Definition:Negative/Complex Number

As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, the concept of a '''negative complex number''', relative to a specified zero, is not defined. However, the '''negative''' of a complex number is defined as follows: Let $z = a + i b$ be a complex number. Then the '''negative of $z$''' is defined as: :$-z = -a - i b$


Definition:Cartesian Product/Coordinate

Let $\displaystyle \prod_{i \mathop \in I} S_i$ be a cartesian product. Let $j \in I$, and let $s = \sequence {s_i}_{i \mathop \in I} \in \displaystyle \prod_{i \mathop \in I} S_i$. Then $s_j$ is called the '''$j$th coordinate of $s$'''. If the indexing set $I$ consists of ordinary numbers $1, 2, \ldots, n$, one speaks about, for example, the '''first''', '''second''', or '''$n$th coordinate'''. For an element $\tuple {s, t} \in S \times T$ of a binary cartesian product, $s$ is the '''first coordinate''', and $t$ is the '''second coordinate'''.


Definition:Distance/Points/Complex Numbers

Let $x, y \in \C$ be complex numbers. Let $\cmod {x - y}$ be the complex modulus of $x - y$. Then the function $d: \C^2 \to \R$: :$\map d {x, y} = \cmod {x - y}$ is called the '''distance between $x$ and $y$'''.


Definition:Argument of Complex Number/Principal Range

It is understood that the argument of a complex number $z$ is unique only up to multiples of $2 k \pi$. With this understanding, we can limit the choice of what $\theta$ can be for any given $z$ by requiring that $\theta$ lie in some half open interval of length $2 \pi$. The most usual of these are: :$\hointr 0 {2 \pi}$ :$\hointl {-\pi} \pi$ but in theory any such interval may be used. This interval is known as the '''principal range'''.


Definition:Substitution (Formal Systems)/Well-Formed Part

Let $\FF$ be a formal language with alphabet $\AA$. Let $\mathbf B$ be a well-formed formula of $\FF$. Let $\mathbf A$ be a well-formed part of $\mathbf B$. Let $\mathbf A'$ be another well-formed formula. Then the '''substitution of $\mathbf A'$ for $\mathbf A$ in $\mathbf B$''' is the collation resulting from $\mathbf B$ by replacing all occurrences of $\mathbf A$ in $\mathbf B$ by $\mathbf A'$. It is denoted as $\map {\mathbf B} {\mathbf A' \mathbin {//} \mathbf A}$.


Definition:Complex Number as Vector

Let $z = x + i y$ be a complex number. Then $z$ can be considered as a vector $OP$ in the complex plane such that: : its initial point is the origin : its terminal point $P$ is the point $\tuple {x, y}$. Two vectors which have the same magnitude and direction, but different initial points, are considered '''equal'''.


Definition:Spherical Representation of Complex Number

Let $\PP$ be the complex plane. Let $\mathbb S$ be the unit sphere which is tangent to $\PP$ at $\tuple {0, 0}$ (that is, where $z = 0$). Let the diameter of $\mathbb S$ perpendicular to $\PP$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\mathbb S$ and $S$ be referred to as the '''south pole''' of $\mathbb S$. Let $A$ be a point on $P$. Let the line $NA$ be constructed. :900px Then $NA$ passes through a point of $\mathbb S$. Thus any complex number can be represented by a point on the surface of the unit sphere. The point $N$ on $\mathbb S$ corresponds to the point at infinity. Thus any point on the surface of the unit sphere corresponds to a point on the extended complex plane.


Definition:Complex Point at Infinity

The zero in the set of complex numbers $\C$ has no inverse for multiplication. That is, the expression: :$\dfrac 1 0$ has no meaning. The '''point at infinity''' is the element added to $\C$ in order to allow $\C$ to be closed under division: :$\forall x, y \in \C: \dfrac x y \in \C$ The set $\C$ with that point added is known as the extended complex plane. Conceptually, it can be imagined as a point which is '''''at infinity''''' in '''''all''''' directions. It can also be considered as the $N$ point on the Riemann sphere which does not map to the complex plane. This point can be denoted $\infty$.


Definition:Definable Truth Function

Let $f: \Bbb B^n \to \Bbb B$ be a truth function. Let $S$ be a set of truth functions. Then $f$ is '''definable from $S$''' iff there exist: :a truth function $g: \Bbb B^m \to \Bbb B$, obtained by composition of truth functions from $S$ :an injection $i: \Bbb B^n \to \Bbb B^m$ such that: :$f = g \circ i$ {{expand|example}}


Definition:Satisfiable/Boolean Interpretations

Let $\mathbf A$ be a WFF of propositional logic. $\mathbf A$ is called '''satisfiable (for boolean interpretations)''' iff: :$v \left({\mathbf A}\right) = T$ for some boolean interpretation $v$ for $\mathbf A$. In terms of validity, this can be rendered: :$v \models_{\mathrm{BI}} \mathbf A$ that is, $\mathbf A$ is valid in the boolean interpretation $v$ of $\mathbf A$.


Definition:Falsifiable/Boolean Interpretations

Let $\mathbf A$ be a WFF of propositional logic. $\mathbf A$ is called '''falsifiable (for boolean interpretations)''' iff: :$v \left({\mathbf A}\right) = F$ for some boolean interpretation $v$ for $\mathbf A$. In terms of validity, this can be rendered: :$v \not\models_{\mathrm{BI}} \mathbf A$ that is, $\mathbf A$ is invalid in the boolean interpretation $v$ of $\mathbf A$.


Definition:Extended Real Number Line/Definition 1

The '''extended real number line''' $\overline \R$ is defined as: :$\overline \R := \R \cup \set {+\infty, -\infty}$ that is, the set of real numbers together with two auxiliary symbols: :$+\infty$, '''positive infinity''' :$-\infty$, '''negative infinity''' such that: :$\forall x \in \R: x < +\infty$ :$\forall x \in \R: -\infty < x$


Definition:Extended Real Number Line/Definition 2

The '''extended real number line''' $\overline \R$ is the order completion of the set of real numbers $\R$. The greatest element of $\overline \R$ is often denoted by $+\infty$ and its least element by $-\infty$.


Definition:Extended Complex Plane

The '''extended complex plane''' $\overline \C$ is defined as: :$\overline \C := \C \cup \set \infty$ that is, the set of complex numbers together with the point at infinity.


Definition:Stereographic Projection

Let $\PP$ be a the plane. Let $\mathbb S$ be a sphere which is tangent to $\PP$ at the origin $\tuple {0, 0}$. Let the diameter of $\mathbb S$ perpendicular to $\PP$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\mathbb S$ and $S$ be referred to as the '''south pole''' of $\mathbb S$. Let $A$ be a point on $P$. Let the line $NA$ be constructed. :900px Then $NA$ passes through a point of $\mathbb S$. Thus any point on $P$ can be represented by a point on $\mathbb S$. With this construction, the point $N$ on $\mathbb S$ maps to no point on $\mathbb S$.


Definition:Dot Product/Complex/Definition 2

The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \circ z_2 = \cmod {z_1} \, \cmod{z_2} \cos \theta$ where: :$\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\theta$ denotes the angle between $z_1$ and $z_2$.


Definition:Dot Product/Complex/Definition 1

The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \circ z_2 = x_1 x_2 + y_1 y_2$


Definition:Dot Product/Complex

Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers. === Definition 1 === {{:Definition:Dot Product/Complex/Definition 1}} === Definition 2 === {{:Definition:Dot Product/Complex/Definition 2}} === Definition 3 === {{:Definition:Dot Product/Complex/Definition 3}} === Definition 4 === {{:Definition:Dot Product/Complex/Definition 4}}


Definition:Dot Product/Complex/Definition 3

The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \circ z_2 := \map \Re {\overline {z_1} z_2}$ where: :$\map \Re z$ denotes the real part of a complex number $z$ :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\overline {z_1} z_2$ denotes complex multiplication.


Definition:Dot Product/Complex/Definition 4

The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \circ z_2 := \dfrac {\overline {z_1} z_2 + z_1 \overline {z_2} } 2$ where: :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\overline {z_1} z_2$ denotes complex multiplication.


Definition:Vector Cross Product/Definition 1

The '''vector cross product''', denoted $\mathbf a \times \mathbf b$, is defined as: :$\mathbf a \times \mathbf b = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k\\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end{vmatrix}$ where $\begin{vmatrix} \ldots \end{vmatrix}$ is interpreted as a determinant. More directly: : $\mathbf a \times \mathbf b = \mathbf i \paren {a_j b_k - a_k b_j} - \mathbf j \paren {a_i b_k - a_k b_i} + \mathbf k \paren {a_i b_j - a_j b_i}$


Definition:Vector Cross Product/Definition 2

The '''vector cross product''', denoted $\mathbf a \times \mathbf b$, is defined as: :$\mathbf a \times \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \sin \theta \hat {\mathbf n}$ where: :$\norm {\mathbf a}$ denotes the length of $\mathbf a$ :$\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction :$\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right hand rule.


Definition:Orthogonal (Linear Algebra)/Set

Let $S = \set {u_1, \ldots, u_n}$ be a subset of $V$. Then $S$ is an '''orthogonal set''' {{iff}} its elements are pairwise orthogonal: :$\forall i \ne j: \innerprod {u_i}, {u_j} = 0$


Definition:Vector Cross Product/Complex

Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers. === Definition 1 === {{:Definition:Vector Cross Product/Complex/Definition 1}} === Definition 2 === {{:Definition:Vector Cross Product/Complex/Definition 2}} === Definition 3 === {{:Definition:Vector Cross Product/Complex/Definition 3}} === Definition 4 === {{:Definition:Vector Cross Product/Complex/Definition 4}}


Definition:Vector Cross Product/Complex/Definition 2

The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \times z_2 = \cmod {z_1} \, \cmod {z_2} \sin \theta$ where: :$\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.


Definition:Vector Cross Product/Complex/Definition 1

The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \times z_2 = x_1 y_2 - y_1 x_2$


Definition:Vector Cross Product/Complex/Definition 3

The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \times z_2 := \map \Im {\overline {z_1} z_2}$ where: :$\map \Im z$ denotes the imaginary part of a complex number $z$ :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\overline {z_1} z_2$ denotes complex multiplication.


Definition:Vector Cross Product/Complex/Definition 4

The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \times z_2 := \dfrac {\overline {z_1} z_2 - z_1 \overline {z_2}} {2 i}$ where: :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\overline {z_1} z_2$ denotes complex multiplication.


Definition:Bound of Real-Valued Function

Let $S$ be a set. Let $f: S \to \R$ be a real-valued function. Let $f$ be bounded. Then $B$ is a '''bound''' for $f$ {{iff}}: :$\forall x \in S: B \ge \size {\map f x}$


Definition:Bounded Ordered Set/Unbounded

Let $\left({S, \preceq}\right)$ be an ordered set. A subset $T \subseteq S$ is '''unbounded (in $S$)''' {{iff}} it is not bounded.


Definition:Bounded Above Set/Unbounded

Let $\struct {S, \preceq}$ be an ordered set. A subset $T \subseteq S$ is '''unbounded above (in $S$)''' {{iff}} it is not bounded above.


Definition:Bounded Below Set/Unbounded

Let $\struct {S, \preceq}$ be an ordered set. A subset $T \subseteq S$ is '''unbounded below (in $S$)''' {{iff}} it is not bounded below.


Definition:Bounded Mapping/Unbounded

$f$ is '''unbounded''' {{iff}} it is neither bounded above or bounded below.


Definition:Bounded Sequence/Unbounded

A sequence which is not bounded is '''unbounded'''.


Definition:Decision Procedure/Refutation Procedure

Given a decision procedure for satisfiability, one can craft a decision procedure for tautologies in the following way: Suppose one wanted to decide if a propositional formula $\mathbf A$ is a tautology. Then apply the given procedure to decide if its negation $\neg \mathbf A$ is satisfiable. Now: :If $\neg \mathbf A$ is not satisfiable, then by Tautology iff Negation is Unsatisfiable, $\mathbf A$ is a tautology. :If $\neg \mathbf A$ is satisfiable, then by Satisfiable iff Negation is Falsifiable, $\mathbf A$ is falsifiable, so cannot be a tautology. Hence we have crafted a decision procedure for tautologies. Such a procedure is called a '''refutation procedure''', because it proceeds by refuting, i.e. proving unsatisfiability of the negation of the formula at hand.


Definition:Bounded Below Mapping/Unbounded

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Then $f$ is '''unbounded below (in $T \ $)''' {{iff}} there exists no $L \in S$ such that: :$\forall x \in S: L \preceq \map f x$


Definition:Upper Bound of Sequence

A special case of an upper bound of a mapping is an '''upper bound of a sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Let $\sequence {x_n}$ be bounded above in $T$ by $H \in T$. Then $H$ is an '''upper bound of $\sequence {x_n}$'''.


Definition:Lower Bound of Sequence

A special case of a lower bound of a mapping is a '''lower bound of a sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Let $\sequence {x_n}$ be bounded below in $T$ by $L \in T$. Then $L$ is a '''lower bound of $\sequence {x_n}$'''.


Definition:Upper Bound of Number

When considering the upper bound of a set of numbers, it is commonplace to ignore the set and instead refer just to the number itself. Thus the construction: :''The set of numbers which fulfil the propositional function $P \left({n}\right)$ is bounded above with the upper bound $N$'' would be reported as: :''The number $n$ such that $P \left({n}\right)$ has the upper bound $N$''. This construct obscures the details of what is actually being stated. Its use on {{ProofWiki}} is considered an abuse of notation and so discouraged. This also applies in the case where it is the upper bound of a mapping which is under discussion.


{{:Definition:Infimum of Mapping/Real-Valued Function/Definition 1}}


Definition:Bounded Mapping/Real-Valued/Definition 1

$f$ is '''bounded on $S$''' {{iff}}: :$f$ is bounded above on $S$ and also: :$f$ is bounded below on $S$.


Definition:Bounded Mapping/Real-Valued/Definition 2

$f$ is '''bounded on $S$''' {{iff}}: :$\exists K \in \R_{\ge 0}: \forall x \in S: \size {\map f x} \le K$ where $\size {\map f x}$ denotes the absolute value of $\map f x$.


Definition:Bounded Above Mapping/Real-Valued

Let $f: S \to \R$ be a real-valued function. $f$ is '''bounded above on $S$''' by the upper bound $H$ {{iff}}: :$\forall x \in S: \map f x \le H$


Definition:Bounded Above Mapping/Real-Valued/Unbounded

Let $f: S \to \R$ be a real-valued function. Then $f$ is '''unbounded above on $S$''' {{iff}} it is not bounded above on $S$: :$\neg \exists H \in \R: \forall x \in S: \map f x \le H$


Definition:Bounded Above Mapping/Unbounded

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Then $f$ is '''unbounded above on $S$''' {{iff}} it is not bounded above on $S$: :$\neg \exists H \in T: \forall x \in S: \map f x \preceq H$


Definition:Bounded Below Mapping/Real-Valued

Let $f: S \to \R$ be a real-valued function. Then $f$ is '''bounded below on $S$''' by the lower bound $L$ {{iff}}: :$\forall x \in S: L \le \map f x$


Definition:Bounded Below Mapping/Real-Valued/Unbounded Below

Let $f: S \to \R$ be a real-valued function. Then $f$ is '''unbounded below on $S$''' {{iff}} it is not bounded below on $S$: :$\neg \exists L \in \R: \forall x \in S: L \le \map f x$


Definition:Bound of Subset of Real Numbers

Let $S \subseteq \R$ be a subset of the real numbers $\R$. Let $S$ be bounded. Then $K$ is a '''bound''' for $S$ {{iff}}: :$\forall x \in S: \size x \le K$


Definition:Bounded Ordered Set/Real Numbers/Definition 1

Let $T \subseteq \R$ be both bounded below and bounded above in $\R$. Then $T$ is '''bounded in $\R$'''.


