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§22.10 Maclaurin Series

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§22.10(i) Maclaurin Series in $z$

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§22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$

… ►Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …

… ►and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. … ►Formulas (18.17.12) and (18.17.13) are fractional generalizations of the differentiation formulas given in (Erdélyi et al., 1953b, §10.9(15)). … ►In particular, in case of exponential Fourier transforms, we may assume

y\in\mathbb{R}

. … ►In (18.17.21_1) the branch choice of

\sqrt{2c-1}

for

0<c<\frac{1}{2}

is unimportant because on the right-hand side only even powers of

\sqrt{2c-1}

occur after expansion of the Hermite polynomial by (18.5.13). … ►Many of the Fourier transforms given in §18.17(v) have analytic continuations to Laplace transforms. …

… ►With

\psi\left(x\right)

denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of

K\left(k\right)

and

E\left(k\right)

near the singularity at

k=1

is given by the following convergent series: ►

19.12.1

K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left% (\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln\left(\frac{1}{k^{% \prime}}\right)+d(m)\right),

0<|k^{\prime}|<1

,

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $d(m)$ : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

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19.12.2

E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2\right)_{m}}m!}{k^{\prime% }}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)-\frac{1}{(2m+1)(2m+% 2)}\right),

|k^{\prime}|<1

,

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $d(m)$ : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

… ►Asymptotic approximations for

\Pi\left(\phi,\alpha^{2},k\right)

, with different variables, are given in Karp et al. (2007). … ►

19.12.6

R_{C}\left(x,y\right)=\frac{\pi}{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(% \sqrt{\frac{x}{y}}\right)\right),

x/y\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 and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

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… ► … ►

Operations

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§1.9(vii) Inversion of Limits

►

Double Sequences and Series

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Term-by-Term Integration

…

… ►where the inverse sine has its principal value when

-K\leq x\leq K

and is defined by continuity elsewhere. … ►

Fourier Series

… ►In Equations (22.16.21)–(22.16.23),

-K<x<K.

… ►In Equations (22.16.24)–(22.16.26),

-2K<x<2K

. … ►With

E\left(k\right)

and

K\left(k\right)

as in §19.2(ii) and

x\in\mathbb{R}

, …

§18.3 Definitions

… ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of

x-1

for Jacobi polynomials, in powers of

x

for the other cases). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for

n=0,1,\ldots,6

are given in §18.5(iv). … ►In consequence, additional properties are included in Chapter 14. … ►For

\nu\in\mathbb{R}

and

N>-\tfrac{1}{2}

a finite system of Jacobi polynomials

P^{(-N-1+\mathrm{i}\nu,-N-1-\mathrm{i}\nu)}_{n}\left(\mathrm{i}x\right)

(called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on

(-\infty,\infty)

with

w(x)=\left(1+x^{2}\right)^{-N-1}{\mathrm{e}}^{2\nu\operatorname{arctan}x}

. …

… ►

A. R. DiDonato and A. H. Morris (1986) Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.

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External Links:: ISSN 0098-3500, Document, ZentralBlatt
Referenced by:: §8.12, §8.25(iii).
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

… ►

P. Dienes (1931) The Taylor Series. Oxford University Press, Oxford.

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Notes:: Reprinted by Dover Publications Inc., New York, 1957
External Links:: MathReview, ZentralBlatt
Referenced by:: §1.9(iii).
Encodings:: BibTeX

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P. G. L. Dirichlet (1837) Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, pp. 45–81 (German).

