Index [ ] · PDF fileJackson’s theorem, 63 ... oscillations of the error curve, 290 binomial coefﬁcient gamma functions, 27 Pochhammer symbol, 27 ... ascending power series, 178,

Index

Abramowitz functioncomputed by Clenshaw’s method,

74absolute error, 356Airy function

contour integral for, 166Airy functions

algorithm, 359asymptotic estimate of, 18asymptotic expansions, 81, 360Chebyshev expansions, 80, 85computing

complex arguments, 359Gauss quadrature, 145scaled functions, 359zeros, 224

connection formulas, 360, 361contour integral for, 264differential equation, 249, 359relation with hypergeometric

function, 28used in uniform asymptotic

expansion, 250Airy-type asymptotic expansion

for modified Bessel functions ofpurely imaginary order, 375

for parabolic cylinder functions,383

obtained from integrals, 249, 264algorithm

for Airy functions, 359for computing zeros of Bessel

functions, 385for modified Bessel functions, 370for oblate spheroidal harmonics,

365

for parabolic cylinder functions,378

for prolate spheroidal harmonics,364

for Scorer functions, 361for toroidal harmonics, 366of Remes, 290

analytic continuation of generalizedhypergeometric function, 27

anomalous behavior of recursions, 118a warning, 122confluent hypergeometric

functions, 120exponential integrals, 121first order inhomogeneous

equation, 121modified Bessel functions, 118

anti-Miller algorithm, 110, 112associated Legendre functions

computation for �z > 0, 363asymptotic expansion

uniform, 237asymptotic expansions

alternative asymptoticrepresentation for �(z), 49

alternative expansionfor �(z), 49for �(a, z), 47

convergent asymptoticrepresentation, 46

converging factor, 40exponentially improved, 39

for �(a, z), 39exponentially small remainders, 38hyperasymptotics, 40of exponential integral, 37, 38

405

Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed.

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

406 Index

of incomplete gamma function�(a, z), 37

of modified Bessel function Kν(z),43

of Poincaré type, 34of the exponential integral, 34Stokes phenomenon, 40to compute zeros, 199, 200

of Airy functions, 224of Bessel functions, 233of Bessel functions with

McMahon expansions, 200of error functions, 229of orthogonal polynomials, 234of parabolic cylinder functions,

233of Scorer functions, 227

transforming into factorial series,44

uniform, 239for the incomplete gamma

functions, 240upper bound for remainder, 39

for log�(z), 39Wagner’s modification, 48Watson’s lemma, 36

asymptotic inversionof distribution functions, 317of incomplete beta functions, 318of incomplete gamma functions,

312of the incomplete beta function

error function case, 322incomplete gamma function

case, 324symmetric case, 319

backsubstitution in Olver’s method, 117backward recurrence algorithm, see also

Miller algorithmfor computing continued fractions,

181backward sweep, 215base-2 floating-point arithmetic, 356Bernoulli numbers and polynomials,

131, 331

order estimate, 336Bessel functions

Airy-type expansions, 250algorithms for computing, 369computing zeros, 197, 204, 385

asymptotic expansions, 200, 233asymptotic expansions of Airy

type, 204eigenvalue problems, 208, 212McMahon expansions, 200, 204

differential equation, 19, 24J0(x) computation

Chebyshev expansion, 83numerical inversion of Laplace

transform, 349the trapezoidal rule, 128

Jν(z) as hypergeometric function,28

Neumann function Yν(z), 25recurrence relations, 96recursion for Jν(z) and Yν(z), 87series expansion for Jν(z), 24Wronskian, 255

Bessel polynomials, 348best approximation, 51

Jackson’s theorem, 63polynomial, 290

versus Chebyshev series, 291rational, 290

oscillations of the error curve,290

binomial coefficientgamma functions, 27Pochhammer symbol, 27

bisection method, 191, 193, 195order of convergence, 194

Bolzano’s theorem, 193Boole’s summation method, 336boundary value problem

for differential equations in thecomplex plane

Taylor-series method, 293Bühring’s analytic continuation formula

for hypergeometric functions,31



Index 407

Carlson’s symmetric elliptic integrals,345

Casorati determinant, 89its use in anti-Miller algorithm, 110

Cauchy’s form for the remainder ofTaylor’s formula, 16

Cauchy’s inequality, 16Cauchy–Riemann equations, 162chaotic behavior in the complex plane,

197characteristic equation, 92Chebyshev equioscillation theorem, 63Chebyshev expansion

computing coefficients, 69convergence properties, 68

analytic functions, 68for Airy functions, 80, 85for error function, 83for J-Bessel functions, 83for Kummer U-function, 84of a function, 66

Chebyshev interpolation, 62computing the polynomial, 64of the second kind, 65

Chebyshev polynomial, 56, 140Chebyshev polynomials

as particular case of Jacobipolynomials, 62

discrete orthogonality relation, 59economization of power series, 80equidistant zeros and extrema, 61expansion of a function, 66minimax approximation, 58of the first kind, 56of the second, third, and fourth

kinds, 60orthogonality relation, 59polynomial representation, 59shifted polynomial T ∗n (x), 60

Chebyshev sumevaluated by Clenshaw’s method,

75Christoffel numbers for Gauss

quadrature, 136classical orthogonal polynomials, 140

Clenshaw’s methodfor evaluating a Chebyshev sum,

65, 75error analysis, 76modification, 78

for solving differential equations,70

for the Abramowitz function, 74for the J-Bessel function, 72

Clenshaw–Curtis quadratures, 62, 296,297

compact operator in a Hilbert space, 209complementary error function

as normal distribution function, 242computed by numerical inversion

of Laplace transform, 350contour integral, 350in uniform asymptotic

approximations, 242complex Gauss quadrature formula, 348

nodes and weights, 349complex orthogonal polynomials, 348compound trapezoidal rule, 126condition of TTRRs, 88confluent hypergeometric functions

anomalous behavior of recursion,120

Chebyshev expansion forU−function, 84

differential equation, 19integral representation for

U(a, c, z), 43M in terms of hypergeometric

function, 28recurrence relations, 96, 99

conical functionscomputing zeros, 211, 223recurrence relation, 103, 211

conjugate harmonic functions, 162continued fraction, 173

computing, 181backward recurrence algorithm,

181forward recurrence algorithm,

181



408 Index

forward series recurrencealgorithm, 181

modified Lentz algorithm, 183Steed’s algorithm, 181

contractions, 175convergence, 175, 179equivalence transformations, 175even and odd part, 175for incomplete beta function, 189for incomplete gamma function,

176for incomplete gamma function

�(a, z), 186for ratios of Gauss hypergeometric

function, 187for special functions, 185Jacobi fraction, J-fraction, 179linear transformations, 174nth convergent, nth approximant,

174numerical evaluation, 181of Gauss, 188recursion for convergents, 174relation with

ascending power series, 178, 179Padé approximant, 278Padé approximants, 179three-term recurrence relation,

95Stieltjes fraction, S-fraction, 178theorems on convergence, 180value of the, 174

contour integrals in the complex planequadrature for, 157

convergence propertiesChebyshev expansion, 68

analytic functions, 68continued fraction, 175

convergent power series, 15converging factor for asymptotic

expansion, 40Coulomb wave functions

recurrence relations, 98cylinder functions, 233, see also Bessel

functions

degree of exactness, 124, 132difference equation

first order inhomogeneous, 112second order homogeneous, 87

differential equationFrobenius method, 22fundamental system of solutions,

21homogeneous linear second order,

292in the complex plane

Taylor-series method, 292Taylor-series method for

boundary value problem, 293Taylor-series method for initial

value problem, 292inhomogeneous linear second

order, 292of Airy functions, 359of Bessel functions, 19of confluent hypergeometric

functions, 19of Gauss hypergeometric functions,

18of Hermite polynomials, 19of Legendre functions, 19, 363of modified Bessel functions, 370

of purely imaginary order, 372of parabolic cylinder functions, 19,

377of Whittaker functions, 19singular point, 19

irregular, 19regular, 19

Taylor expansion method, 291Dini–Lipschitz continuity, 64discrete cosine transform, 66dominant solution of a recurrence

relation, 90double factorial, 364dual algorithm for computing toroidal

harmonics, 369

economization of power series, 80eigenvalue problem

for Bessel functions, 208, 212, 213



Index 409

for compact infinite matrix, 210for conical functions, 211for minimal solutions of three-term

recurrence relations, 207for orthogonal polynomials, 205

elliptic integralother forms, 347

elliptic integralsCarlson’s symmetric forms, 345incomplete of the first kind, 344incomplete of the second kind, 344of Legendre, 344of the third kind, 345