Definition:Bounded Ordered Set/Real Numbers/Definition 2

Let $T \subseteq \R$ be a subset of $\R$ such that: :$\exists K \in \R: \forall x \in T: \size x \le K$ where $\size x$ denotes the absolute value of $x$. Then $T$ is '''bounded in $\R$'''.


{{:Definition:Bounded Ordered Set/Real Numbers/Definition 1}}


Definition:Upper Bound of Set/Real Numbers

Let $\R$ be the set of real numbers. Let $T$ be a subset of $\R$. An '''upper bound for $T$ (in $\R$)''' is an element $M \in \R$ such that: :$\forall t \in T: t \le M$ That is, $M$ is greater than or equal to every element of $T$.


Definition:Bounded Above Set/Real Numbers

Let $\R$ be the set of real numbers. A subset $T \subseteq \R$ is '''bounded above (in $\R$)''' {{iff}} $T$ admits an upper bound (in $\R$).


Definition:Bounded Below Set/Real Numbers

Let $\R$ be the set of real numbers. A subset $T \subseteq \R$ is '''bounded below (in $\R$)''' {{iff}} $T$ admits a lower bound (in $\R$).


Definition:Lower Bound of Set/Real Numbers

Let $\R$ be the set of real numbers. Let $T$ be a subset of $S$. A '''lower bound for $T$ (in $\R$)''' is an element $m \in \R$ such that: :$\forall t \in T: m \le t$


Definition:Bounded Below Mapping/Real-Valued/Unbounded


Definition:Upper Bound of Mapping/Real-Valued

Let $f: S \to \R$ be a real-valued function. Let $f$ be bounded above in $\R$ by $H \in \R$. Then $H$ is an '''upper bound of $f$'''.


Definition:Lower Bound of Mapping/Real-Valued

Let $f: S \to \R$ be a real-valued function. Let $f$ be bounded below in $T$ by $L \in T$. Then $L$ is a '''lower bound of $f$'''.


Definition:Supremum of Mapping/Real-Valued Function/Definition 1

The '''supremum of $f$ on $S$''' is defined by: :$\displaystyle \sup_{x \mathop \in S} \map f x := \sup f \sqbrk S$ where :$\sup f \sqbrk S$ is the supremum in $\R$ of the image of $S$ under $f$.


Definition:Supremum of Mapping/Real-Valued Function/Definition 2

The '''supremum of $f$ on $S$''' is defined as $\displaystyle \sup_{x \mathop \in S} \map f x := K \in \R$ such that: :$(1): \quad \forall x \in S: \map f x \le K$ :$(2): \quad \exists x \in S: \forall \epsilon \in \R_{>0}: \map f x > K - \epsilon$


Definition:Infimum of Mapping/Real-Valued Function/Definition 1

The '''infimum of $f$ on $S$''' is defined by: :$\displaystyle \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$ where :$\inf f \sqbrk S$ is the infimum in $\R$ of the image of $S$ under $f$.


Definition:Infimum of Mapping/Real-Valued Function/Definition 2

The '''infimum of $f$ on $S$''' is defined as $\displaystyle \inf_{x \mathop \in S} \map f x := k \in \R$ such that: :$(1): \quad \forall x \in S: k \le \map f x$ :$(2): \quad \forall \epsilon \in \R_{>0}: \exists x \in S: \map f x < k + \epsilon$


Definition:Bounded Above Sequence/Real

Let $\sequence {x_n}$ be a real sequence. Then $\sequence {x_n}$ is '''bounded above''' {{iff}}: :$\exists M \in \R: \forall i \in \N: x_i \le M$


Definition:Bounded Below Sequence/Real

Let $\sequence {x_n}$ be a real sequence. Then $\sequence {x_n}$ is '''bounded below''' {{iff}}: : $\exists m \in \R: \forall i \in \N: m \le x_i$


Definition:Bounded Above Sequence/Real/Unbounded

$\sequence {x_n}$ is '''unbounded above''' {{iff}} there exists no $M$ in $\R$ such that: :$\forall i \in \N: x_i \le M$


Definition:Bounded Below Sequence/Real/Unbounded

$\sequence {x_n}$ is '''unbounded below''' {{iff}} there exists no $m$ in $\R$ such that: :$\forall i \in \N: m \le x_i$


Definition:Bounded Sequence/Real

Let $\sequence {x_n}$ be a real sequence. Then $\sequence {x_n}$ is '''bounded''' {{iff}} $\exists m, M \in \R$ such that $\forall i \in \N$: :$m \le x_i$ :$x_i \le M$


Definition:Bounded Sequence/Real/Unbounded

$\sequence {x_n}$ is '''unbounded''' {{iff}} it is not bounded.


Definition:Bounded Mapping/Real-Valued/Unbounded

$f$ is '''unbounded''' {{iff}} it is neither bounded above nor bounded below.


Definition:Bounded Ordered Set/Real Numbers/Unbounded

$T \subseteq \R$ is '''unbounded (in $\R$)''' {{iff}} it is not bounded.


Definition:Bounded Above Set/Real Numbers/Unbounded

$T \subseteq \R$ is '''unbounded above (in $\R$)''' {{iff}} it is not bounded above.


Definition:Bounded Below Set/Real Numbers/Unbounded

$T \subseteq \R$ is '''unbounded below (in $\R$)''' {{iff}} it is not bounded below.


Definition:Lower Bound of Sequence/Real

Let $\sequence {x_n}$ be a real sequence. Let $\sequence {x_n}$ be bounded below in $T$ by $L \in \R$. Then $L$ is a '''lower bound of $\sequence {x_n}$'''.


Definition:Upper Bound of Sequence/Real

Let $\sequence {x_n}$ be a real sequence. Let $\sequence {x_n}$ be bounded above by $H \in \R$. Then $H$ is an '''upper bound of $\sequence {x_n}$'''.


Definition:Infimum of Set/Real Numbers

Let $T \subseteq \R$. A real number $c \in \R$ is the '''infimum of $T$ in $\R$''' {{iff}}: :$(1): \quad c$ is a lower bound of $T$ in $\R$ :$(2): \quad d \le c$ for all lower bounds $d$ of $T$ in $\R$.


Definition:Supremum of Set/Real Numbers

Let $T \subseteq \R$ be a subset of the real numbers. A real number $c \in \R$ is the '''supremum of $T$ in $\R$''' {{iff}}: :$(1): \quad c$ is an upper bound of $T$ in $\R$ :$(2): \quad c \le d$ for all upper bounds $d$ of $T$ in $\R$.


Definition:Bounded Metric Space/Complex

Let $D$ be a subset of the complex plane $\C$. Then '''$D$ is bounded (in $\C$)''' {{iff}} there exists $M \in \R$ such that: : $\forall z \in D: \cmod z \le M$


Definition:Bounded Metric Space/Complex/Unbounded

Let $D$ be a subset of the complex plane $\C$. Then '''$D$ is unbounded (in $\C$)''' {{iff}}: : $\nexists M \in \R: \forall z \in D: \cmod z \le M$ That is, if $D$ is not bounded in $\C$.


