DLMF: Untitled Document

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N. M. Temme (1993) Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.

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External Links:: ISSN 0022-2526, FullText, MathReview (E. Rodney Canfield), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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N. M. Temme (2015) Asymptotic Methods for Integrals. Series in Analysis, Vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.

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External Links:: ISBN 978-981-4612-15-9, MathReview Entry, ZentralBlatt
Referenced by:: §12.9(i), §12.9(i), (13.8.13), (13.8.14), (13.8.15), (13.8.16), §13.8(ii), §13.8(ii), §13.8(iii), §13.8(iii), §14.15(i), §14.15(i), §14.15(iii), §14.15(iii), §15.12(iii), §15.12(iii), §18.15(vi), §18.15(vi), §2.4(v), §2.4(v), §2.4(vi), §2.4(vi), §26.8(vii), §26.8(vii), §33.12(i), §33.12(i), §33.12(ii), §33.12(ii), Profile Nico M. Temme.
Encodings:: BibTeX

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J. S. Thompson (1996) High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.

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External Links:: FullText
Referenced by:: §25.21(vii).
Encodings:: BibTeX

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E. C. Titchmarsh (1962b) The Theory of Functions. 2nd edition, Oxford University Press, Oxford.

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External Links:: ZentralBlatt
Referenced by:: §1.10(ix), §1.10(v), §1.8(i), §1.8(ii), §1.8(iii), §1.9(vii).
Encodings:: BibTeX

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L. N. Trefethen and D. Bau (1997) Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-361-7, MathReview (Moody T. Chu), ZentralBlatt
Referenced by:: §3.2(iii), §3.2(vii).
Encodings:: BibTeX

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J. L. Schiff (1999) The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:: ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt
Referenced by:: §1.14(iii), §1.14(vii).
Encodings:: BibTeX

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M. J. Seaton and G. Peach (1962) The determination of phases of wave functions. Proc. Phys. Soc. 79 (6), pp. 1296–1297.

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External Links:: Document, ZentralBlatt
Referenced by:: §33.23(vii).
Encodings:: BibTeX

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M. J. Seaton (1984) The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

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External Links:: Document
Referenced by:: §33.23(vii).
Encodings:: BibTeX

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J. D. Secada (1999) Numerical evaluation of the Hankel transform. Comput. Phys. Comm. 116 (2-3), pp. 278–294.

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External Links:: ISSN 0010-4655, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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B. Simon (1995) Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.

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External Links:: MathReview Entry
Referenced by:: §1.18(vii).
Encodings:: BibTeX

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… ►These cases are treated in §§12.10(vii)–12.10(viii). … ►

12.10.23

\eta=\tfrac{1}{2}\operatorname{arccos}t-\tfrac{1}{2}t\sqrt{1-t^{2}},

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Defines:: $\eta$ (locally)
Symbols:: $\operatorname{arccos}\NVar{z}$ : arccosine function
Referenced by:: §12.10(vii), §12.14(ix)
Permalink:: http://dlmf.nist.gov/12.10.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §12.10(iv), §12.10 and Ch.12

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§12.10(vii) Negative $a$ , $-2\sqrt{-a}<x<\infty$ . Expansions in Terms of Airy Functions

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12.10.40

\phi(\zeta)=\left(\frac{\zeta}{t^{2}-1}\right)^{\frac{1}{4}}.

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Defines:: $\phi(\zeta)$ : function (locally)
Symbols:: $\zeta$ : change of variable
Referenced by:: §12.10(vii)
Permalink:: http://dlmf.nist.gov/12.10.E40
Encodings:: pMML, png, TeX
See also:: Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.41

t=1+w-\tfrac{1}{10}w^{2}+\tfrac{11}{350}w^{3}-\tfrac{823}{63000}w^{4}+\tfrac{1% \;50653}{242\;55000}w^{5}+\cdots,

|\zeta|<\left(\tfrac{3}{4}\pi\right)^{\frac{2}{3}}.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter and $\zeta$ : change of variable
Referenced by:: §12.11(iii), §12.11(iii)
Permalink:: http://dlmf.nist.gov/12.10.E41
Encodings:: pMML, png, TeX
See also:: Annotations for §12.10(vii), §12.10 and Ch.12

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G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert

W

. Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:: ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

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E. Kamke (1977) Differentialgleichungen: Lösungsmethoden und Lösungen. Teil I. B. G. Teubner, Stuttgart (German).

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External Links:: ISBN 3-519-02017-3, MathReview, ZentralBlatt
Referenced by:: §1.13(vii), §10.13, §4.20, §4.34, §4.7(ii).
Encodings:: BibTeX

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E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §19.12.
Encodings:: BibTeX

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C. Kittel (1996) Introduction to Solid State Physics. 7th Edition edition, John Wiley and Sons, New York.

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External Links:: ISBN 0-471-11181-3
Referenced by:: §1.18(vii).
Encodings:: BibTeX

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K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

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Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §25.21(v).
Encodings:: BibTeX

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J. B. Parkinson (1969) Optical properties of layer antiferromagnets with

\mathrm{K}_{2}\mathrm{NiF}_{4}

structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.

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External Links:: Document
Referenced by:: §19.35(ii).
Encodings:: BibTeX

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R. Piessens and M. Branders (1983) Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.

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External Links:: ISSN 0006-3835, Document, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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R. Piessens and M. Branders (1985) A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.