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Notes:: Reprinted in G. Lejeune Dirichlet’s Werke, Band I, pp. 313–342, Reimer, Berlin, 1889 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.
External Links:: ISBN 0-8284-0225-6, Link, MathReview
Referenced by:: §25.15(i).
Encodings:: BibTeX

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P. G. L. Dirichlet (1849) Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:: Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.
External Links:: ISBN 0-8284-0225-6, Link, MathReview
Referenced by:: §27.11.
Encodings:: BibTeX

…

… ►The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … ►As a consequence, in addition to providing more information about the special functions that were covered in AMS 55, the NIST Handbook includes several special functions that have appeared in the interim in applied mathematics, the physical sciences, and engineering, as well as in other areas. … ►The first section in each of the special function chapters (Chapters 5–36) lists notation that has been adopted for the functions in that chapter. … ►Similarly in the case of confluent hypergeometric functions (§13.2(i)). … ►In the DLMF this information is provided in pop-up windows at the subsection level. …

… ►

Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

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Subsection 25.2(ii) Other Infinite Series

It is now mentioned that (25.2.5), defines the Stieltjes constants $\gamma_{n}$ . Consequently, $\gamma_{n}$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

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Equation (33.6.5)

33.6.5

{H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{{\mathrm{e}}^{\pm\mathrm{i}{% \theta_{\ell}}\left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-\ell\pm\mathrm{i% }\eta\right)}\left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(2\ell% +2\right)_{k}}k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2\mathrm{i}\rho% \right)+\psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(2\ell+2+k\right)% \right)-\sum_{k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!{\left(1-a% \right)_{k}}}(\mp 2\mathrm{i}\rho)^{a-k}\right)

Originally the factor in the denominator on the right-hand side was written incorrectly as $\Gamma\left(-\ell+\mathrm{i}\eta\right)$ . This has been corrected to $\Gamma\left(-\ell\pm\mathrm{i}\eta\right)$ .

Reported by Ian Thompson on 2018-05-17

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Equation (34.3.7)

34.3.7

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ j_{1}&-j_{1}-m_{3}&m_{3}\end{pmatrix}=(-1)^{j_{1}-j_{2}-m_{3}}\left(\frac{(2j_% {1})!(-j_{1}+j_{2}+j_{3})!(j_{1}+j_{2}+m_{3})!(j_{3}-m_{3})!}{(j_{1}+j_{2}+j_{% 3}+1)!(j_{1}-j_{2}+j_{3})!(j_{1}+j_{2}-j_{3})!(-j_{1}+j_{2}-m_{3})!(j_{3}+m_{3% })!}\right)^{\frac{1}{2}}

In the original equation the prefactor of the above 3j symbol read $(-1)^{-j_{2}+j_{3}+m_{3}}$ . It is now replaced by its correct value $(-1)^{j_{1}-j_{2}-m_{3}}$ .

Reported 2014-06-12 by James Zibin.

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Equation (34.4.2)

34.4.2

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!}

Originally the factor $\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_% {1}l_{2}j_{3})$ was missing in this equation.

Reported 2012-12-31 by Yu Lin.

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in inverse factorial series

11—19 of 19 matching pages

11: 22.10 Maclaurin Series

§22.10 Maclaurin Series

§22.10(i) Maclaurin Series in $z$

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$

12: 18.17 Integrals

13: 19.12 Asymptotic Approximations

14: 1.9 Calculus of a Complex Variable

Operations

§1.9(vii) Inversion of Limits

Double Sequences and Series

Term-by-Term Integration

15: 22.16 Related Functions

Fourier Series

16: 18.3 Definitions

§18.3 Definitions

17: Bibliography D

18: Mathematical Introduction

19: Errata

in inverse factorial series

11—19 of 19 matching pages

11: 22.10 Maclaurin Series

§22.10 Maclaurin Series

§22.10(i) Maclaurin Series in z

§22.10(ii) Maclaurin Series in k and k ′

12: 18.17 Integrals

13: 19.12 Asymptotic Approximations

14: 1.9 Calculus of a Complex Variable

Operations

§1.9(vii) Inversion of Limits

Double Sequences and Series

Term-by-Term Integration

15: 22.16 Related Functions

Fourier Series

16: 18.3 Definitions

§18.3 Definitions

17: Bibliography D

18: Mathematical Introduction

19: Errata

§22.10(i) Maclaurin Series in $z$

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$