elementary functions

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1: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►

19.36.1

1-\tfrac{1}{10}E_{2}+\tfrac{1}{14}E_{3}+\tfrac{1}{24}E_{2}^{2}-\tfrac{3}{44}E_% {2}E_{3}-\tfrac{5}{208}E_{2}^{3}+\tfrac{3}{104}E_{3}^{2}+\tfrac{1}{16}E_{2}^{2% }E_{3},

ⓘ

Symbols:: $E_{s}(\mathbf{z})$ : symmetric function
Referenced by:: §19.19, §19.36(i), §19.36(i)
Permalink:: http://dlmf.nist.gov/19.36.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36 and Ch.19

►where the elementary symmetric functions

E_{s}

are defined by (19.19.4). … ►

19.36.2

1-\tfrac{3}{14}E_{2}+\tfrac{1}{6}E_{3}+\tfrac{9}{88}E_{2}^{2}-\tfrac{3}{22}E_{% 4}-\tfrac{9}{52}E_{2}E_{3}+\tfrac{3}{26}E_{5}-\tfrac{1}{16}E_{2}^{3}+\tfrac{3}% {40}E_{3}^{2}+\tfrac{3}{20}E_{2}E_{4}+\tfrac{45}{272}E_{2}^{2}E_{3}-\tfrac{9}{% 68}(E_{3}E_{4}+E_{2}E_{5}).

ⓘ

Symbols:: $E_{s}(\mathbf{z})$ : symmetric function
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.36.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36 and Ch.19

… ►

19.36.4

\begin{aligned} z_{1}&=2.10985\;99098\;8,\\ z_{3}&=2.15673\;49098\;8,\\ Z_{1}&=0.00977\;77253\;5,\end{aligned}\quad\begin{aligned} z_{2}&=2.12548\;490% 98\;8,\\ A&=2.13069\;32432\;1,\\ Z_{2}&=0.00244\;44313\;4,\end{aligned}\\ {Z_{3}=-Z_{1}-Z_{2}=-0.01222\;21566\;9,}\\ {E_{2}=\Sci{-1.25480\;14}{-4},\quad E_{3}=\Sci{-2.9212}{-7}.}

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Symbols:: $E_{s}(\mathbf{z})$ : symmetric function, $Z_{j}$ : components and $A$
Permalink:: http://dlmf.nist.gov/19.36.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36(i), §19.36 and Ch.19

…

2: 19.19 Taylor and Related Series

… ►Define the elementary symmetric function

E_{s}(\mathbf{z})

by ►

19.19.4

\prod_{j=1}^{n}(1+tz_{j})=\sum_{s=0}^{n}t^{s}E_{s}(\mathbf{z}),

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Defines:: $E_{s}(\mathbf{z})$ : symmetric function (locally)
Symbols:: $n$ : nonnegative integer
Referenced by:: §19.36(i)
Permalink:: http://dlmf.nist.gov/19.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.19 and Ch.19

… ►

19.19.5

T_{N}(\mathbf{\tfrac{1}{2}},\mathbf{z})=\sum(-1)^{M+N}{\left(\tfrac{1}{2}% \right)_{M}}\frac{E_{1}^{m_{1}}(\mathbf{z})\cdots E_{n}^{m_{n}}(\mathbf{z})}{m% _{1}!\cdots m_{n}!},

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer, $n$ : nonnegative integer, $N$ : nonnegative integer, $T_{N}(\mathbf{b},\mathbf{z})$ : polynomial, $E_{s}(\mathbf{z})$ : symmetric function and $M$ : sum
Referenced by:: §19.19, §19.36(i)
Permalink:: http://dlmf.nist.gov/19.19.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.19 and Ch.19

… ►The number of terms in

T_{N}

can be greatly reduced by using variables

\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)

with

A

chosen to make

E_{1}(\mathbf{Z})=0

. … ►

E_{1}(\mathbf{Z})=0

|Z_{j}|<1

…

3: 4 Elementary Functions

Chapter 4 Elementary Functions

…

4: Bibliography X

… ►

G. L. Xu and J. K. Li (1994) Variable precision computation of elementary functions. J. Numer. Methods Comput. Appl. 15 (3), pp. 161–171 (Chinese).

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Notes:: English translation in Chinese J. Numer. Math. Appl. 17 (1995), no. 1, pp. 13–24
External Links:: ISSN 1000-3266, MathReview, ZentralBlatt
Referenced by:: §4.48(iii).
Encodings:: BibTeX

5: 17.17 Physical Applications

… ►See Kassel (1995). …

6: 4.1 Special Notation

… ► ►►

$k, m, n$	integers.
…

►It is assumed the user is familiar with the definitions and properties of elementary functions of real arguments

x

. …

7: 19.10 Relations to Other Functions

… ►

§19.10(ii) Elementary Functions

…

8: 4.44 Other Applications

… ►The Einstein functions and Planck’s radiation function are elementary combinations of exponentials, or exponentials and logarithms. …

9: 4.46 Tables

… ►Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4). …

10: Ranjan Roy

… ►

4 Elementary Functions

…