epsilon algorithm of Wynn, 278equidistant interpolation

Runge phenomenon, 54equioscillation property, 67error

absolute, relative, 356bound for fixed point method, 194

error functionsChebyshev expansion, 83computing zeros by using

asymptotic expansions, 229inversion, 330

Euler–Maclaurin formula, 131relation with the trapezoidal rule,

130Euler’s summation formula, 331

limitations, 336exponential function

Padé approximants, 280exponential integral

anomalous behavior of recursion,121

as solution of an inhomogeneouslinear first order differenceequations, 115

asymptotic expansions, 37, 38expansion as factorial series, 45sequence transformations, 288

exponentially improved asymptoticexpansions, 39

factorial series, 44condition for convergence, 44

for exponential integral, 45for incomplete gamma function

�(a, z), 45Fadeeva function, 229fast cosine transforms, 67fast Fourier transform, 69, 298Fejér quadrature, 296

first rule, 297second rule, 297

Filon’s method for oscillatory integrals,303

first order linear inhomogeneousdifference equations, 87

fixed point method, 192, 193, 196based on global strategies, 213error bound, 194Newton–Raphson method, 195order of convergence, 194

fixed point theorem, 193floating-point

IEEE formats, 356IEEE-754 standard for base-2

arithmetic, 356numbers, 356

forward differences, 53forward elimination in Olver’s method,

117forward recurrence algorithm

for computing continued fractions,181

forward series recurrence algorithmfor computing continued fractions,

181fractal, 197Frobenius method, 22fundamental Lagrange interpolation

polynomial, 52fundamental system of solutions, 21

gamma functionalternative asymptotic

representation, 49asymptotic expansion, 243numerical algorithm based on

recursion, 246



410 Index

Gauss hypergeometric functions, 28Bühring’s analytic continuation

formula, 31convergence domains of power

series, 31deriving the continued fraction for

a ratio of, 187differential equation, 18Norlünd’s continued fraction, 105other power series, 30Padé approximants, 283recurrence relations in all

directions, 104recursion for power series

coefficients, 29removable singularities, 33special cases, 29value at z = 1, 29

Gauss quadrature, 132, 135, 191Christoffel numbers, 136computing zeros and weights, 133,

141, 145example for Legendre polynomials,

145for computing Airy functions, 360for computing the Airy function in

the complex plane, 145Gauss–Kronrod, 299Gauss–Lobatto, 299Gauss–Radau, 299generalized Hermite polynomials,

141Golub–Welsch algorithm, 133, 141,

145Hermite polynomials, 145Jacobi matrix

nonorthonormal case, 144orthonormal case, 142

Jacobi polynomials, 145Kronrod nodes, 300Laguerre polynomials, 145Meixner–Pollaczek polynomials,

141orthonormal polynomials, 134other rules, 298Patterson, 301

recursion for orthogonalpolynomials, 143

Stieltjes procedure, 139Gegenbauer polynomial, 140

and Gauss–Kronrod quadrature,300

generalizedhypergeometric function, 27

analytic continuation, 27terminating series, 27

Laguerre polynomial, 140global fixed point methods, 213global strategies for finding zeros, 213Golub–Welsch algorithm for Gauss

quadrature, 133, 141, 145Gram–Schmidt orthogonalization, 134

Hadamard-type expansionsfor modified Bessel function Iν(z),

41Hankel transforms, 303Hermite

interpolation, 53, 54, 136Hermite polynomial

and Gauss–Kronrod quadrature,300

Hermite polynomialsdifferential equation, 19Gauss quadrature, 145generalized for Gauss quadrature,

141special case of parabolic cylinder

functions, 102zeros, 102

hyperasymptotics, 40for the gamma function, 40

hypergeometric functions, see Gausshypergeometric functions

hypergeometric series, 26

IEEE floating-point, see floating-pointill-conditioned problem, 357incomplete beta function

asymptotic inversion, 318error function case, 322



Index 411

incomplete gamma functioncase, 324

symmetric case, 319deriving the continued fraction, 189in terms of the Gauss

hypergeometric function, 189incomplete gamma functions

as solution of inhomogeneouslinear first order differenceequation, 114

asymptotic expansions, 237alternative representation, 47for �(a, z), 37, 238simpler uniform expansions, 247uniform, 242

asymptotic inversion, 312, 329continued fraction, 176

computing �(a, z), 177, 181, 182for �(a, z), 186

expansion as factorial series, 45normalized functions P(a, z) and

Q(a, z), 241numerical algorithm based on

uniform expansion, 245Padé approximant, 284

indicial equation, 22inhomogeneous

Airy functions, 359linear difference equations, 112linear first order difference equation

condition of the recursion, 113for exponential integrals, 115for incomplete gamma function,

