value at infinity

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1: 7.2 Definitions

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Values at Infinity

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Values at Infinity

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2: 6.2 Definitions and Interrelations

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Values at Infinity

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3: 6.4 Analytic Continuation

… ►Analytic continuation of the principal value of $E_1\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\mathrm{\infty }$. …

4: 4.23 Inverse Trigonometric Functions

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Table 4.23.1: Inverse trigonometric functions: principal values at 0, $\pm 1$, $\pm \mathrm{\infty }$.

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$x$	$\arcsin x$	$\arccos x$	$\arctan x$	$\mathrm{arccsc}x$	$\mathrm{arcsec}x$	$\mathrm{arccot}x$
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5: 10.9 Integral Representations

… ►Also, ${(t^2-1)}^{\nu -\frac{1}{2}}$ is continuous on the path, and takes its principal value at the intersection with the interval $(1,\mathrm{\infty })$. …

6: 12.14 The Function $W(a,x)$

… ► $W(a,x)$ and $W(a,-x)$ form a numerically satisfactory pair of solutions when . ►

§12.14(ii) Values at $z=0$ and Wronskian

… ►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. … ►Then as $x\to \mathrm{\infty }$ … ►

Negative $a$,

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7: 4.13 Lambert $W$-Function

… ► $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. …

8: 1.4 Calculus of One Variable

… ►For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are analytic functions of their variables, and thus $C^{\mathrm{\infty }}$, and well defined for all values of these variables; possible exceptions being at boundary points. …

9: 4.17 Special Values and Limits

§4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
0	0	1	0	$\mathrm{\infty }$	1	$\mathrm{\infty }$
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$\pi$	0	$-1$	$0$	$\mathrm{\infty }$	$-1$	$\mathrm{\infty }$

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10: 4.31 Special Values and Limits

§4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

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$z$	0	$\frac{1}{2}\pi \mathrm{i}$	$\pi \mathrm{i}$	$\frac{3}{2}\pi \mathrm{i}$	$\mathrm{\infty }$
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$\tanh z$	$0$	$\mathrm{\infty }\mathrm{i}$	$0$	$-\mathrm{\infty }\mathrm{i}$	$1$
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$\mathrm{sech}z$	$1$	$\mathrm{\infty }$	$-1$	$\mathrm{\infty }$	$0$
$\coth z$	$\mathrm{\infty }$	$0$	$\mathrm{\infty }$	$0$	$1$

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