Definition:Compact Space/Metric Space/Complex

Let $D$ be a subset of the complex plane $\C$. Then $D$ is '''compact (in $\C$)''' {{iff}}: : $D$ is closed in $\C$ and : $D$ is bounded in $\C$.


Definition:Interior Point (Topology)/Definition 1

Let $h \in H$. Then $h$ is an '''interior point''' of $H$ {{iff}}: :$h \in H^\circ$ where $H^\circ$ denotes the interior of $H$.


Definition:Interior Point (Topology)/Definition 2

Let $h \in H$. $h$ is an '''interior point''' of $H$ {{iff}} $h$ has an open neighborhood $N_h$ such that $N_h \subseteq H$.


Definition:Theory

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a set of $\mathcal L$-formulas. Then $\mathcal F$ is an '''$\mathcal L$-theory''' iff, for every $\phi \in \mathcal L$: :$\mathcal F \models_{\mathscr M} \phi \implies \phi \in \mathcal F$ where $\models_{\mathscr M}$ denotes $\mathscr M$-semantic consequence. === Theory of Set of Formulas === {{:Definition:Theory/Set of Formulas}}


Definition:Axiomatization

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathcal L$-theory. An '''axiomatization of $\mathcal F$''' is a subset $\mathcal A \subseteq \mathcal F$ such that: :$\mathcal F = \left\{{\phi \in \mathcal L: \mathcal A \models_{\mathscr M} \phi}\right\}$ That is, all of $\mathcal F$ is a semantic consequence of $\mathcal A$. === Axiom === {{:Definition:Axiomatization/Axiom}}


Definition:Axiomatization/Axiom

Let $\mathcal A$ be an axiomatization of $\mathcal F$. Then a formula $\phi \in \mathcal A$ is called an '''axiom'''.


Definition:Finitely Axiomatizable

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathcal L$-theory. Then $\mathcal F$ is called '''finitely axiomatizable''' iff there exists a finite axiomatization $\mathcal A$ of $\mathcal F$.


Definition:Closed Set/Metric Space/Definition 1

'''$H$ is closed (in $M$)''' {{iff}} its complement $A \setminus H$ is open in $M$.


Definition:Closed Set/Metric Space/Definition 2

'''$H$ is closed (in $M$)''' {{iff}} every limit point of $H$ is also a point of $H$.


Definition:Closed Set/Topology

{{:Definition:Closed Set/Topology/Definition 1}}


Definition:Literal/Positive

A '''positive literal''' is an atom $p$ of propositional logic.


Definition:Literal/Negative

A '''negative literal''' is the negation $\neg p$ of an atom $p$ of propositional logic.


Definition:Closed Set/Complex Analysis

Let $S \subseteq \C$ be a subset of the complex plane. '''$S$ is closed (in $\C$)''' {{iff}} every limit point of $S$ is also a point of $S$. That is: {{iff}} $S$ contains all its limit points.


Definition:Interior Point (Complex Analysis)

Let $S \subseteq \C$ be a subset of the complex plane. Let $z \in S$. $z$ is an '''interior point''' of $S$ {{iff}} $z$ has an $\epsilon$-neighborhood $\map {N_\epsilon} z$ such that $\map {N_\epsilon} z \subseteq S$.


Definition:Boundary Point (Complex Analysis)

Let $S \subseteq \C$ be a subset of the complex plane. Let $z_0 \in \C$. $z_0$ is a '''boundary point''' of $S$ {{iff}} every $\epsilon$-neighborhood $\map {N_\epsilon} {z_0}$ of $z_0$ contains points of $\C$ in $S$ and also points of $\C$ which are not in $S$.


Definition:Exterior Point (Complex Analysis)/Definition 1

$z_0$ is an '''exterior point''' of $S$ {{iff}} $z_0$ has an $\epsilon$-neighborhood which is disjoint from $S$.


Definition:Exterior Point (Complex Analysis)/Definition 2

$z_0$ is an '''exterior point''' of $S$ {{iff}}: :$z_0$ is not an interior point of $S$ and: :$z_0$ is not a boundary point of $S$.


Let $S \subseteq \C$ be a subset of the complex plane. Let $z_0 \in \C$. === Definition 1 === {{:Definition:Exterior Point (Complex Analysis)/Definition 1}} === Definition 2 === {{:Definition:Exterior Point (Complex Analysis)/Definition 2}}


Definition:Open Set/Complex Analysis/Definition 1

Let $S \subseteq \C$ be a subset of the set of complex numbers. Let: : $\forall z_0 \in S: \exists \epsilon \in \R_{>0}: N_{\epsilon} \left({z_0}\right) \subseteq S$ where $N_{\epsilon} \left({z_0}\right)$ is the $\epsilon$-neighborhood of $z_0$ for $\epsilon$. Then $S$ is an '''open set (of $\C$)''', or '''open (in $\C$)'''.


Definition:Open Set/Complex Analysis/Definition 2

Let $S \subseteq \C$ be a subset of the set of complex numbers. Then $S$ is an '''open set (of $\C$)''', or '''open (in $\C$)''' {{iff}} every point of $S$ is an interior point.


Definition:Region/Plane

A point set $R$ in the plane is a '''region''' {{iff}}: :$(1): \quad$ Each point of $R$ is the center of a circle all of whose elements consist of points of $R$ :$(2): \quad$ Each point of $R$ can be joined by a curve consisting entirely of points of $R$.


Definition:Interior (Complex Analysis)

Let $H \subseteq \C$ be a subset of the complex plane. The '''interior of $H$''' is the subset of $H$ which consists of the interior points of $H$.


Let $D \subseteq \C$ be a subset of the set of complex numbers. === Definition 1 === {{:Definition:Connected Set (Complex Analysis)/Definition 1}} === Definition 2 === {{:Definition:Connected Set (Complex Analysis)/Definition 2}}


Definition:Open Region/Complex

Let $D \subseteq \C$ be a subset of the set of complex numbers. $D$ is an '''open region''' {{iff}} $D$ is: :$(1): \quad$ An open set and :$(2): \quad$ Connected.


Definition:Open Region/Plane

An '''open region''' is a region without its boundary, i.e. the  interior of such a region.


Definition:Closed Region

=== Complex Analysis === {{:Definition:Closed Region/Complex}} === Closed Region in the Plane === {{:Definition:Closed Region/Plane}} Category:Definitions/Complex Analysis Category:Definitions/Analytic Geometry bek3de33ql83xa4rml7jp62suzqxwta


Definition:Closed Region/Plane

A '''closed region''' is a region complete with its boundary.


Definition:Closed Region/Complex

Let $D \subseteq \C$ be a subset of the set of complex numbers. $D$ is a '''closed region''' {{iff}} $D$ the closure of an open region.


Definition:Region/Complex

Let $D \subseteq \C$ be a subset of the set of complex numbers. $D$ is a '''region of $\C$''' {{iff}}: :$(1): \quad$ $D$ is non-empty :$(2): \quad$ $D$ is path-connected.