114minimal and dominant solutions,

113linear first order difference

equations, 112second order difference equations,

115example, 115Olver’s method, 116subdominant solution, 115superminimal solution, 115

initial value problemfor differential equations in the

complex planeTaylor-series method, 292

inner product of polynomials, 133interpolation

by orthogonal polynomials, 65Chebyshev, 62

of the second kind, 65Hermite, 53, 54, 136Lagrange, 52, 54Runge phenomenon, 54

interpolation polynomialfundamental Lagrange, 52

inversionof complementary error function,

309of error function, 330of incomplete beta functions, 318of incomplete gamma functions,

312, 329irregular singular point of a differential

equation, 19

Jackson’s theorem, 63Jacobi continued fraction, 179Jacobi matrix, 205

Gauss quadraturenonorthonormal case, 144

used in Gauss quadrature, 142Jacobi polynomial, 60, 140

as hypergeometric series, 60Gauss quadrature, 145zeros, 191

Julia set, 197

Kummer functions, see confluenthypergeometric functions

Lagrangeinterpolation, 52

formula for the error, 125fundamental polynomials, 124polynomial, 54

remainder of Taylor’s formula, 16



412 Index

Laguerre polynomial, 140computing zeros, 221

Laguerre polynomialsGauss quadrature, 145

Lambert’sW -function, 312Laplace transform

inversion by using Padéapproximations, 352

numerical inversion, 347, 349Lebesgue constants for Fourier series,

291Legendre functions

associated functions, 363associated functions Pµν (z), Q

µν (z),

363differential equation, 19, 363oblate spheroidal harmonics, 363prolate spheroidal harmonics, 363recurrence relations, 103

for conical functions, 103with respect to the degree, 104with respect to the order, 103

toroidal harmonics, 363Legendre polynomial, 140

example for Gauss quadrature, 145Legendre’s elliptic integrals, 344Levin’s sequence transformation, 287linear

differential equationsregular and singular points, 19solved by Taylor expansion, 291

homogeneous three-termrecurrence relation, 87

independent solutions of arecurrence relation, 89

inhomogeneous first orderdifference equations, 87

Liouville–Green approximation, 26Liouville transformation, 25local strategies for finding zeros, 197logarithmic derivative of the gamma

function, 33Longman’s method for computing

oscillatory integrals, 303

machine-ε, 356

Maclaurin series, 16mathematical libraries for computing

special functions, 355McMahon expansions for zeros of

Bessel functions, 200, 204Meixner–Pollaczek polynomials, 141method of Taylor series

for differential equations in thecomplex plane, 292

Miller algorithmcondition for convergence, 108estimating starting value N, 110for computing modified Bessel

functions In+1/2(x), 106numerical stability, 109numerical stability of the

normalizing sum, 109when a function value is known,

105with a normalizing sum, 107

minimal solution of a recurrencerelation, 90

how to compute by backwardrecursion, 105

minimax approximation, 51Jackson’s theorem, 63

modified Bessel functionsalgorithm, 370anomalous behavior of recursion,

118asymptotic expansion for Kν(z), 43Chebyshev expansions for K0(x)

and K1(x), 370differential equation, 370expansion for Kν(z) in terms of

confluent hypergeometricfunctions, 43

of integer and half-integer orders,370

of purely imaginary order, 372Airy-type asymptotic

expansions, 375algorithms for Kia(x), Lia(x),

372asymptotic expansions, 374



Index 413

continued fraction for Kia(x),373

differential equation, 372nonoscillating integral

representations, 375scaled functions, 372series expansions, 373Wronskian relation, 372

Padé approximants to Kν(z), 352recurrence relation, 97, 370spherical, 370

algorithm, 371notation, 370recurrence relation, 371

trapezoidal rule for K0(x), 153modified Lentz algorithm


modulus of continuity, 63monic

orthogonal polynomials, 134orthonormal polynomials, 134

Newton’s binomial formula, 27Newton’s divided difference formula, 53Newton–Raphson method, 191, 193, 195

high order inversion, 196, 327order of convergence, 195

nodes of a quadrature rule, 124nonlinear differential equations, 25Norlünd’s continued fraction for Gauss

hypergeometric functions, 105normalized incomplete gamma function

asymptotic estimate, 42normalized incomplete gamma functions

asymptotic estimate for P(a, z), 41Hadamard-type expansions, 41relation with chi-square probability

functions, 240uniform asymptotic expansions,

242numerical condition, 357numerical inversion of Laplace

transforms, 347, 349by deforming the contour, 350

complex Gauss quadrature formula,348

to compute Bessel function J0(x),349

to compute the complementaryerror function, 350

numerical stability, 357numerically unstable method, 357

oblate spheroidal harmonics, 363algorithm, 365recurrence relations, 365scaled functions, 364

