Symbols:A

The (celestial) altitude of $X$ is defined as the angle subtended by the the arc of the vertical circle through $X$ between the celestial horizon and $X$ itself.

The $\LaTeX$ code for $a$ is a .

Azimuth (Astronomy)

$A$

Let $X$ be a point on the celestial sphere.

The spherical angle between the principal vertical circle and the vertical circle on which $X$ lies is the azimuth of $X$.

The azimuth is usually measured in degrees, $0 \degrees$ to $180 \degrees$ either west or east, depending on whether $X$ lies on the eastern or western hemisphere of the celestial sphere.

The symbol for azimuth (in the context of astronomy) is $A$.

The $\LaTeX$ code for $A$ is A .

Ampere

$\mathrm A$

The ampere is the SI base unit of electric current.

It is defined as being:

The constant current which will produce a force of attraction whose value is $2 \times 10^{–7}$ newtons per metre of length between two straight, parallel conductors of infinite length and of infinitesimal circular cross-section placed one metre apart in a vacuum.

The symbol for the ampere is $\mathrm A$.

Its $\LaTeX$ code is \mathrm A .

Angstrom

$\mathring {\mathrm A}$

The angstrom is a metric unit of length.

$\ds $

$\ds 1$

angstrom

$\ds $

$=$

$\ds 10^{-1}$

nanometres

$\ds $

$=$

$\ds 10^{-4}$

micrometres

$\ds $

$=$

$\ds 10^{-7}$

millimetres

$\ds $

$=$

$\ds 10^{-8}$

centimetres

$\ds $

$=$

$\ds 10^{-10}$

metres

The symbol for the angstrom is $\mathring {\mathrm A}$.

The $\LaTeX$ code for $\mathring {\mathrm A}$ is \mathring {\mathrm A} .

Alternating Group

$A_n$

Let $S_n$ denote the symmetric group on $n$ letters.

For any $\pi \in S_n$, let $\map \sgn \pi$ be the sign of $\pi$.

The kernel of the mapping $\sgn: S_n \to C_2$ is called the alternating group on $n$ letters and denoted $A_n$.

The $\LaTeX$ code for $A_n$ is A_n .

Airy Function of the First Kind

$\map \Ai x$

An Airy function of the first kind is an Airy function which is of the form:

$\ds \map {\Ai} x = \dfrac 1 \pi \int_0^\infty \map \cos {\dfrac {t^3} 3 + x t} \rd t$

The $\LaTeX$ code for $\map \Ai x$ is \map \Ai x .

Automorphism Group

$\Aut S$

Let $\struct {S, *}$ be an algebraic structure.

Let $\mathbb S$ be the set of automorphisms of $S$.

Then the algebraic structure $\struct {\mathbb S, \circ}$, where $\circ$ denotes composition of mappings, is called the automorphism group of $S$.

The structure $\struct {S, *}$ is usually a group. However, this is not necessary for this definition to be valid.

The automorphism group of an algebraic structure $S$ is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\Aut S$.

The $\LaTeX$ code for $\Aut S$ is \Aut S .

arc-

Symbols:A/arc-

Arccosine

Real Arccosine Function

From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\closedint 0 \pi$.

From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$.

Therefore, let $g: \closedint 0 \pi \to \closedint {-1} 1$ be the restriction of $\cos x$ to $\closedint 0 \pi$.

Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\closedint {-1} 1$.

Thus:

The domain of arccosine is $\closedint {-1} 1$

The image of arccosine is $\closedint 0 \pi$.

arccos

$\arccos$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arccosine function is $\arccos$.

The $\LaTeX$ code for $\arccos$ is \arccos .

acos

$\operatorname {acos}$

A variant symbol used to denote the arccosine function is $\operatorname {acos}$.

The $\LaTeX$ code for $\operatorname {acos}$ is \operatorname {acos} .

Arccosecant

Arccosecant Function

From Shape of Cosecant Function, we have that $\csc x$ is continuous and strictly decreasing on the intervals $\hointr {-\dfrac \pi 2} 0$ and $\hointl 0 {\dfrac \pi 2}$.

From the same source, we also have that:

$\csc x \to + \infty$ as $x \to 0^+$

$\csc x \to - \infty$ as $x \to 0^-$

Let $g: \hointr {-\dfrac \pi 2} 0 \to \hointl {-\infty} {-1}$ be the restriction of $\csc x$ to $\hointr {-\dfrac \pi 2} 0$.

Let $h: \hointl 0 {\dfrac \pi 2} \to \hointr 1 \infty$ be the restriction of $\csc x$ to $\hointl 0 {\dfrac \pi 2}$.

Let $f: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0 \to \R \setminus \openint {-1} 1$:

$\map f x = \begin{cases}

\map g x & : -\dfrac \pi 2 \le x < 0 \\ \map h x & : 0 < x \le \dfrac \pi 2 \end{cases}$

From Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\hointl {-\infty} {-1}$.

From Inverse of Strictly Monotone Function, $\map h x$ admits an inverse function, which will be continuous and strictly decreasing on $\hointr 1 \infty$.

As both the domain and range of $g$ and $h$ are disjoint, it follows that:

$\map {f^{-1} } x = \begin {cases}

\map {g^{-1} } x & : x \le -1 \\ \map {h^{-1} } x & : x \ge 1 \end {cases}$

This function $\map {f^{-1} } x$ is called arccosecant of $x$.

Thus:

The domain of the arccosecant is $\R \setminus \openint {-1} 1$

The image of the arccosecant is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0$.

arccsc

$\arccsc$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arccosecant function is $\arccsc$.

The $\LaTeX$ code for $\arccsc$ is \arccsc .

arccosec

$\operatorname {arccosec}$

A variant symbol used to denote the arccosecant function is $\operatorname {arccosec}$.

The $\LaTeX$ code for $\operatorname {arccosec}$ is \operatorname {arccosec} .

acosec

$\operatorname {acosec}$

A variant symbol used to denote the arccosecant function is $\operatorname {acosec}$.

The $\LaTeX$ code for $\operatorname {acosec}$ is \operatorname {acosec} .

acsc

$\operatorname {acsc}$

A variant symbol used to denote the arccosecant function is $\operatorname {acsc}$.

The $\LaTeX$ code for $\operatorname {acsc}$ is \operatorname {acsc} .

Arccotangent

Arccotangent Function

From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$.

From the same source, we also have that:

$\cot x \to + \infty$ as $x \to 0^+$

$\cot x \to - \infty$ as $x \to \pi^-$

Let $g: \openint 0 \pi \to \R$ be the restriction of $\cot x$ to $\openint 0 \pi$.

Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\R$.

This function is called arccotangent of $x$ and is written $\arccot x$.

Thus:

The domain of the arccotangent is $\R$

The image of the arccotangent is $\openint 0 \pi$.

arccot

$\arccot$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arccotangent function is $\arccot$.

The $\LaTeX$ code for $\arccot$ is \arccot .

acot

$\operatorname {acot}$

A variant symbol used to denote the arccotangent function is $\operatorname {acot}$.

Its $\LaTeX$ code is \operatorname {acot} .

actn

$\operatorname {actn}$

A variant symbol used to denote the arccotangent function is $\operatorname {actn}$.

Its $\LaTeX$ code is \operatorname {actn} .

Area Hyperbolic Cosine

The principal branch of the real inverse hyperbolic cosine function is defined as:

$\forall x \in S: \map \arcosh x := \map \ln {x + \sqrt {x^2 - 1} }$

where:

$\ln$ denotes the natural logarithm of a (strictly positive) real number.

$\sqrt {x^2 - 1}$ specifically denotes the positive square root of $x^2 - 1$

That is, where $\map \arcosh x \ge 0$.

arcosh

$\arcosh$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic cosine function is $\arcosh$.

The $\LaTeX$ code for $\arcosh$ is \arcosh .

acosh

$\operatorname {acosh}$

A variant symbol used to denote the area hyperbolic cosine function is $\operatorname {acosh}$.

Its $\LaTeX$ code is \operatorname {acosh} .

Area Hyperbolic Cosecant

The inverse hyperbolic cosecant $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:

$\forall x \in \R_{\ne 0}: \map \arcsch x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$

where:

$\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$

$\ln$ denotes the natural logarithm of a (strictly positive) real number.

arcsch

$\arcsch$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic cosecant function is $\arcsch$.

