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21—26 of 26 matching pages

21: Errata

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Subsection 19.2(ii) and Equation (19.2.9)

The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9

{K^{\prime}}\left(k\right)=\begin{cases}K\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ K\left(k^{\prime}\right)\mp 2\mathrm{i}K\left(-k\right),&\tfrac{1}{2}\pi<\pm% \operatorname{ph}k<\pi,\end{cases}

{E^{\prime}}\left(k\right)=\begin{cases}E\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ E\left(k^{\prime}\right)\mp 2\mathrm{i}(K\left(-k\right)-E\left(-k\right)),&% \tfrac{1}{2}\pi<\pm\operatorname{ph}k<\pi\end{cases}

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Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, $\mathsf{P}_{n},\mathsf{Q}_{n},P_{n},Q_{n},\boldsymbol{Q}_{n}$ and the Laguerre polynomial, $L_{n}$ , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

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Equation (14.6.6)

14.6.6

\mathsf{P}^{-m}_{\nu}\left(x\right)=\left(1-x^{2}\right)^{-m/2}\int_{x}^{1}\!% \dots\!\int_{x}^{1}\mathsf{P}_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^% {m}

The right-hand side has been corrected by replacing the Legendre function $P_{\nu}\left(x\right)$ with the Ferrers function $\mathsf{P}_{\nu}\left(x\right)$ .

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Equations (22.14.16), (22.14.17)

22.14.16

\int_{0}^{K\left(k\right)}\ln\left(\operatorname{sn}\left(t,k\right)\right)\,% \mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)-\tfrac{1}{2}K\left(k% \right)\ln k,

22.14.17

\int_{0}^{K\left(k\right)}\ln\left(\operatorname{cn}\left(t,k\right)\right)\,% \mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)+\tfrac{1}{2}K\left(k% \right)\ln\left(k^{\prime}/k\right)

Originally, a factor of $\pi$ was missing from the terms containing the $-\tfrac{1}{4}{K^{\prime}}\left(k\right)$ .

Reported by Fred Hucht on 2020-08-06

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Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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22: 26.14 Permutations: Order Notation

… ►A descent of a permutation is a pair of adjacent elements for which the first is larger than the second. …The major index is the sum of all positions that mark the first element of a descent: … ►In this subsection

S\left(n,k\right)

is again the Stirling number of the second kind (§26.8), and

B_{m}

is the

m

th Bernoulli number (§24.2(i)). … ►

26.14.7

\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{n-k}(-1)^{n-k-j}j!\genfrac{(}{)}{0.0% pt}{}{n-j}{k}S\left(n,j\right),

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Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(iii), §26.14 and Ch.26

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26.14.12

S\left(n,m\right)=\frac{1}{m!}\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}% \genfrac{(}{)}{0.0pt}{}{k}{n-m},

n\geq m

n\geq 1

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Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.14.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(iii), §26.14 and Ch.26

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23: 2.11 Remainder Terms; Stokes Phenomenon

… ►First, it is impossible to bound the tail by majorizing its terms. … ►Taking

m=10

in (2.11.2), the first three terms give us the approximation … ►The first of these two references also provides an introduction to the powerful Borel transform theory. … ►Subtraction of this result from the sum of the first 5 terms in (2.11.25) yields 0. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

24: Bibliography R

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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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K. Reinsch and W. Raab (2000) Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(i), §19.36(ii).
Encodings:: BibTeX

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R. Roy (2011) Sources in the development of mathematics. Cambridge University Press, Cambridge.

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Notes:: Infinite Series and Products from the Fifteenth to the Twenty-first Century
External Links:: ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani), ZentralBlatt
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

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25: Bibliography C

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B. C. Carlson (1964) Normal elliptic integrals of the first and second kinds. Duke Math. J. 31 (3), pp. 405–419.

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External Links:: Document, MathReview (S. Hitotumatu), ZentralBlatt
Referenced by:: §19.22(iii), §19.23, §19.33(i), §19.33(i).
Encodings:: BibTeX

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B. C. Carlson (1977a) Elliptic integrals of the first kind. SIAM J. Math. Anal. 8 (2), pp. 231–242.

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External Links:: Document, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §19.29(iii).
Encodings:: BibTeX

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R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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W. J. Cody (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

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Notes:: Double-precision Fortran, maximum accuracy 24S.
External Links:: ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

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26: Bibliography S

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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R. Szmytkowski (2006) On the derivative of the Legendre function of the first kind with respect to its degree. J. Phys. A 39 (49), pp. 15147–15172.

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External Links:: ISSN 0305-4470, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

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R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

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External Links:: ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

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R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

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External Links:: ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

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R. Szmytkowski (2012) On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.

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External Links:: ISSN 0022-247X, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX