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22: 36.11 Leading-Order Asymptotics

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36.11.1

t_{1}(\mathbf{x})<t_{2}(\mathbf{x})<\cdots<t_{j_{\max}}(\mathbf{x}),

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Symbols:: $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11 and Ch.36

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36.11.2

\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).

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Symbols:: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $t$ : variable, $K$ : codimension and $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11 and Ch.36

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23: Bibliography T

… ►

G. Taubmann (1992) Parabolic cylinder functions

U(n,x)

for natural

n

and positive

x

. Comput. Phys. Commun. 69, pp. 415–419.

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Notes:: Double-precision Fortran, minimum accuracy 14S, maximum accuracy 21D.
External Links:: Document, Code
Referenced by:: 4th item.
Encodings:: BibTeX

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N. M. Temme (1978) The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:: Algol procedures, maximum accuracy 14S.
External Links:: FullText, ZentralBlatt
Referenced by:: §12.21(ii).
Encodings:: BibTeX

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H. C. Thacher Jr. (1963) Algorithm 165: Complete elliptic integrals. Comm. ACM 6 (4), pp. 163–164.

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Notes:: Includes Algol program, maximum accuracy 10D. For certification see same journal 6 (1963), pp. 166–167
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

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W. J. Thompson (1997) Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners. John Wiley & Sons Inc., New York.

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Notes:: With CD-ROM containing a large collection of mathematical function software written in Fortran 90 and Mathematica (an edition with the same software in C and Mathematica exists also). The functions are computed for real variables only. Maximum accuracy 12D.
External Links:: ISBN 0-471-18171-4, MathReview, ZentralBlatt
Referenced by:: §19.36(ii), § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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24: 19.27 Asymptotic Approximations and Expansions

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19.27.13

R_{J}\left(x,y,z,p\right)=\frac{3}{2\sqrt{z}p}\left(\ln\left(\frac{8z}{a+g}% \right)-2R_{C}\left(1,\frac{p}{z}\right)+O\left(\left(\frac{a}{z}+\frac{a}{p}% \right)\ln\frac{p}{a}\right)\right),

\max(x,y)/\min(z,p)\to 0

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ and $g$
Referenced by:: §19.12, §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.27.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §19.27(v), §19.27 and Ch.19

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19.27.14

R_{J}\left(x,y,z,p\right)=\frac{3}{\sqrt{yz}}R_{C}\left(x,p\right)-\frac{6}{yz% }R_{G}\left(0,y,z\right)+O\left(\frac{\sqrt{x+2p}}{yz}\right),

\max(x,p)/\min(y,z)\to 0

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.27.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §19.27(v), §19.27 and Ch.19

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19.27.16

R_{J}\left(x,y,z,p\right)=(3/\sqrt{x})R_{C}\left((h+p)^{2},2(b+h)p\right)+O% \left(\frac{1}{x^{3/2}}\ln\frac{x}{b+h}\right),

\max(y,z,p)/x\to 0

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $b$ and $h$
Permalink:: http://dlmf.nist.gov/19.27.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §19.27(v), §19.27 and Ch.19

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25: 3.11 Approximation Techniques

… ►A sufficient condition for

p_{n}(x)

to be the minimax polynomial is that

\left|\epsilon_{n}(x)\right|

attains its maximum at

n+2

distinct points in

[a,b]

and

\epsilon_{n}(x)

changes sign at these consecutive maxima. … ►(Thus the

m_{j}

are approximations to

m

, where

\pm m

is the maximum value of

\left|\epsilon_{n}(x)\right|

[a,b]

.) … ►More precisely, it is known that for the interval

[a,b]

, the ratio of the maximum value of the remainder …to the maximum error of the minimax polynomial

p_{n}(x)

is bounded by

1+L_{n}

, where

L_{n}

is the

n

th Lebesgue constant for Fourier series; see §1.8(i). … ►and

\pm m

is the maximum of

\left|\epsilon_{k,\ell}(x)\right|

[a,b]

. …

26: Bibliography D

… ►

Delft Numerical Analysis Group (1973) On the computation of Mathieu functions. J. Engrg. Math. 7, pp. 39–61.

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Notes:: Variable-precision Algol program, maximum precision 15D
External Links:: Document, MathReview (C. J. Bouwkamp), ZentralBlatt
Referenced by:: §28.36(ii), §28.36(iii).
Encodings:: BibTeX

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G. Delic (1979a) Chebyshev expansion of the associated Legendre polynomial

P^{M}_{L}(x)

. Comput. Phys. Comm. 18 (1), pp. 63–71.

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Notes:: Double-precision Fortran, maximum accuracy 25D
External Links:: Document, Code
Referenced by:: 3rd item.
Encodings:: BibTeX

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A. R. DiDonato and A. H. Morris (1987) Algorithm 654: Fortran subroutines for computing the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 13 (3), pp. 318–319.

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Notes:: Single-precision Fortran, maximum accuracy 14D.
External Links:: ISSN 0098-3500, Document, GAMS (module 654; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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A. R. DiDonato and A. H. Morris (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.

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Notes:: Single-precision Fortran, maximum accuracy 14D.
External Links:: ISSN 0098-3500, Document, GAMS (module 708; package TOMS), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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E. Dorrer (1968) Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

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Notes:: Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39
External Links:: ISSN 0001-0782, Document
Referenced by:: §8.28(iv).
Encodings:: BibTeX

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27: 6.16 Mathematical Applications

… ►The first maximum of

\frac{1}{2}\operatorname{Si}\left(x\right)

for positive

x

occurs at

x=\pi

and equals

(1.1789\dots)\times\frac{1}{4}\pi

; compare Figure 6.3.2. …

28: 14.21 Definitions and Basic Properties

… ►The generating function expansions (14.7.19) (with

\mathsf{P}

replaced by

P

) and (14.7.22) apply when

|h|<\min\left|z\pm\left(z^{2}-1\right)^{1/2}\right|

; (14.7.21) (with

\mathsf{P}

replaced by

P

) applies when

|h|>\max\left|z\pm\left(z^{2}-1\right)^{1/2}\right|

29: 19.34 Mutual Inductance of Coaxial Circles

… ►is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. …

30: 27.4 Euler Products and Dirichlet Series

… ►

27.4.11

\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)% \zeta\left(s-\alpha\right),

\Re s>\max(1,1+\Re\alpha)

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, $\Re$ : real part and $n$ : positive integer
A&S Ref:: 24.3.3 I.B
Permalink:: http://dlmf.nist.gov/27.4.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §27.4 and Ch.27

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