The $\LaTeX$ code for $\arcsch$ is \arcsch .

acsch

$\operatorname {acsch}$

A variant symbol used to denote the area hyperbolic cosecant function is $\operatorname {acsch}$.

The $\LaTeX$ code for $\operatorname {acsch}$ is \operatorname {acsch} .

acosech

$\operatorname {acosech}$

A variant symbol used to denote the area hyperbolic cosecant function is $\operatorname {acosech}$.

The $\LaTeX$ code for $\operatorname {acosech}$ is \operatorname {acosech} .

Area Hyperbolic Cotangent

The inverse hyperbolic cotangent $\arcoth: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \arcoth x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.

arcoth

$\arcoth$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic cotangent function is $\arcoth$.

The $\LaTeX$ code for $\arcoth$ is \arcoth .

acoth

$\operatorname {acoth}$

A variant symbol used to denote the area hyperbolic cotangent function is $\operatorname {acoth}$.

The $\LaTeX$ code for $\operatorname {acoth}$ is \operatorname {acoth} .

actnh

$\operatorname {actnh}$

A variant symbol used to denote the area hyperbolic cotangent function is $\operatorname {actnh}$.

The $\LaTeX$ code for $\operatorname {actnh}$ is \operatorname {actnh} .

adj

$\adj {\mathbf A}$

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $\mathbf C$ be its cofactor matrix.

The adjugate matrix of $\mathbf A$ is the transpose of $\mathbf C$:

$\adj {\mathbf A} = \mathbf C^\intercal$

The $\LaTeX$ code for $\adj {\mathbf A}$ is \adj {\mathbf A} .

aln

$\operatorname {aln}$

The antilogarithm of the natural logarithm.

Its $\LaTeX$ code is \operatorname {aln} .

alog

$\operatorname {alog}_b$

Let $x \in \R_{>0}$ be a strictly positive real number.

Let $b \in \R_{>1}$ be a real number which is greater than $1$.

Let $y = \log_b x$ be the logarithm of $x$ base $b$.

Then $x$ is the antilogarithm of $y$ base $b$.

The $\LaTeX$ code for $\operatorname {alog}_b$ is \operatorname {alog}_b .

Amplitude of Incomplete Elliptic Integral of the First Kind

The parameter $\phi$ of $u = \map F {k, \phi}$ is called the amplitude of $u$.

am

$\am$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the amplitude of the incomplete elliptic integral of the first kind is $\am$.

The $\LaTeX$ code for $\am$ is \am .

amp

$\operatorname {amp} $

A variant symbol used to denote the amplitude of the incomplete elliptic integral of the first kind is $\operatorname {amp}$.

The $\LaTeX$ code for $\operatorname {amp}$ is \operatorname {amp} .

Ann

$\operatorname {Ann}$

Let $B: R \times \Z$ be a bilinear mapping defined as:

$B: R \times \Z: \tuple {r, n} \mapsto n \cdot r$

where $n \cdot r$ defined as an integral multiple of $r$:

$n \cdot r = r + r + \cdots \paren n \cdots r$

Note the change of order of $r$ and $n$:

$\map B {r, n} = n \cdot r$

Let $D \subseteq R$ be a subring of $R$.

Then the annihilator of $D$ is defined as:

$\map {\mathrm {Ann} } D = \set {n \in \Z: \forall d \in D: n \cdot d = 0_R}$

or, when $D = R$:

$\map {\mathrm {Ann} } R = \set {n \in \Z: \forall r \in R: n \cdot r = 0_R}$

Its $\LaTeX$ code is \operatorname {Ann} .

arg

$\arg$

The argument of a complex number.

Its $\LaTeX$ code is \arg .

Arg

$\operatorname {Arg}$

The principal argument of a complex number.

Its $\LaTeX$ code is \Arg .

Arcsine

Arcsine Function

From Shape of Sine Function, we have that $\sin x$ is continuous and strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From Sine of Half-Integer Multiple of Pi:

$\map \sin {-\dfrac {\pi} 2} = -1$

and:

$\sin \dfrac {\pi} 2 = 1$

Therefore, let $g: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \to \closedint {-1} 1$ be the restriction of $\sin x$ to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

Thus from Inverse of Strictly Monotone Function, $g \paren x$ admits an inverse function, which will be continuous and strictly increasing on $\closedint {-1} 1$.

