by using Padé method of Taylor series approximations, 352 for differential equations in the numerical inversion, 347, 349 complex plane, 292 Lebesgue constants for Fourier series, Miller algorithm 291 condition for convergence, 108 Legendre functions estimating starting value N,110 associated functions, 363 for computing modified Bessel associated functions P µ(z), Qµ(z), ν ν functions In+1/2(x), 106 363 numerical stability, 109 differential equation, 19, 363 numerical stability of the oblate spheroidal harmonics, 363 normalizing sum, 109 prolate spheroidal harmonics, 363 when a function value is known, recurrence relations, 103 105 for conical functions, 103 with a normalizing sum, 107 with respect to the degree, 104 minimal solution of a recurrence with respect to the order, 103 relation, 90 toroidal harmonics, 363 how to compute by backward Legendre polynomial, 140 recursion, 105 example for Gauss quadrature, 145 minimax approximation, 51 Legendre’s elliptic integrals, 344 Jackson’s theorem, 63 Levin’s sequence transformation, 287 modified Bessel functions linear algorithm, 370 differential equations regular and singular points, 19 anomalous behavior of recursion, solved by Taylor expansion, 291 118 homogeneous three-term asymptotic expansion for Kν(z),43 recurrence relation, 87 Chebyshev expansions for K0(x) independent solutions of a and K1(x), 370 recurrence relation, 89 differential equation, 370 inhomogeneous first order expansion for Kν(z) in terms of difference equations, 87 confluent hypergeometric Liouville–Green approximation, 26 functions, 43 Liouville transformation, 25 of integer and half-integer orders, local strategies for finding zeros, 197 370 logarithmic derivative of the gamma of purely imaginary order, 372 function, 33 Airy-type asymptotic Longman’s method for computing expansions, 375 oscillatory integrals, 303 algorithms for Kia(x), Lia(x), 372 machine-, 356 asymptotic expansions, 374

Index 413

continued fraction for Kia(x), complex Gauss quadrature formula, 373 348 differential equation, 372 to compute Bessel function J0(x), nonoscillating integral 349 representations, 375 to compute the complementary scaled functions, 372 error function, 350 series expansions, 373 numerical stability, 357 Wronskian relation, 372 numerically unstable method, 357 Padé approximants to Kν(z), 352 recurrence relation, 97, 370 oblate spheroidal harmonics, 363 spherical, 370 algorithm, 365 algorithm, 371 recurrence relations, 365 notation, 370 scaled functions, 364 recurrence relation, 371 Olver’s method for inhomogeneous second order difference trapezoidal rule for K0(x), 153 modified Lentz algorithm equations, 116 for computing continued fractions, order of convergence 183 asymptotic error constant, 194 modulus of continuity, 63 fixed point methods, 194 monic ordinary differential equation, see orthogonal polynomials, 134 differential equation orthonormal polynomials, 134 orthogonal basis with respect to inner product, 134 orthogonal polynomials Newton’s binomial formula, 27 computing zeros by using Newton’s divided difference formula, 53 asymptotic expansions, 234 Newton–Raphson method, 191, 193, 195 on complex contour, 348 high order inversion, 196, 327 zeros, 135 order of convergence, 195 orthogonality with respect to inner nodes of a quadrature rule, 124 product, 134 nonlinear differential equations, 25 oscillatory integrals, 301 Norlünd’s continued fraction for Gauss asymptotic expansion, 301 hypergeometric functions, 105 convergence acceleration schemes, normalized incomplete gamma function 303 asymptotic estimate, 42 Filon’s method, 303 normalized incomplete gamma functions general forms, 303 asymptotic estimate for P(a, z),41 Hankel transforms, 303 Hadamard-type expansions, 41 Longman’s method for computing, relation with chi-square probability 303 functions, 240 overflow threshold, 356 uniform asymptotic expansions, 242 Padé approximants, 276 numerical condition, 357 continued fractions, 278 numerical inversion of Laplace diagonal elements in the table, 277 transforms, 347, 349 generating the lower triangular part by deforming the contour, 350 in the table, 278

