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1: 7.23 Tables

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Abramowitz and Stegun (1964, Chapter 7) includes $\mathrm{erf}x$, $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [0,2]$, 10D; $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [2,10]$, 8S; $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$, $x^{-2}\in [0,0.25]$, 7D; $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=1(1)6,10,11$, $x\in [0,5]$, 6S; $F\left(x\right)$, $x\in [0,2]$, 10D; $xF\left(x\right)$, $x^{-2}\in [0,0.25]$, 9D; $C\left(x\right)$, $S\left(x\right)$, $x\in [0,5]$, 7D; $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in [0,1]$, $x^{-1}\in [0,1]$, 15D.

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Finn and Mugglestone (1965) includes the Voigt function $H(a,u)$, $u\in [0,22]$, $a\in [0,1]$, 6S.

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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2: 29.9 Stability

… ►If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h$ lies in one of the closed intervals with endpoints $a_{\nu }^m\left(k^2\right)$ and $b_{\nu }^m\left(k^2\right)$, $m=1,2,\mathrm{\dots }$. If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h\in [b_{\nu }^m\left(k^2\right),a_{\nu }^m\left(k^2\right)]$ for some $m=1,2,\mathrm{\dots },\nu$.

3: 18.32 OP’s with Respect to Freud Weights

… ►All of these forms appear in applications, see §18.39(iii) and Table 18.39.1, albeit sometimes with $x\in [0,\mathrm{\infty })$, where the term half-Freud weight is used; or on $x\in [-1,1]$ or $[0,1]$, where the term Rys weight is employed, see Rys et al. (1983). For (generalized) Freud weights on a subinterval of $[0,\mathrm{\infty })$ see also Levin and Lubinsky (2005).

4: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

… ►Jacobian elliptic functions with real moduli in the intervals $(-\mathrm{\infty },0)$ and $(1,\mathrm{\infty })$, or with purely imaginary moduli are related to functions with moduli in the interval $[0,1]$ by the following formulas. … ►For proofs of these results and further information see Walker (2003).

5: Mathematical Introduction

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$ℂ$	complex plane (excluding infinity).
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${f(z)|}_C=0$	$f(z)$ is continuous at all points of a simple closed contour $C$ in $ℂ$.
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$[a,b]$	closed interval in $ℝ$, or closed straight-line segment joining $a$ and $b$ in $ℂ$.

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$(a,b]$ or $[a,b)$	half-closed intervals.
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6: About Color Map

… ►Mathematically, we scale the height to $h$ lying in the interval $[0,4]$ and the components are computed as follows … ►Specifically, by scaling the phase angle in $[0,2\pi )$ to $q$ in the interval $[0,4)$, the hue (in degrees) is computed as …

7: 4.37 Inverse Hyperbolic Functions

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4.37.17 $$\mathrm{arcsinh}\left(\mathrm{i}y\right)=\frac{1}{2}\pi \mathrm{i}\pm \ln \left({(y^2-1)}^{1/2}+y\right),$$ $y\in [1,\mathrm{\infty })$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b)$: half-closed interval, $\in$: element of, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.37.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.18 $$\mathrm{arcsinh}\left(\mathrm{i}y\right)=-\frac{1}{2}\pi \mathrm{i}\pm \ln \left({(y^2-1)}^{1/2}-y\right),$$ $y\in (-\mathrm{\infty },-1]$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.37.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

… ►It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on $(-\mathrm{\infty },1]$. … ►

4.37.22 $$\mathrm{arccosh}x=\pm \ln \left(\mathrm{i}{(1-x^2)}^{1/2}+x\right),$$ $x\in (-1,1]$,

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Symbols:

$\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.37.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.24 $$\mathrm{arctanh}z=\frac{1}{2}\ln \left(\frac{1+z}{1-z}\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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8: 5.14 Multidimensional Integrals

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5.14.4 $${\int }_{{[0,1]}^n}{t}_1{t}_2\mathrm{\cdots }{t}_m{|\mathrm{\Delta }(t_1,\mathrm{\dots },t_n)|}^{2c}\prod _{k=1}^n{t}_k^{a-1}{(1-t_k)}^{b-1}d{t}_k=\frac{1}{{(\mathrm{\Gamma }\left(1+c\right))}^n}\prod _{k=1}^m\frac{a+(n-k)c}{a+b+(2n-k-1)c}\prod _{k=1}^n\frac{\mathrm{\Gamma }\left(a+(n-k)c\right)\mathrm{\Gamma }\left(b+(n-k)c\right)\mathrm{\Gamma }\left(1+kc\right)}{\mathrm{\Gamma }\left(a+b+(2n-k-1)c\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer, $a$: real or complex variable, $b$: real or complex variable and $\mathrm{\Delta }$: product

Permalink:

http://dlmf.nist.gov/5.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

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5.14.5 $${\int }_{{[0,\mathrm{\infty })}^n}{t}_1{t}_2\mathrm{\cdots }{t}_m{|\mathrm{\Delta }(t_1,\mathrm{\dots },t_n)|}^{2c}\prod _{k=1}^n{t}_k^{a-1}{\mathrm{e}}^{-t_k}d{t}_k=\prod _{k=1}^m(a+(n-k)c)\frac{{\prod }_{k=1}^n\mathrm{\Gamma }\left(a+(n-k)c\right)\mathrm{\Gamma }\left(1+kc\right)}{{(\mathrm{\Gamma }\left(1+c\right))}^n},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $[a,b)$: half-closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer, $a$: real or complex variable and $\mathrm{\Delta }$: product

Permalink:

http://dlmf.nist.gov/5.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

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5.14.7 $\mathrm{\Re }b>-1/n$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer and $b$: real or complex variable

Referenced by:

§5.14, §5.20

Permalink:

http://dlmf.nist.gov/5.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

9: 22.18 Mathematical Applications

… ►With $k\in [0,1]$ the mapping $z\to w=\mathrm{sn}(z,k)$ gives a conformal map of the closed rectangle $[-K,K]\times [0,K^{\prime }]$ onto the half-plane $\mathrm{\Im }w\ge 0$, with $0,\pm K,\pm K+\mathrm{i}{K}^{\prime },\mathrm{i}{K}^{\prime }$ mapping to $0,\pm 1,\pm k^{-2},\mathrm{\infty }$ respectively. The half-open rectangle $(-K,K)\times [-K^{\prime },K^{\prime }]$ maps onto $ℂ$ cut along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$. …

10: 7.20 Mathematical Applications

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Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by $x=C\left(t\right)$, $y=S\left(t\right)$, $t\in [0,\mathrm{\infty })$. Magnify

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