Definition:Supremum of Sequence

A special case of a supremum of a mapping is a '''supremum of a sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Let $\set {x_n: n \in \N}$ admit a supremum. Then the '''supremum''' of $\sequence {x_n}$) is defined as: :$\displaystyle \map \sup {\sequence {x_n} } = \map \sup {\set {x_n: n \in \N} }$


Definition:Infimum of Sequence

A special case of an infimum of a mapping is an '''infimum of a sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Let $\set {x_n: n \in \N}$ admit an infimum. Then the '''infimum''' of $\sequence {x_n}$) is defined as: :$\displaystyle \map \inf {\sequence {x_n} } = \map \inf {\set {x_n: n \in \N} }$


Definition:Semantic Tableau

'''Semantic tableaus''' are the labeled trees which can be obtained in the course of executing the Semantic Tableau Algorithm. === Semantic Tableau Algorithm === {{:Semantic Tableau Algorithm}} === Completed Tableau === {{:Definition:Semantic Tableau/Completed|Completed Tableau}} === Closed Tableau === {{:Definition:Semantic Tableau/Closed|Closed Tableau}} === Open Tableau === {{:Definition:Semantic Tableau/Open|Open Tableau}}


Definition:Semantic Tableau/Completed

Let $T$ be a semantic tableau. Then $T$ is '''completed''' iff all of its leaves are marked.


Definition:Semantic Tableau/Closed

Let $T$ be a semantic tableau. Then $T$ is '''closed''' iff all of its leaves are marked closed.


Definition:Semantic Tableau/Open

Let $T$ be a semantic tableau. Then $T$ is '''open''' iff all of its leaves are marked open.


Definition:Marked Leaf

Let $T$ be a semantic tableau. Let $t$ be a leaf of $T$. Then $t$ is '''marked''' iff it has occurred in '''Step 3''' of the Semantic Tableau Algorithm. Otherwise, it is called '''unmarked'''. === Marked Closed Leaf === {{:Definition:Marked Leaf/Closed}} === Marked Open Leaf === {{:Definition:Marked Leaf/Open}}


Definition:Alpha-Formula

Let $\mathbf A$ be a WFF of propositional logic that is not a literal. Then $\mathbf A$ is an '''$\alpha$-formula''' {{iff}} either: :$\mathbf A = \neg \neg \mathbf A_1$ for some WFF $\mathbf A_1$, or: :$\mathbf A$ is semantically equivalent to a conjunction $\mathbf A_1 \land \mathbf A_2$ for some WFFs $\mathbf A_1, \mathbf A_2$. === Table of $\alpha$-Formulas === {{:Definition:Alpha-Formula/Table}}


Definition:Alpha-Formula/Table

From Classification of $\alpha$-Formulas, we obtain the following table of $\alpha$-formulas $\mathbf A$ and corresponding $\mathbf A_1$ and $\mathbf A_2$: ::$\begin{array}{ccc} \hline \mathbf A & \mathbf A_1 & \mathbf A_2\\ \hline \neg\neg \mathbf A_1 & \mathbf A_1 & \\ \mathbf A_1 \land \mathbf A_2 & \mathbf A_1 & \mathbf A_2 \\ \neg \left({\mathbf A_1 \lor \mathbf A_2}\right) & \neg \mathbf A_1 & \neg \mathbf A_2 \\ \neg \left({\mathbf A_1 \implies \mathbf A_2}\right) & \mathbf A_1 & \neg \mathbf A_2 \\ \neg \left({\mathbf A_1 \mathbin \uparrow \mathbf A_2}\right) & \mathbf A_1 & \mathbf A_2 \\ \mathbf A_1 \mathbin \downarrow \mathbf A_2 & \neg \mathbf A_1 & \neg \mathbf A_2 \\ \mathbf A_1 \iff \mathbf A_2 & \mathbf A_1 \implies \mathbf A_2 & \mathbf A_2 \implies \mathbf A_1 \\ \neg \left({\mathbf A_1 \oplus \mathbf A_2}\right) & \mathbf A_1 \implies \mathbf A_2 & \mathbf A_2 \implies \mathbf A_1 \\ \hline \end{array}$


Definition:Beta-Formula/Table

From Classification of $\beta$-Formulas, we obtain the following table of $\beta$-formulas $\mathbf B$ and corresponding $\mathbf B_1$ and $\mathbf B_2$: ::$\begin{array}{ccc} \hline \mathbf B & \mathbf B_1 & \mathbf B_2\\ \hline \neg \left({\mathbf B_1 \land \mathbf B_2}\right) & \neg \mathbf B_1 & \neg \mathbf B_2 \\ \mathbf B_1 \lor \mathbf B_2 & \mathbf B_1 & \mathbf B_2 \\ \mathbf B_1 \implies \mathbf B_2 & \neg \mathbf B_1 & \mathbf B_2 \\ \mathbf B_1 \mathbin \uparrow \mathbf B_2 & \neg \mathbf B_1 & \neg \mathbf B_2 \\ \neg \left({\mathbf B_1 \mathbin \downarrow \mathbf B_2}\right) & \mathbf B_1 & \mathbf B_2 \\ \neg \left({\mathbf B_1 \iff \mathbf B_2}\right) & \neg \left({\mathbf B_1 \implies \mathbf B_2}\right) & \neg \left({\mathbf B_2 \implies \mathbf B_1}\right) \\ \mathbf B_1 \oplus \mathbf B_2 & \neg \left({\mathbf B_1 \implies \mathbf B_2}\right) & \neg \left({\mathbf B_2 \implies \mathbf B_1}\right) \\ \hline \end{array}$


Definition:Beta-Formula

Let $\mathbf B$ be a WFF of propositional logic that is not a literal. Then $\mathbf B$ is a '''$\beta$-formula''' iff: :$\mathbf B$ is semantically equivalent to a disjunction $\mathbf B_1 \lor \mathbf B_2$ for some WFFs $\mathbf B_1, \mathbf B_2$. === Table of $\beta$-Formulas === {{:Definition:Beta-Formula/Table}}


Definition:Triangle (Geometry)/Right-Angled/Opposite

In a right-angled triangle, for a given non-right angled vertex, the opposite side is referred to as '''the opposite'''. In the above figure: :the '''opposite''' to vertex $A$ is side $a$ :the '''opposite''' to vertex $C$ is side $c$.


Definition:Theory/Set of Formulas

Let $\mathcal F$ be a set of $\mathcal L$-formulas. Then the '''$\mathcal L$-theory of $\mathcal F$''', denoted $T \left({\mathcal F}\right)$ is the set: :$\left\{{\phi \in \mathcal L: \mathcal F \models_{\mathscr M} \phi}\right\}$ where $\models_{\mathscr M}$ denotes $\mathscr M$-semantic consequence.


Definition:Lemniscate of Bernoulli

=== Geometric Definition === {{:Definition:Lemniscate of Bernoulli/Geometric Definition}} === Cartesian Definition === {{:Definition:Lemniscate of Bernoulli/Cartesian Definition}} === Polar Definition === {{:Definition:Lemniscate of Bernoulli/Polar Definition}} === Parametric Definition === {{:Definition:Lemniscate of Bernoulli/Parametric Definition}}


Definition:Gentzen Proof System

'''Gentzen proof systems''' are a class of proof systems for propositional and predicate logic. Their characteristics include: * The presence of few axioms and many rules of inference. * Use of formal, sequent-like notation involving the turnstile $\vdash$. * Proofs whose structure can be viewed as rooted trees.


