# 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 $1-\tfrac{1}{10}E_{2}+\tfrac{1}{14}E_{3}+\tfrac{1}{24}E_{2}^{2}-\tfrac{3}{44}E_% {2}E_{3}-\tfrac{5}{208}E_{2}^{3}+\tfrac{3}{104}E_{3}^{2}+\tfrac{1}{16}E_{2}^{2% }E_{3},$
where the elementary symmetric functions $E_{s}$ are defined by (19.19.4). …
19.36.2 $1-\tfrac{3}{14}E_{2}+\tfrac{1}{6}E_{3}+\tfrac{9}{88}E_{2}^{2}-\tfrac{3}{22}E_{% 4}-\tfrac{9}{52}E_{2}E_{3}+\tfrac{3}{26}E_{5}-\tfrac{1}{16}E_{2}^{3}+\tfrac{3}% {40}E_{3}^{2}+\tfrac{3}{20}E_{2}E_{4}+\tfrac{45}{272}E_{2}^{2}E_{3}-\tfrac{9}{% 68}(E_{3}E_{4}+E_{2}E_{5}).$
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}.}
##### 2: 19.19 Taylor and Related Series
Define the elementary symmetric function $E_{s}(\mathbf{z})$ by
19.19.4 $\prod_{j=1}^{n}(1+tz_{j})=\sum_{s=0}^{n}t^{s}E_{s}(\mathbf{z}),$
19.19.5 $T_{N}(\mathbf{\tfrac{1}{2}},\mathbf{z})=\sum(-1)^{M+N}{\left(\tfrac{1}{2}% \right)_{M}}\frac{E_{1}^{m_{1}}(\mathbf{z})\cdots E_{n}^{m_{n}}(\mathbf{z})}{m% _{1}!\cdots m_{n}!},$
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 $E_{1}(\mathbf{Z})=0$. …
$E_{1}(\mathbf{Z})=0$ , $|Z_{j}|<1$.
##### 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).
• ##### 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 $x$. …
##### 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). …