Olver’s method for inhomogeneoussecond order differenceequations, 116

order of convergenceasymptotic error constant, 194fixed point methods, 194

ordinary differential equation, seedifferential equation

orthogonal basis with respect to innerproduct, 134

orthogonal polynomialscomputing zeros by using

asymptotic expansions, 234on complex contour, 348zeros, 135

orthogonality with respect to innerproduct, 134

oscillatory integrals, 301asymptotic expansion, 301convergence acceleration schemes,

303Filon’s method, 303general forms, 303Hankel transforms, 303Longman’s method for computing,

303overflow threshold, 356

Padé approximants, 276continued fractions, 278diagonal elements in the table, 277generating the lower triangular part

in the table, 278



414 Index

how to compute, 278by Wynn’s cross rule, 278

Luke’s examples for specialfunctions, 283

normality , 277relation with continued fractions,

179table, 277to the exponential function, 280to the Gauss hypergeometric

function, 283to the incomplete gamma functions,

284to the modified Bessel function

Kν(z), 352Wynn’s cross rule for, 278

parabolic cylinder functions, 377algorithm, 378

Maclaurin series, 379regions in (a, x)−plane, 378

asymptotic expansions for large x,380

computing zeros by usingasymptotic expansions, 233

contour integral for, 168definition, 101differential equation, 19, 377integral representations, 384oscillatory behavior, 102recurrence relation for U(a, x), 385relation with Hermite polynomials,

102scaled functions, 377three-term recurrence relations, 101uniform Airy-type asymptotic

expansion, 383uniform asymptotic expansions in

elementary functions, 381Wronskian relation, 101

Perron’s theorem, 92, 93intuitive form, 92

Pincherle’s theorem, 95plasma-dispersion function, 229Pochhammer symbol, 27polynomial

Stieltjes, 300

polynomial approximationminimax, 51

Jackson’s theorem, 63poorly conditioned problem, 357power series

of Bessel function Jν(z), 24, 28of confluent hypergeometric

M-function, 28of Gauss hypergeometric functions,

28of hypergeometric type, 26of the Airy functions, 18of the exponential function, 17

primal algorithmfor computing toroidal harmonics,

366prolate spheroidal harmonics, 363

algorithm, 364recurrence relation for Pmn (x), 365recurrence relation forQmn (x), 365scaled functions, 364

quadraturecharacteristic function for the error,

149Clenshaw–Curtis, 296, 297degree of exactness, 124double exponential formulas, 156erf-rule, 155Fejér, 296

first rule, 297second rule, 297

for contour integrals in the complexplane, 157

Gauss–Kronrod, 299Gauss–Lobatto, 299Gauss quadrature, 132Gauss–Radau, 299other Gauss rules, 298Patterson, 301Romberg quadrature, 294simple trapezoidal rule, 124Simpson’s rule, 295tanh-rule, 154the trapezoidal rule on R, 147transforming the variable, 153



Index 415

weight function, 132weights, 124

quotient-difference algorithm, 178

recurrence relationfor Bessel functions, 96for computing modified Bessel

function Kν(z), 100for confluent hypergeometric

functions, 99in all directions, 99in the (++) direction, 100in the (+ 0) direction, 99in the (0+) direction, 100

for Coulomb wave functions, 98for Legendre functions, 103for modified Bessel functions, 97,

370for modified spherical Bessel

functions, 371for parabolic cylinder functions,

101, 385for prolate spheroidal harmonics

Pmn (x), 365for prolate spheroidal harmonics

Qmn (x), 365for toroidal harmonics, 367

recurrent trapezoidal rule, 129regular point a differential equation, 19regular singular point of a differential

equation, 19relative error, 356Remes’ second algorithm of, 290repeated nodes in Hermite interpolation,

54reverting asymptotic series, 226Riccati–Bessel functions

difference-differential system, 213zeros, 213

Romberg quadrature, 294Runge phenomenon, 54

saddle point method, 158saddle point, 158

scalar product of polynomials, 133scaling functions, 358

to enlarge the domain ofcomputation, 358

to obtain higher accuracy, 358Schwarzian derivative, 26Scorer functions

algorithm for Hi(z), 361asymptotic expansion for Gi(z),

362asymptotic expansion for Hi(z),

362computation for complex

arguments, 359computing scaled functions, 359,

363computing zeros by using

asymptotic expansions, 227connection formulas for Gi(z), 362integral representation for Hi(z),