<section notitle="true" name="tc"> This instance of a Gentzen proof system is used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Let $\mathcal L$ be the language of propositional logic. The Gentzen system applies to sets of propositional formulae. The intuition behind the system is that a set $U$ represents its disjunction. $\mathscr G$ has the following axioms and rules of inference: === Axioms === A set $U$ of propositional formulae is an axiom of $\mathscr G$ iff $U$ contains a complementary pair of literals. The invocation of an axiom may be denoted by: :$\vdash U$ === Rules of Inference === Let $U_1, U_2$ be sets of propositional formulae. $\mathscr G$ has two rules of inference, the $\alpha$-rule and the $\beta$-rule:<ref>In {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}}, the meanings of $\alpha$-formula and $\beta$-formula are interchanged at this point compared to their earlier appearances. This is done because of a later connection with semantic tableaus, but can serve only to confuse. Caution needs to be exercised when using this source, because of the context-dependent meaning of these two terms.</ref> ==== $\alpha$-Rule ==== {{:Definition:Gentzen Proof System/Instance 1/Alpha-Rule}} ==== $\beta$-Rule ==== {{:Definition:Gentzen Proof System/Instance 1/Beta-Rule}} Invocations of these rules in a proof can be denoted as: :$(\alpha) \dfrac {\vdash U_1, \mathbf A_1 \hspace{3em} \vdash U_2, \mathbf A_2}{\vdash U_1, U_2, \mathbf A} \hspace{3em}  (\beta) \dfrac {\vdash U_1, \mathbf B_1, \mathbf B_2}{\vdash U_1, \mathbf B}$ This notation suppresses the set notation as a matter of convenience. </section> {{NamedforDef|Gerhard Karl Erich Gentzen|cat = Gentzen}}


Definition:Tableau Proof (Formal Systems)

A '''tableau proof''' for a proof system is a technique for presenting a logical argument in a straightforward, standard form. On {{ProofWiki}}, the proof system is usually natural deduction. A '''tableau proof''' is a sequence of lines specifying the order of premises, assumptions, inferences and conclusion in support of an argument. Each line of a '''tableau proof''' has a particular format. It consists of the following parts: * '''Line:''' The line number of the proof. This is a simple numbering from 1 upwards. * '''Pool:''' The list of all the lines containing the pool of assumptions for the formula introduced on this line. * '''Formula:''' The propositional formula introduced on this line. * '''Rule:''' The justification for introducing this line. This should be the rule of inference being used to derive this line. * '''Depends on:''' The lines (if any) upon which this line ''directly'' depends. For premises and assumptions, this field will be empty. Optionally, a comment may be added to explicitly point out possible intricacies. If any assumptions are discharged on a certain line, for the sake of clarity it is preferred that such be mentioned explicitly in a comment. At the end of a tableau proof, the only lines upon which the proof depends may be those which contain the premises.


Definition:Tableau Proof (Formal Systems)/Length

The '''length''' of a tableau proof is the number of lines it has.


Definition:Variable/Complex

A '''complex variable''' is a symbol which can stand for any one of a set of complex numbers.


Definition:Independent Variable/Complex Function

Let $f: \C \to \C$ be a complex function. Let $\map f z = w$. Then $z$ is referred to as an '''independent variable (of $f$)'''.


Definition:Dependent Variable/Complex Function

Let $f: \C \to \C$ be a complex function. Let $\map f z = w$. Then $w$ is referred to as the '''dependent variable (of $f$)'''.


Definition:Left-Total Relation/Multifunction

In the field of complex analysis, a '''left-total relation''' is usually referred to as a '''multifunction'''.


Definition:Left-Total Relation/Multifunction/Branch

Let $D \subseteq \C$ be a subset of the complex numbers. Let $f: D \to \C$ be a multifunction on $D$. Let $\family {S_i}_{i \mathop \in I}$ be a partitioning of the codomain of $f$ such that: :$\forall i \in I: f \restriction_{D \times S_i}$ is a mapping. Then each $f \restriction_{D \times S_i}$ is a '''branch''' of $f$.


Definition:Distinct/Plural

Two objects $x$ and $y$ are '''distinct''' {{iff}} $x \ne y$. If $x$ and $y$ are '''distinct''', then that means they can be '''distinguished''', or '''identified as being different from each other'''.


Definition:Distinct/Plural/Pairwise Distinct

A set of objects is '''pairwise distinct''' if each pair of elements of that set is distinct.


Definition:Distinct/Singular

Let $x \in S$ be an element of a set of objects $S$. $x$ is '''distinguished''' from the other elements of $S$ if is endowed with a property that the other elements of $S$ are specifically deemed not to possess. Such an element is identified as being '''distinct''' from the others.


Definition:Distinct/Indistinguishable

Two objects are '''indistinguishable''' if they can not (in a particular context) be told apart from each other. So, two objects may be '''distinct''' but (at a given level) '''indistinguishable''', like identical twins.


Definition:Left-Total Relation/Multifunction/Branch/Principal Branch

Let $D \subseteq \C$ be a subset of the complex numbers. Let $f: D \to \C$ be a multifunction on $D$. Let $\sequence {S_i}_{i \in I}$ be a partitioning of the codomain of $f$ into branches. It is usual to distinguish one such branch of $f$ from the others, and label it the '''principal branch''' of $f$.


Definition:Real Function/Two Variables

Let $S, T \subseteq \R$ be subsets of the set of real numbers $\R$. Let $f: S \times T \to \R$ be a mapping. Then $f$ is defined as a '''(real) function of two (independent) variables'''.  The expression: :$z = \map f {x, y}$ means: :(The dependent variable) $z$ is a function of (the independent variables) $x$ and $y$.


Definition:Complex Transformation

A '''complex transformation''' is a mapping on the complex plane $f: \C \to \C$ which is specifically ''not'' a multifunction. Let $z = x + i y$ be a complex variable. Let $w = u + i v = \map f z$. Then $w$ can be expressed as: :$u + i v = \map f {x + i y}$ such that: :$u = \map u {x, y}$ and: :$v = \map v {x, y}$ are real functions of two variables. Thus a point $P = \tuple {x, y}$ in the complex plane is '''transformed''' to a point $P' = \tuple {\map u {x, y}, \map v {x, y} }$ by $f$. Thus $P'$ is the image of $P$ under $f$.


Definition:Curvilinear Coordinate System/Complex Plane

Let $u + i v = \map f {x + i y}$ be a complex transformation. Let $P = \tuple {x, y}$ be a point in the complex plane. Then $\tuple {\map u {x, y}, \map v {x, y} }$ are the '''curvilinear coordinates of $P$ under $f$'''.


Definition:Complex Curvilinear Coordinates/Coordinate Curves

Let $c_1$ and $c_2$ be constants. The curves: :$\map u {x, y} = c_1$ :$\map v {x, y} = c_2$ are the '''coordinate curves''' of $f$.


Definition:Rational Function/Real

Let $P: \R \to \R$ and $Q: \R \to \R$ be polynomial functions on the set of real numbers. Let $S$ be the set $\R$ from which all the roots of $Q$ have been removed. That is: : $S = \R \setminus \left\{{x \in \R: Q \left({x}\right) = 0}\right\}$. Then the equation $y = \dfrac {P \left({x}\right)} {Q \left({x}\right)}$ defines a function from $S$ to $\R$. Such a function is a '''rational function'''.


Definition:Rational Function/Complex

Let $P: \C \to \C$ and $Q: \C \to \C$ be polynomial functions on the set of complex numbers. Let $S$ be the set $\C$ from which all the roots of $Q$ have been removed. That is: :$S = \C \setminus \set {z \in \C: \map Q z = 0}$ Then the equation $y = \dfrac {\map P z} {\map Q z}$ defines a function from $S$ to $\C$. Such a function is a '''rational (algebraic) function'''.


Definition:Power (Algebra)/Real Number/Complex

Let $x \in \R$ be a real number such that $x > 0$. Let $r \in \C$ be ''any'' complex number. Then we define $x^r$ as: :$x^r := \map \exp {r \ln x}$ where $\exp$ denotes the complex exponential function.


Definition:Natural Logarithm/Complex/Definition 1

Let $z = r e^{i \theta}$ be a complex number expressed in exponential form such that $z \ne 0$. The '''complex natural logarithm''' of $z \in \C_{\ne 0}$ is the multifunction defined as: :$\map \ln z := \set {\map \ln r + i \paren {\theta + 2 k \pi}: k \in \Z}$ where $\map \ln r$ is the natural logarithm of the (strictly) positive real number $r$.