This function is called arcsine of $x$.

Thus:

The domain of arcsine is $\closedint {-1} 1$

The image of arcsine is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

arcsin

$\arcsin$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arcsine function is $\arcsin$.

The $\LaTeX$ code for $\arcsin$ is \arcsin .

asin

$\operatorname {asin}$

A variant symbol used to denote the arcsine function is $\operatorname {asin}$.

The $\LaTeX$ code for $\operatorname {asin}$ is \operatorname {asin} .

Arcsecant

Arcsecant Function

From Shape of Secant Function, we have that $\sec x$ is continuous and strictly increasing on the intervals $\hointr 0 {\dfrac \pi 2}$ and $\hointl {\dfrac \pi 2} \pi$.

From the same source, we also have that:

$\sec x \to + \infty$ as $x \to \dfrac \pi 2^-$

$\sec x \to - \infty$ as $x \to \dfrac \pi 2^+$

Let $g: \hointr 0 {\dfrac \pi 2} \to \hointr 1 \to$ be the restriction of $\sec x$ to $\hointr 0 {\dfrac \pi 2}$.

Let $h: \hointl {\dfrac \pi 2} \pi \to \hointl \gets {-1}$ be the restriction of $\sec x$ to $\hointl {\dfrac \pi 2} \pi$.

Let $f: \closedint 0 \pi \setminus \dfrac \pi 2 \to \R \setminus \openint {-1} 1$:

$\map f x = \begin{cases}

\map g x & : 0 \le x < \dfrac \pi 2 \\ \map h x & : \dfrac \pi 2 < x \le \pi \end{cases}$

From Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\hointr 1 \to$.

From Inverse of Strictly Monotone Function, $\map h x$ admits an inverse function, which will be continuous and strictly increasing on $\hointl \gets {-1}$.

As both the domain and range of $g$ and $h$ are disjoint, it follows that:

$\map {f^{-1} } x = \begin{cases}

\map {g^{-1} } x & : x \ge 1 \\ \map {h^{-1} } x & : x \le -1 \end{cases}$

This function $\map {f^{-1} } x$ is called arcsecant of $x$.

Thus:

The domain of arcsecant is $\R \setminus \openint {-1} 1$

The image of arcsecant is $\closedint 0 \pi \setminus \dfrac \pi 2$.

arcsec

$\arcsec$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arcsecant function is $\arcsec$.

The $\LaTeX$ code for $\arcsec$ is \arcsec .

asec

$\operatorname {asec}$

A variant symbol used to denote the arcsecant function is $\operatorname {asec}$.

The $\LaTeX$ code for $\operatorname {asec}$ is \operatorname {asec} .

Arctangent

Arctangent Function

From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From the same source, we also have that:

$\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$

$\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$

Let $g: \openint {-\dfrac \pi 2} {\dfrac \pi 2} \to \R$ be the restriction of $\tan x$ to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\R$.

This function is called arctangent of $x$ and is written $\arctan x$.

Thus:

The domain of arctangent is $\R$

The image of arctangent is $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

arctan

$\arctan$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arctangent function is $\arctan$.

The $\LaTeX$ code for $\arctan$ is \arctan .

atan

$\operatorname {atan}$

A variant symbol used to denote the arctangent function is $\operatorname {atan}$.

The $\LaTeX$ code for $\operatorname {atan}$ is \operatorname {atan} .

atn

$\operatorname {atn}$

A variant symbol used to denote the arctangent function is $\operatorname {atn}$.

The $\LaTeX$ code for $\operatorname {atn}$ is \operatorname {atn} .

Area Hyperbolic Sine

The inverse hyperbolic sine $\arsinh: \R \to \R$ is a real function defined on $\R$ as:

$\forall x \in \R: \map \arsinh x := \map \ln {x + \sqrt {x^2 + 1} }$

where:

$\ln$ denotes the natural logarithm of a (strictly positive) real number

$\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$.

arsinh

$\arsinh$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic sine function is $\arsinh$.

The $\LaTeX$ code for $\arsinh$ is \arsinh .

asinh

$\operatorname {asinh}$

A variant symbol used to denote the area hyperbolic sine function is $\operatorname {asinh}$.

