subtraction

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1: About Color Map

… ►The conventional CMYK color wheel (not to be confused with the traditional Artist’s color wheel) places the additive colors (red, green, blue) and the subtractive colors (yellow, cyan, magenta) at multiples of 60 degrees. …

2: 26.18 Counting Techniques

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26.18.1

\left|S\setminus(A_{1}\cup A_{2}\cup\cdots\cup A_{n})\right|=\left|S\right|+% \sum_{t=1}^{n}(-1)^{t}\sum_{1\leq j_{1}<j_{2}<\cdots<j_{t}\leq n}\left|A_{j_{1% }}\cap A_{j_{2}}\cap\cdots\cap A_{j_{t}}\right|.

ⓘ

Symbols:: $\left|\NVar{A}\right|$ : number of elements of a finite set $A$ , $\cap$ : intersection, $\setminus$ : set subtraction, $\cup$ : union, $j$ : nonnegative integer, $n$ : nonnegative integer, $A_{k}$ : subsets and $S$ : set
Permalink:: http://dlmf.nist.gov/26.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.18 and Ch.26

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3: 4.37 Inverse Hyperbolic Functions

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4.37.16

\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z\right),

z/i\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)

;

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.19

\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right),

z\in\mathbb{C}\setminus(-\infty,1)

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction and $z$ : complex variable
Referenced by:: §4.37(iv), §4.37(iii), §4.37(iv)
Permalink:: http://dlmf.nist.gov/4.37.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.21

\operatorname{arccosh}z=2\ln\left(\left(\frac{z+1}{2}\right)^{1/2}+\left(\frac% {z-1}{2}\right)^{1/2}\right),

z\in\mathbb{C}\setminus(-\infty,1)

;

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E21
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

\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),

z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)

;

ⓘ

Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : 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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4: Mathematical Introduction

… ► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$\setminus$	set subtraction.
…

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5: 23.2 Definitions and Periodic Properties

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23.2.4

\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right),

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Defines:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function
Symbols:: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\in$ : element of, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice and $z$ : complex
Keywords:: modular functions, Weierstrass $\wp$ -function
Referenced by:: §23.9, Erratum (V1.0.5) for Equation (23.2.4)
Permalink:: http://dlmf.nist.gov/23.2.E4
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: Originally the denominator $(z-w)^{2}$ was given incorrectly as $(z-w^{2})$ .
Reported 2012-02-16 by James D. Walker
See also:: Annotations for §23.2(ii), §23.2 and Ch.23

►

23.2.5

\zeta\left(z\right)=\frac{1}{z}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(\frac% {1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}\right),

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Defines:: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function
Symbols:: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\zeta\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass zeta function, $\in$ : element of, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice and $z$ : complex
Keywords:: modular functions, Weierstrass zeta function
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.2(ii), §23.2 and Ch.23

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23.2.6

\sigma\left(z\right)=z\prod_{w\in\mathbb{L}\setminus\{0\}}\left(\left(1-\frac{% z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right)\right).

ⓘ

Defines:: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function
Symbols:: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\sigma\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass sigma function, $\in$ : element of, $\exp\NVar{z}$ : exponential function, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice and $z$ : complex
Keywords:: modular functions, Weierstrass sigma function
Referenced by:: §23.2(iii)
Permalink:: http://dlmf.nist.gov/23.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §23.2(ii), §23.2 and Ch.23

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6: 25.12 Polylogarithms

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25.12.2

\operatorname{Li}_{2}\left(z\right)=-\int_{0}^{z}t^{-1}\ln\left(1-t\right)\,% \mathrm{d}t,

z\in\mathbb{C}\setminus(1,\infty)

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Symbols:: $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $x$ : real variable and $z$ : complex variable
Keywords:: definite integral, integral representation
Source:: Maximon (2003, (2.1), p. 2807)
A&S Ref:: 27.7.1 (is a modified form with $z=1-x$ )
Referenced by:: §25.12(i)
Permalink:: http://dlmf.nist.gov/25.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

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25.12.3

\operatorname{Li}_{2}\left(z\right)+\operatorname{Li}_{2}\left(\frac{z}{z-1}% \right)=-\frac{1}{2}(\ln\left(1-z\right))^{2},

z\in\mathbb{C}\setminus[1,\infty)

ⓘ

Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\ln\NVar{z}$ : principal branch of logarithm function, $\setminus$ : set subtraction, $x$ : real variable and $z$ : complex variable
Keywords:: connection formula
Source:: Maximon (2003, (3.4), p. 2808)
A&S Ref:: 27.7.5 (is a modified form with $z=1-x$ )
Permalink:: http://dlmf.nist.gov/25.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

►

25.12.4

\operatorname{Li}_{2}\left(z\right)+\operatorname{Li}_{2}\left(\frac{1}{z}% \right)=-\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln\left(-z\right))^{2},

z\in\mathbb{C}\setminus[0,\infty)

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\ln\NVar{z}$ : principal branch of logarithm function, $\setminus$ : set subtraction and $z$ : complex variable
Keywords:: connection formula
Source:: Maximon (2003, (3.2), p. 2808)
Referenced by:: §25.12(ii)
Permalink:: http://dlmf.nist.gov/25.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

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7: 23.9 Laurent and Other Power Series

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23.9.1

c_{n}=(2n-1)\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-2n},

n=2,3,4,\dots

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Symbols:: $\in$ : element of, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice, $n$ : integer and $c_{n}$
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/23.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

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8: 23.3 Differential Equations

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23.3.1

g_{2}=60\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-4},

ⓘ

Symbols:: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\in$ : element of, $\setminus$ : set subtraction and $\mathbb{L}$ : lattice
A&S Ref:: 18.1.1
Referenced by:: §23.3(i)
Permalink:: http://dlmf.nist.gov/23.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.3(i), §23.3 and Ch.23

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23.3.2

g_{3}=140\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-6}.

ⓘ

Symbols:: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\in$ : element of, $\setminus$ : set subtraction and $\mathbb{L}$ : lattice
A&S Ref:: 18.1.1
Referenced by:: §23.3(i)
Permalink:: http://dlmf.nist.gov/23.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.3(i), §23.3 and Ch.23

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9: 4.23 Inverse Trigonometric Functions

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4.23.19

\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+iz\right),

z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)

;

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Referenced by:: §4.23(iv)
Permalink:: http://dlmf.nist.gov/4.23.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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4.23.22

\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((1-z^{2})^{1/2}+iz\right),

z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)

;

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Referenced by:: §4.23(iv)
Permalink:: http://dlmf.nist.gov/4.23.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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4.23.23

\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z}{2}\right)^{1/2}+i\left(% \frac{1-z}{2}\right)^{1/2}\right),

z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)

;

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Referenced by:: §4.23(iv)
Permalink:: http://dlmf.nist.gov/4.23.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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4.23.26

\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i+z}{i-z}\right),

z/i\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)

;

ⓘ

Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Referenced by:: §4.23(iv)
Permalink:: http://dlmf.nist.gov/4.23.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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10: 20.6 Power Series

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