Definition:Natural Logarithm/Notation

The notation for the natural logarithm function is misleadingly inconsistent throughout the literature. It is written variously as: :$\ln z$ :$\log z$ :$\Log z$ :$\log_e z$ The first of these is commonly encountered, and is the preferred form on {{ProofWiki}}. However, many who consider themselves serious mathematicians believe this notation to be unsophisticated. The second and third are ambiguous (it doesn't tell you which base it is the logarithm of). While the fourth option is more verbose than the others, there is no confusion about exactly what is meant.


Definition:Natural Logarithm/Complex/Definition 2

Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''complex natural logarithm''' of $z$ is the multifunction defined as: :$\map \ln z := \set {w \in \C: e^w = z}$


Definition:Natural Logarithm/Positive Real/Definition 1

Let $x \in \R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$\displaystyle \ln x := \int_1^x \frac {\d t} t$


Definition:Natural Logarithm/Positive Real/Definition 2

Let $x \in \R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$\ln x := y \in \R: e^y = x$ where $e$ is Euler's number.


Definition:Gentzen Proof System/Instance 1/Alpha-Rule

$(\alpha)$: For any $\alpha$-formula $\mathbf A$ and associated $\mathbf A_1, \mathbf A_2$ from the table of $\alpha$-formulas: ::Given $U_1 \cup \left\{{\mathbf A_1}\right\}$ and $U_2 \cup \left\{{\mathbf A_2}\right\}$, one may infer $U_1 \cup U_2 \cup \left\{{\mathbf A}\right\}$.


Definition:Gentzen Proof System/Instance 1/Beta-Rule

$(\beta)$: For any $\beta$-formula $\mathbf B$ and associated $\mathbf B_1, \mathbf B_2$ from the table of $\beta$-formulas: :: Given $U_1 \cup \left\{{\mathbf B_1, \mathbf B_2}\right\}$, one may infer $U_1 \cup \left\{{\mathbf B}\right\}$.


Definition:Gentzen Proof System/Instance 1/Alpha-Rule/Notation

In a tableau proof, the $\alpha$-rule can be used as follows: {{begin-axiom}} {{axiom|lc = '''Pool:'''        |t  = Empty. }} {{axiom|lc = '''Formula:'''         |t  = $U_1 \cup U_2 \cup \left\{ {\mathbf A}\right\}$. }} {{axiom|lc = '''Description:'''         |t  = $\alpha$-Rule. }} {{axiom|lc = '''Depends on:'''         |t  = The lines containing $U_1 \cup \left\{ {\mathbf A_1}\right\}$ and $U_2 \cup \left\{ {\mathbf A_2}\right\}$. }} {{axiom|lc = '''Abbreviation:'''         |t  = $\alpha \circ$, where $\circ$ is the binary logical connective such that $\mathbf A {{=}} \mathbf A_1 \circ \mathbf A_2$ or $\mathbf A {{=}} \neg \left({\mathbf A_1 \circ \mathbf A_2}\right)$, or $\neg \neg$ in the case that $\mathbf A = \neg \neg \mathbf A_1$. }} {{end-axiom}}


Definition:Gentzen Proof System/Instance 1/Beta-Rule/Notation

In a tableau proof, the $\beta$-rule can be used as follows: {{begin-axiom}} {{axiom|lc = '''Pool:'''        |t  = Empty. }} {{axiom|lc = '''Formula:'''         |t  = $U_1 \cup \left\{ {\mathbf B}\right\}$. }} {{axiom|lc = '''Description:'''         |t  = $\beta$-Rule. }} {{axiom|lc = '''Depends on:'''         |t  = The line containing $U_1 \cup \left\{ {\mathbf B_1, \mathbf B_2}\right\}$. }} {{axiom|lc = '''Abbreviation:'''         |t  = $\beta \circ$, where $\circ$ is the binary logical connective such that $\mathbf B {{=}} \mathbf B_1 \circ \mathbf B_2$ or $\mathbf B {{=}} \neg \left({\mathbf B_1 \circ \mathbf B_2}\right)$. }} {{end-axiom}}


Definition:Natural Logarithm/Complex/Principal Branch

The principal branch of the complex natural logarithm is usually defined in one of two ways: :$\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointr 0 {2 \pi}$ :$\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointl {-\pi} \pi$ It is important to specify which is in force during a particular exposition.


Definition:Principle

A '''principle''' is a theorem which is deemed to be particularly important. The word has the same force about it as does '''law''', the two words being used interchangeably.


Definition:General Logarithm/Positive Real

Let $x \in \R_{>0}$ be a strictly positive real number. Let $a \in \R_{>0}$ be a strictly positive real number such that $a \ne 1$. The '''logarithm to the base $a$ of $x$''' is defined as: :$\log_a x := y \in \R: a^y = x$ where $a^y = e^{y \ln a}$ as defined in Powers of Real Numbers.


Definition:General Logarithm/Complex

Let $z \in \C_{\ne 0}$ be a non-zero complex number. Let $a \in \R_{>0}$ be a strictly positive real number such that $a \ne 1$. The '''logarithm to the base $a$ of $z$''' is defined as: :$\log_a z := \set {y \in \C: a^y = z}$ where $a^y = e^{y \ln a}$ as defined in Powers of Complex Numbers.


Definition:Inverse Sine/Real/Arcsine

{{:Graph of Arcsine Function|Graph}} 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 $x$''' and is written $\arcsin x$. Thus: :The domain of $\arcsin x$ is $\closedint {-1} 1$ :The image of $\arcsin x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.


Definition:Inverse Sine/Real

Let $x \in \R$ be a real number such that $-1 \le x \le 1$. The '''inverse sine of $x$''' is the multifunction defined as: :$\sin^{-1} \left({x}\right) := \left\{{y \in \R: \sin \left({y}\right) = x}\right\}$ where $\sin \left({y}\right)$ is the sine of $y$.


Definition:Sine/Real Function

The real function $\sin: \R \to \R$ is defined as: {{begin-eqn}} {{eqn | l = \sin x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots       | c =  }} {{end-eqn}}


Definition:Sine/Complex Function

The complex function $\sin: \C \to \C$ is defined as: {{begin-eqn}} {{eqn | l = \sin z       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!} + \cdots       | c =  }} {{end-eqn}}


Definition:Inverse Sine/Complex

{{:Definition:Inverse Sine/Complex/Definition 1}}


Definition:Hilbert Proof System

'''Hilbert proof systems''' are a class of proof systems for propositional and predicate logic. Their characteristics include: * The presence of many axioms; * Typically only Modus Ponendo Ponens as a rule of inference.


Definition:Hilbert Proof System/Instance 1

<section notitle="true" name="tc"> This instance of a Hilbert proof system is used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Let $\LL$ be the language of propositional logic. $\mathscr H$ has the following axioms and rules of inference: === Axioms === Let $\mathbf A, \mathbf B, \mathbf C$ be WFFs. The following three WFFs are axioms of $\mathscr H$: {{begin-axiom}} {{axiom | lc= '''Axiom 1:'''         | m = \mathbf A \implies \paren {\mathbf B \implies \mathbf A} }} {{axiom | lc= '''Axiom 2:'''         | m = \paren {\mathbf A \implies \paren {\mathbf B \implies \mathbf C} } \implies \paren {\paren {\mathbf A \implies \mathbf B} \implies \paren {\mathbf A \implies \mathbf C} } }} {{axiom | lc= '''Axiom 3:'''         | m = \paren {\neg \mathbf B \implies \neg \mathbf A} \implies \paren {\mathbf A \implies \mathbf B} }} {{end-axiom}} === Rules of Inference === The sole rule of inference is Modus Ponendo Ponens: :From $\mathbf A$ and $\mathbf A \implies \mathbf B$, one may infer $\mathbf B$. </section> {{NamedforDef|David Hilbert}}


Definition:Derived Rule

Let $\mathcal L$ be a formal language. Let $\mathscr P$ be a proof system for $\mathcal L$. A '''derived rule''' for $\mathscr P$ is a rule of inference $R$ such that: :The proof system $\mathscr P'$, obtained by adding $R$ to $\mathscr P$, is equivalent to $\mathscr P$. That is, $R$ is a '''derived rule''' iff it does not extend the collection of $\mathscr P$-theorems.