The $\LaTeX$ code for $\operatorname {asinh}$ is \operatorname {asinh} .

Area Hyperbolic Secant

The principal branch of the real inverse hyperbolic secant function is defined as:

$\forall x \in S: \map \arsech x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$

where:

$\ln$ denotes the natural logarithm of a (strictly positive) real number.

$\sqrt {1 - x^2}$ specifically denotes the positive square root of $x^2 - 1$

That is, where $\map \arsech x \ge 0$.

arsech

$\arsech$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic secant function is $\arsech$.

The $\LaTeX$ code for $\arsech$ is \arsech .

asech

$\operatorname {asech}$

A variant symbol used to denote the area hyperbolic secant function is $\operatorname {asech}$.

The $\LaTeX$ code for $\operatorname {asech}$ is \operatorname {asech} .

Area Hyperbolic Tangent

The inverse hyperbolic tangent $\artanh: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \map \artanh x := \dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.

artanh

$\artanh$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic tangent function is $\artanh$.

The $\LaTeX$ code for $\artanh$ is \artanh .

atanh

$\operatorname {atanh}$

A variant symbol used to denote the area hyperbolic tangent function is $\operatorname {atanh}$.

The $\LaTeX$ code for $\operatorname {atanh}$ is \operatorname {atanh} .

Standard Atmosphere

The standard atmosphere is a unit of pressure.

It is defined as being:

The amount of pressure equal to exactly $101 \, 325$ pascals.

$\ds $

$\ds 1$

standard atmosphere

$\ds $

$=$

$\ds 101 \, 325$

pascals

$\ds $

$\approx$

$\ds 760$

millimetres of mercury

$\ds $

$\approx$

$\ds 14.70$

pounds per square inch

atm

$\mathrm {atm}$

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the standard atmosphere is $\mathrm {atm}$.

The $\LaTeX$ code for $\mathrm {atm}$ is \mathrm {atm} .

Int atm

$\mathrm {Int \, atm}$

A variant symbol used to denote the standard atmosphere is $\mathrm {Int \, atm}$.

This reflects its variant name of international atmosphere.

The $\LaTeX$ code for $\mathrm {Int \, atm}$ is \mathrm {Int \, atm} .

Abampere

$\mathrm {abA}$

The symbol for the abampere is $\mathrm {abA}$.

Its $\LaTeX$ code is \mathrm {abA} .

Abcoulomb

$\mathrm {abC}$

The symbol for the abcoulomb is $\mathrm {abC}$.

Its $\LaTeX$ code is \mathrm {abC} .

Abvolt

$\mathrm {abV}$

The symbol for the abvolt is $\mathrm {abV}$.

Its $\LaTeX$ code is \mathrm {abV} .

Abohm

$\mathrm {ab \Omega}$

The symbol for the abohm is $\mathrm {ab \Omega}$, where $\Omega$ is the Greek letter Omega.

Its $\LaTeX$ code is \mathrm {ab \Omega} .

Abhenry

$\mathrm {abH}$

The symbol for the abhenry is $\mathrm {abH}$.

Its $\LaTeX$ code is \mathrm {abH} .

Abfarad

$\mathrm {abF}$

The symbol for the abfarad is $\mathrm {abF}$.

Its $\LaTeX$ code is \mathrm {abF} .

Atomic Mass Unit

$\mathrm {amu}$ or $\mathrm {AMU}$

The symbol for the atomic mass unit is $\mathrm {amu}$ or $\mathrm {AMU}$.

The $\LaTeX$ code for $\mathrm {amu}$ is \mathrm {amu} .

The $\LaTeX$ code for $\mathrm {AMU}$ is \mathrm {AMU} .

Bohr Radius

$a_0$

The symbol for the Bohr radius is $a_0$.

The $\LaTeX$ code for $a_0$ is a_0 .

Astronomical Unit

$\mathrm {AU}$ or $\mathrm {au}$

The symbol for the astronomical unit is $\mathrm {AU}$ or $\mathrm {au}$.

The $\LaTeX$ code for $\mathrm {AU}$ is \mathrm {AU} .

The $\LaTeX$ code for $\mathrm {au}$ is \mathrm {au} .

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