414 Index

how to compute, 278 polynomial approximation by Wynn’s cross rule, 278 minimax, 51 Luke’s examples for special Jackson’s theorem, 63 functions, 283 poorly conditioned problem, 357 normality , 277 power series relation with continued fractions, of Bessel function Jν(z), 24, 28 179 of confluent hypergeometric table, 277 M-function, 28 to the exponential function, 280 of Gauss hypergeometric functions, to the Gauss hypergeometric 28 function, 283 of hypergeometric type, 26 to the incomplete gamma functions, of the Airy functions, 18 284 of the exponential function, 17 to the modified Bessel function primal algorithm Kν(z), 352 for computing toroidal harmonics, Wynn’s cross rule for, 278 366 parabolic cylinder functions, 377 prolate spheroidal harmonics, 363 algorithm, 378 algorithm, 364 m Maclaurin series, 379 recurrence relation for Pn (x), 365 − m regions in (a, x) plane, 378 recurrence relation for Qn (x), 365 asymptotic expansions for large x, scaled functions, 364 380 computing zeros by using quadrature asymptotic expansions, 233 characteristic function for the error, contour integral for, 168 149 definition, 101 Clenshaw–Curtis, 296, 297 differential equation, 19, 377 degree of exactness, 124 integral representations, 384 double exponential formulas, 156 oscillatory behavior, 102 erf-rule, 155 recurrence relation for U(a, x), 385 Fejér, 296 relation with Hermite polynomials, first rule, 297 102 second rule, 297 scaled functions, 377 for contour integrals in the complex three-term recurrence relations, 101 plane, 157 uniform Airy-type asymptotic Gauss–Kronrod, 299 expansion, 383 Gauss–Lobatto, 299 uniform asymptotic expansions in Gauss quadrature, 132 elementary functions, 381 Gauss–Radau, 299 Wronskian relation, 101 other Gauss rules, 298 Perron’s theorem, 92, 93 Patterson, 301 intuitive form, 92 Romberg quadrature, 294 Pincherle’s theorem, 95 simple trapezoidal rule, 124 plasma-dispersion function, 229 Simpson’s rule, 295 Pochhammer symbol, 27 tanh-rule, 154 polynomial the trapezoidal rule on R, 147 Stieltjes, 300 transforming the variable, 153

Index 415

weight function, 132 to enlarge the domain of weights, 124 computation, 358 quotient-difference algorithm, 178 to obtain higher accuracy, 358 Schwarzian derivative, 26 recurrence relation Scorer functions for Bessel functions, 96 algorithm for Hi(z), 361 for computing modified Bessel asymptotic expansion for Gi(z), function Kν(z), 100 362 for confluent hypergeometric asymptotic expansion for Hi(z), functions, 99 362 in all directions, 99 computation for complex in the (++) direction, 100 arguments, 359 in the (+ 0) direction, 99 computing scaled functions, 359, in the (0 +) direction, 100 363 for Coulomb wave functions, 98 computing zeros by using for Legendre functions, 103 asymptotic expansions, 227 for modified Bessel functions, 97, connection formulas for Gi(z), 362 370 integral representation for Hi(z), for modified spherical Bessel 361 functions, 371 power series for Gi(z), 362 for parabolic cylinder functions, secant method, 191 101, 385 second algorithm of Remes, 290 for prolate spheroidal harmonics second order homogeneous linear m Pn (x), 365 difference equation, 87 for prolate spheroidal harmonics sequence transformations, 286 m Qn (x), 365 for asymptotic expansion of for toroidal harmonics, 367 exponential integral, 288 recurrent trapezoidal rule, 129 Levin’s transformation, 287 regular point a differential equation, 19 numerical examples, 288 regular singular point of a differential of asymptotic series, 288 equation, 19 of power series, 288 relative error, 356 Weniger’s transformation, 287 Remes’ second algorithm of, 290 with remainder estimates, 287 repeated nodes in Hermite interpolation, Simpson’s rule, 125, 295 54 software survey for computing special reverting asymptotic series, 226 functions, 355 Riccati–Bessel functions sources of errors difference-differential system, 213 due to discretization, 357 zeros, 213 due to fixed-length representations, Romberg quadrature, 294 357 Runge phenomenon, 54 due to truncation, 357 in computations, 357 saddle point method, 158 special functions computing saddle point, 158 Airy functions of complex scalar product of polynomials, 133 arguments, 359 scaling functions, 358 mathematical libraries, 355