361power series for Gi(z), 362

secant method, 191second algorithm of Remes, 290second order homogeneous linear

difference equation, 87sequence transformations, 286

for asymptotic expansion ofexponential integral, 288

Levin’s transformation, 287numerical examples, 288of asymptotic series, 288of power series, 288Weniger’s transformation, 287with remainder estimates, 287

Simpson’s rule, 125, 295software survey for computing special

functions, 355sources of errors

due to discretization, 357due to fixed-length representations,

357due to truncation, 357in computations, 357

special functions computingAiry functions of complex

arguments, 359mathematical libraries, 355



416 Index

ratios of Bessel functions, 218Scorer functions of complex

arguments, 359software survey, 355

stability of a numerical method, 357Steed’s algorithm


steepest descent path, 158Stieltjes

continued fraction, 178procedure for recurrence relations,

139Stirling numbers

definitions, 337explicit representations, 337generating functions, 337of the first kind

uniform asymptotic expansion,343

of the second kind, 44uniform asymptotic expansion,

338Stokes phenomenon, 40subdominant solution

of inhomogeneous second orderdifference equation, 115

superminimal solutionof inhomogeneous second order

difference equations, 115symmetric elliptic integrals, 345

Taylor series, 16Cauchy’s formula for remainder, 16Lagrange’s formula for remainder,

16Taylor’s formula for remainder, 16

Taylor-series methodfor boundary value problem in the

complex plane, 293for initial value problem in the

complex plane, 292testing of software

for computing functions, 358by comparison with existing

algorithms, 358

by extended precisionalgorithms, 358

by verification of functionalrelations, 358

consistency between differentmethods, 358

three-term recurrence relation, see alsorecurrence relation

anomalous behavior, 118confluent hypergeometric

functions, 120exponential integrals, 121modified Bessel functions, 118

backward recursion, 91condition of, 88dominant solution, 90forward recursion, 91linear homogeneous, 87linearly independent solutions, 89minimal solution, 89, 90relation with continued fractions,

95scaled form, 94with constant coefficients, 92

toroidal harmonics, 363algorithm, 366asymptotic expansion for PM−1/2(x),

368dual algorithm, 369primal algorithm, 366recurrence relation, 367relation with elliptic integrals, 367scaled functions, 364series expansion for PM−1/2(x), 367

trapezoidal rule, 350simple rule, 124compound rule, 126Euler’s summation formula, 130for computing Scorer functions,

362for computing the Bessel function

J0(x), 128for computing the Bessel function

K0(x), 153for computing the complementary

error function, 350



Index 417

on R, 147recursive computation, 129with exponentially small error, 151

TTRR, see three-term recurrencerelation

turning point of a differential equation,249

underflow threshold, 356

Wagner’s modification of asymptoticexpansions, 48

Watson’s lemma, 36weight function for numerical

quadrature, 132weights of a quadrature rule, 124Weniger’s sequence transformation, 287Whittaker functions

differential equation, 19WKB approximation, 26Wronskian, 21

for Airy functions, 254for Bessel functions, 255for modified Bessel functions of

purely imaginary order, 372for parabolic cylinder functions,

101Wynn’s cross rule

for Padé approximants, 278Wynn’s epsilon algorithm, 278

zeros of functions, 191Airy functions, 224

asymptotic approximations, 197,200

Bessel functions, 204, 233, 385complex zeros, 197eigenvalue problem, 208, 212from Airy-type asymptotic

expansions, 204McMahon expansions, 200, 204

bisection method, 191, 193complex zeros, 197computation based on asymptotic

approximations, 199conical functions, 211, 223cylinder functions, 233eigenvalue problem for orthogonal

polynomials, 205error functions, 229fixed point method, 193fixed point methods and

asymptotics, 199global strategies, 204, 213Jacobi polynomials, 191Laguerre polynomials, 221local strategies, 197matrix methods, 204Newton–Raphson method, 191, 193orthogonal polynomials, 135, 234parabolic cylinder functions, 233Riccati–Bessel functions, 213Scorer functions, 227secant method, 191



Documents

Index [ ] · PDF fileJackson’s theorem, 63 ... oscillations of the error curve, 290 binomial coefﬁcient gamma functions, 27 Pochhammer symbol, 27 ... ascending power series, 178,