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Notes:: International conference on special functions: theory and computation (Turin, 1984)
External Links:: ISSN 0373-1243, MathReview, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

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A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt
Referenced by:: §1.14(vii).
Encodings:: BibTeX

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M. Puoskari (1988) A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

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External Links:: ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

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A. J. MacLeod (2002a) Asymptotic expansions for the zeros of certain special functions. J. Comput. Appl. Math. 145 (2), pp. 261–267.

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External Links:: ISSN 0377-0427, Document, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.70, §11.4(vii), §6.13.
Encodings:: BibTeX

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C. S. Meijer (1946) On the

G

-function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §16.11(ii).
Encodings:: BibTeX

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K. S. Miller and B. Ross (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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External Links:: ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).
Encodings:: BibTeX

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L. M. Milne-Thomson (1950) Jacobian Elliptic Function Tables. Dover Publications Inc., New York.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §22.20(vii).
Encodings:: BibTeX

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L. Moser and M. Wyman (1958a) Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.

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External Links:: Document, MathReview (N. J. Fine), ZentralBlatt
Referenced by:: §26.1, §26.8(vii).
Encodings:: BibTeX

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§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals

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	Open Source				With Book		Commercial
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7.25(vii) $\mathcal{F}\left(z\right)$ , $G\left(z\right)$ , $z\in\mathbb{C}$								✓	✓	✓
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8.28(vii) $E_{p}\left(z\right)$ , $z,p\in\mathbb{C}$	✓		✓	✓				✓	✓	✓
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11 Struve and Related Functions
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25.21(vii) Fermi–Dirac, Bose–Einstein		✓			✓	✓			✓
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30 Spheroidal Wave Functions
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… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►

Table 3.6.1: Weber function

w_{n}=\mathbf{E}_{n}\left(1\right)

computed by Olver’s algorithm.

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$n$	$p_{n}$		$e_{n}$		$e_{n}/(p_{n}p_{n+1})$		$w_{n}$
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11	$0.29154\;738$	$\times 10^{10}$	$0.37225\;201$	$\times 10^{10}$	$0.19952\;026$	$\times 10^{-10}$	$0.58373\;946$	$\times 10^{-1}$
…

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§3.6(vii) Linear Difference Equations of Other Orders

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3.6.17

a_{n}w_{n+1}-b_{n}w_{n}=d_{n}.

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Symbols:: $w_{n}$ : sequence, $a_{n}$ : coefficient, $b_{n}$ : coefficient and $d_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.6.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §3.6(vii), §3.6 and Ch.3

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3.6.18

a_{n,k}w_{n+k}+a_{n,k-1}w_{n+k-1}+\dots+a_{n,0}w_{n}=d_{n},

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Symbols:: $w_{n}$ : sequence, $a_{n}$ : coefficient and $d_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.6.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §3.6(vii), §3.6 and Ch.3

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§1.9(vii) Inversion of Limits

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1.9.64

\left|z_{m,n}-z\right|<\epsilon

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Symbols:: $z$ : variable, $m$ : nonnegative integer, $n$ : nonnegative integer, $\epsilon$ : positive number and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.9.E64
Encodings:: pMML, png, TeX
See also:: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

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1.9.66

z_{p,q}=\sum^{p}_{m=0}\sum^{q}_{n=0}\zeta_{m,n}.

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Symbols:: $z$ : variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $\zeta_{p,q}$ : sum
Permalink:: http://dlmf.nist.gov/1.9.E66
Encodings:: pMML, png, TeX
See also:: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

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1.9.69

\int^{b}_{a}\sum^{\infty}_{n=0}\left|f_{n}(t)\right|\,\mathrm{d}t<\infty,

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Referenced by:: §1.9(vii)
Permalink:: http://dlmf.nist.gov/1.9.E69
Encodings:: pMML, png, TeX
See also:: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

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1.9.71

\int^{b}_{a}\sum^{\infty}_{n=0}f_{n}(t)\,\mathrm{d}t=\sum^{\infty}_{n=0}\int^{% b}_{a}f_{n}(t)\,\mathrm{d}t.

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer
Referenced by:: §1.9(vii)
Permalink:: http://dlmf.nist.gov/1.9.E71
Encodings:: pMML, png, TeX
See also:: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

.卡塔尔世界杯奖金_『网址:687.vii』世界杯竞猜_b5p6v3_2022年11月30日21时49分53秒_3vjzhlfvb.cc

11—20 of 231 matching pages

11: Bibliography T

12: Bibliography S

13: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10(vii) Negative $a$ , $-2\sqrt{-a}<x<\infty$ . Expansions in Terms of Airy Functions

14: Bibliography K

15: Bibliography P

16: Bibliography M

17: 25.21 Software

§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals

18: Software Index

19: 3.6 Linear Difference Equations

§3.6(vii) Linear Difference Equations of Other Orders

20: 1.9 Calculus of a Complex Variable

§1.9(vii) Inversion of Limits

.卡塔尔世界杯奖金_『网址:687.vii』世界杯竞猜_b5p6v3_2022年11月30日21时49分53秒_3vjzhlfvb.cc

11—20 of 231 matching pages

11: Bibliography T

12: Bibliography S

13: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10(vii) Negative a , − 2 ⁢ − a < x < ∞ . Expansions in Terms of Airy Functions

14: Bibliography K

15: Bibliography P

16: Bibliography M

17: 25.21 Software

§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals

18: Software Index

19: 3.6 Linear Difference Equations

§3.6(vii) Linear Difference Equations of Other Orders

20: 1.9 Calculus of a Complex Variable

§1.9(vii) Inversion of Limits

§12.10(vii) Negative $a$ , $-2\sqrt{-a}<x<\infty$ . Expansions in Terms of Airy Functions