Abramowitz Function Computed by Clenshaw's Method, 74 Absolute
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.
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