416 Index

ratios of Bessel functions, 218 by extended precision Scorer functions of complex algorithms, 358 arguments, 359 by verification of functional software survey, 355 relations, 358 stability of a numerical method, 357 consistency between different Steed’s algorithm methods, 358 for computing continued fractions, three-term recurrence relation, see also 181 recurrence relation steepest descent path, 158 anomalous behavior, 118 Stieltjes confluent hypergeometric continued fraction, 178 functions, 120 procedure for recurrence relations, exponential integrals, 121 139 modified Bessel functions, 118 Stirling numbers backward recursion, 91 definitions, 337 condition of, 88 explicit representations, 337 dominant solution, 90 generating functions, 337 forward recursion, 91 of the first kind linear homogeneous, 87 uniform asymptotic expansion, linearly independent solutions, 89 343 minimal solution, 89, 90 of the second kind, 44 relation with continued fractions, uniform asymptotic expansion, 95 338 scaled form, 94 Stokes phenomenon, 40 with constant coefficients, 92 subdominant solution toroidal harmonics, 363 of inhomogeneous second order algorithm, 366 M difference equation, 115 asymptotic expansion for P−1/2(x), superminimal solution 368 of inhomogeneous second order dual algorithm, 369 difference equations, 115 primal algorithm, 366 symmetric elliptic integrals, 345 recurrence relation, 367 relation with elliptic integrals, 367 Taylor series, 16 scaled functions, 364 M Cauchy’s formula for remainder, 16 series expansion for P−1/2(x), 367 Lagrange’s formula for remainder, trapezoidal rule, 350 16 simple rule, 124 Taylor’s formula for remainder, 16 compound rule, 126 Taylor-series method Euler’s summation formula, 130 for boundary value problem in the for computing Scorer functions, complex plane, 293 362 for initial value problem in the for computing the Bessel function complex plane, 292 J0(x), 128 testing of software for computing the Bessel function for computing functions, 358 K0(x), 153 by comparison with existing for computing the complementary algorithms, 358 error function, 350

Index 417

on R, 147 asymptotic approximations, 197, recursive computation, 129 200 with exponentially small error, 151 Bessel functions, 204, 233, 385 TTRR, see three-term recurrence complex zeros, 197 relation eigenvalue problem, 208, 212 turning point of a differential equation, from Airy-type asymptotic 249 expansions, 204 McMahon expansions, 200, 204 underflow threshold, 356 bisection method, 191, 193 complex zeros, 197 Wagner’s modification of asymptotic computation based on asymptotic expansions, 48 approximations, 199 Watson’s lemma, 36 conical functions, 211, 223 weight function for numerical cylinder functions, 233 quadrature, 132 eigenvalue problem for orthogonal weights of a quadrature rule, 124 polynomials, 205 Weniger’s sequence transformation, 287 error functions, 229 Whittaker functions fixed point method, 193 differential equation, 19 fixed point methods and WKB approximation, 26 asymptotics, 199 Wronskian, 21 global strategies, 204, 213 for Airy functions, 254 Jacobi polynomials, 191 for Bessel functions, 255 Laguerre polynomials, 221 for modified Bessel functions of local strategies, 197 purely imaginary order, 372 matrix methods, 204 for parabolic cylinder functions, Newton–Raphson method, 191, 193 101 orthogonal polynomials, 135, 234 Wynn’s cross rule parabolic cylinder functions, 233 for Padé approximants, 278 Riccati–Bessel functions, 213 Wynn’s epsilon algorithm, 278 Scorer functions, 227 secant method, 191 zeros of functions, 191 Airy functions, 224

From "Numerical Methods for Special Functions" by Amparo Gil, Javier Segura, and Nico Temme