List of Figures

1 Algebraic and Analytic Methods
2 Asymptotic Approximations
- 2.10.1 Contour $\mathscr{C}$ .
- 2.11.1 $|\operatorname{erfc}\left(\sqrt{50}\,c(\theta)\right)|$ .
3 Numerical Methods
- $\begin{picture}(152.0,38.0)(-1.0,-1.0)\put(0.0,31.0){\makebox(2.0,6.0){\small 1% }} \put(0.0,26.0){\framebox(2.0,6.0){$s$}} \put(3.0,31.0){\makebox(8.0,6.0){\small 8}} \put(3.0,26.0){\framebox(8.0,6.0){$E$}} \put(12.0,31.0){\makebox(23.0,6.0){\small 23 bits}} \put(12.0,26.0){\framebox(23.0,6.0){$f$}} \put(133.0,31.0){$N=32$,} \put(135.0,27.0){$p=24$} \put(0.0,18.0){\makebox(2.0,6.0){\small 1}} \put(0.0,13.0){\framebox(2.0,6.0){$s$}} \put(3.0,18.0){\makebox(11.0,6.0){\small 11}} \put(3.0,13.0){\framebox(11.0,6.0){$E$}} \put(15.0,18.0){\makebox(52.0,6.0){\small 52 bits}} \put(15.0,13.0){\framebox(52.0,6.0){$f$}} \put(133.0,17.0){$N=64$,} \put(135.0,13.0){$p=53$} \put(0.0,5.0){\makebox(2.0,6.0){\small 1}} \put(0.0,0.0){\framebox(2.0,6.0){$s$}} \put(3.0,5.0){\makebox(15.0,6.0){\small 15}} \put(3.0,0.0){\framebox(15.0,6.0){$E$}} \put(19.0,5.0){\makebox(112.0,6.0){\small 112 bits}} \put(19.0,0.0){\framebox(112.0,6.0){$f$}} \put(133.0,4.0){$N=128$,} \put(135.0,0.0){$p=113$} \end{picture}$ 3.1.1 Representation of data in the binary interchange formats for binary32, binary64 and binary128.
- 3.11.1 Error of the minimax rational approximation to $J_{0}\left(x\right)$
4 Elementary Functions
- 4.2.1 Branch cut for $\ln z$ and $z^{\alpha}$ .
- 4.3.1 $\ln x$ and $e^{x}$ .
- (i) $z$ -plane (ii) $w$ -plane A B C $\overline{\mathrm{C}}$ D $\overline{\mathrm{D}}$ E $\overline{\mathrm{E}}$ F $z$ 0 $r$ $r+i\pi$ $r-i\pi$ $i\pi$ $-i\pi$ $-r+i\pi$ $-r-i\pi$ $-r$ $w$ 1 $e^{r}$ $-e^{r}+i0$ $-e^{r}-i0$ $-1+i0$ $-1-i0$ $-e^{-r}+i0$ $-e^{-r}-i0$ $e^{-r}$ 4.3.2 Conformal mapping of exponential and logarithm.
- 4.3.3 $\ln\left(x+iy\right)$ .
- 4.3.4 $e^{x+iy}$ .
- 4.13.1 Branches $\mathrm{Wp}\left(x\right)$ , $\mathrm{Wm}\left(x\right)$ of the Lambert $W$ -function.
- 4.15.1 $\sin x$ and $\cos x$ .
- 4.15.2 $\operatorname{Arcsin}x$ and $\operatorname{Arccos}x$ .
- 4.15.3 $\tan x$ and $\cot x$ .
- 4.15.4 $\operatorname{arctan}x$ and $\operatorname{arccot}x$ .
- 4.15.5 $\csc x$ and $\sec x$ .
- 4.15.6 $\operatorname{arccsc}x$ and $\operatorname{arcsec}x$ .
- (i) $z$ -plane (ii) $w$ -plane A B C $\mathrm{\overline{C}}$ D $\mathrm{\overline{D}}$ E $\mathrm{\overline{E}}$ F $z$ $0$ $\tfrac{1}{2}\pi$ $\tfrac{1}{2}\pi+ir$ $\tfrac{1}{2}\pi-ir$ $i r$ $-ir$ $-\tfrac{1}{2}\pi+ir$ $-\tfrac{1}{2}\pi-ir$ $-\tfrac{1}{2}\pi$ $w$ $0$ $1$ $\cosh r+i0$ $\cosh r-i0$ $i\sinh r$ $-i\sinh r$ $-\cosh r+i0$ $-\cosh r-i0$ $-1$ 4.15.7 Conformal mapping of sine and inverse sine.
- 4.15.8 $\sin\left(x+iy\right)$ .
- 4.15.9 $\operatorname{arcsin}\left(x+iy\right)$ .
- 4.15.10 $\tan\left(x+iy\right)$ .
- 4.15.11 $\operatorname{arctan}\left(x+iy\right)$ .
- 4.15.12 $\csc\left(x+iy\right)$ .
- 4.15.13 $\operatorname{arccsc}\left(x+iy\right)$ .
- 4.16.1 Quadrants for the angle $\theta$ .
- (i) $\operatorname{arcsin}z$ and $\operatorname{arccos}z$ (ii) $\operatorname{arctan}z$ (iii) $\operatorname{arccsc}z$ and $\operatorname{arcsec}z$ (iv) $\operatorname{arccot}z$ 4.23.1 Branch cuts for the inverse trigonometric functions.
- 4.29.1 $\sinh x$ and $\cosh x$ .
- 4.29.2 $\operatorname{arcsinh}x$ and $\operatorname{arccosh}x$ .
- 4.29.3 $\tanh x$ and $\coth x$ .
- 4.29.4 $\operatorname{arctanh}x$ and $\operatorname{arccoth}x$ .
- 4.29.5 $\operatorname{csch}x$ and $\operatorname{sech}x$ .
- 4.29.6 $\operatorname{arccsch}x$ and $\operatorname{arcsech}x$ .
- (i) $\operatorname{arcsinh}z$ (ii) $\operatorname{arccosh}z$ (iii) $\operatorname{arctanh}z$ (iv) $\operatorname{arccsch}z$ (v) $\operatorname{arcsech}z$ (vi) $\operatorname{arccoth}z$ 4.37.1 Branch cuts for the inverse hyperbolic functions.
- 4.42.1 Planar right triangle.
- 4.42.2 Planar triangle.
- 4.42.3 Spherical triangle.
5 Gamma Function
6 Exponential, Logarithmic, Sine, and Cosine Integrals
7 Error Functions, Dawson’s and Fresnel Integrals
8 Incomplete Gamma and Related Functions
9 Airy and Related Functions
10 Bessel Functions
11 Struve and Related Functions
12 Parabolic Cylinder Functions
13 Confluent Hypergeometric Functions
- 13.4.1 Contour of integration in (13.4.11).
- 13.7.1 Regions for error bounds of $U$ .
14 Legendre and Related Functions
15 Hypergeometric Function
16 Generalized Hypergeometric Functions & Meijer G-Function
- Case (i) Case (ii) Case (iii) 16.17.1 Integration path $L$ for the Meijer $G$ -function.
18 Orthogonal Polynomials
19 Elliptic Integrals