Definition:Cosine/Real Function

The real function $\cos: \R \to \R$ is defined as: {{begin-eqn}} {{eqn | l = \cos x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }       | c =  }} {{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \cdots       | c =  }} {{end-eqn}}


Definition:Cosine/Complex Function

The complex function $\cos: \C \to \C$ is defined as: {{begin-eqn}} {{eqn | l = \cos z       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots       | c =  }} {{end-eqn}}


Definition:Inverse Sine/Complex/Definition 1

Let $z \in \C$ be a complex number. The '''inverse sine of $z$''' is the multifunction defined as: :$\sin^{-1} \paren z := \set {w \in \C: \sin \paren w = z}$ where $\sin \paren w$ is the sine of $w$.


Definition:Inverse Sine/Complex/Definition 2

Let $z \in \C$ be a complex number. The '''inverse sine of $z$''' is the multifunction defined as: :$\sin^{-1} \paren z := \set {\dfrac 1 i \ln \paren {i z + \sqrt {\cmod {1 - z^2} } \exp \paren {\dfrac i 2 \arg \paren {1 - z^2} } } + 2 k \pi: k \in \Z}$ where: : $\sqrt {\cmod {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ : $\arg \paren {1 - z^2}$ denotes the argument of $1 - z^2$ : $\ln$ is the complex natural logarithm considered as a  multifunction.


Definition:Inverse Sine/Complex/Arcsine

The principal branch of the complex inverse sine function is defined as: :$\map \arcsin z = \dfrac 1 i \, \map \Ln {i z + \sqrt {1 - z^2} }$ where: : $\Ln$ denotes the principal branch of the complex natural logarithm : $\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.


Definition:Inverse Cosine/Real

Let $x \in \R$ be a real number such that $-1 \le x \le 1$. The '''inverse cosine of $x$''' is the multifunction defined as: :$\cos^{-1} \left({x}\right) := \left\{{y \in \R: \cos \left({y}\right) = x}\right\}$ where $\cos \left({y}\right)$ is the cosine of $y$.


Definition:Inverse Cosine/Complex

{{:Definition:Inverse Cosine/Complex/Definition 1}}


Definition:Inverse Cosine/Complex/Definition 1

Let $z \in \C$ be a complex number. The '''inverse cosine of $z$''' is the multifunction defined as: :$\cos^{-1} \left({z}\right) := \left\{{w \in \C: \cos \left({w}\right) = z}\right\}$ where $\cos \left({w}\right)$ is the cosine of $w$.


Definition:Inverse Cosine/Complex/Definition 2

Let $z \in \C$ be a complex number. The '''inverse cosine of $z$''' is the multifunction defined as: :$\cos^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({z + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)} }\right) + 2 k \pi: k \in \Z}\right\}$ where: : $\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$ : $\ln$ denotes the complex natural logarithm considered as a  multifunction.


Definition:Inverse Cosine/Complex/Arccosine

The principal branch of the complex inverse cosine function is defined as: :$\map \arccos z = \dfrac 1 i \, \map \Ln {z + \sqrt {z^2 - 1} }$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.


Definition:Inverse Tangent/Real

Let $x \in \R$ be a real number. The '''inverse tangent of $x$''' is the multifunction defined as: :$\tan^{-1} \left({x}\right) := \left\{{y \in \R: \tan \left({y}\right) = x}\right\}$ where $\tan \left({y}\right)$ is the tangent of $y$.


Definition:Inverse Tangent/Complex

{{:Definition:Inverse Tangent/Complex/Definition 1}}


Definition:Inverse Tangent/Complex/Definition 1

The '''inverse tangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \tan^{-1} \left({z}\right) := \left\{{w \in \C: \tan \left({w}\right) = z}\right\}$ where $\tan \left({w}\right)$ is the tangent of $w$.


Definition:Inverse Tangent/Complex/Definition 2

The '''inverse tangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \tan^{-1} \paren z := \set {\dfrac 1 {2 i} \ln \paren {\dfrac {i - z} {i + z} } + k \pi: k \in \Z}$ where $\ln$ denotes the complex natural logarithm as a multifunction.


Definition:Inverse Tangent/Complex/Arctangent

The principal branch of the complex inverse tangent function is defined as: :$\map \arctan z := \dfrac 1 {2 i} \, \map \Ln {\dfrac {i - z} {i + z} }$ where $\Ln$ denotes the principal branch of the complex natural logarithm.


Definition:Inverse Cotangent/Complex

{{:Definition:Inverse Cotangent/Complex/Definition 1}}


Definition:Inverse Cotangent/Complex/Definition 1

The '''inverse cotangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \cot^{-1} \left({z}\right) := \left\{{w \in \C: \cot \left({w}\right) = z}\right\}$ where $\cot \left({w}\right)$ is the cotangent of $w$.


Definition:Inverse Cotangent/Complex/Definition 2

The '''inverse cotangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \cot^{-1} \left({z}\right) := \left\{{\dfrac 1 {2 i} \ln \left({\dfrac {z + i} {z - i}}\right) + k \pi: k \in \Z}\right\}$ where $\ln$ denotes the complex natural logarithm as a multifunction.


Definition:Inverse Cotangent/Complex/Arccotangent

The principal branch of the complex inverse cotangent function is defined as: :$\map \arccot z := \dfrac 1 {2 i} \, \map \Ln {\dfrac {z + i} {z - i} }$ where $\Ln$ denotes the principal branch of the complex natural logarithm.


Definition:Inverse Secant/Real

Let $x \in \R$ be a real number such that $x \le -1$ or $x \ge 1$. The '''inverse secant of $x$''' is the multifunction defined as: :$\map {\sec^{-1} } x := \set {y \in \R: \map \sec y = x}$ where $\map \sec y$ is the secant of $y$.


Definition:Inverse Secant/Complex

{{:Definition:Inverse Secant/Complex/Definition 1}}


Definition:Inverse Secant/Complex/Definition 1

Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''inverse secant of $z$''' is the multifunction defined as: :$\map {\sec^{-1} } z := \set {w \in \C: \map \sec w = z}$ where $\map \sec w$ is the secant of $w$.


Definition:Inverse Secant/Complex/Definition 2

Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''inverse secant of $z$''' is the multifunction defined as: :$\sec^{-1} z := \set {\dfrac 1 i \map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi: k \in \Z}$ where: :$\sqrt {\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ :$\map \arg {1 - z^2}$ denotes the argument of $1 - z^2$ :$\ln$ denotes the complex natural logarithm as a multifunction.


Definition:Inverse Secant/Complex/Arcsecant

The principal branch of the complex inverse secant function is defined as: :$\forall z \in \C_{\ne 0}: \map \arcsec z := \dfrac 1 i \, \map \Ln {\dfrac {1 + \sqrt {1 - z^2} } z}$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.


Definition:Inverse Cosecant/Complex

{{:Definition:Inverse Cosecant/Complex/Definition 1}}


Definition:Inverse Cosecant/Complex/Definition 1

Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''inverse cosecant of $z$''' is the multifunction defined as: :$\csc^{-1} \left({z}\right) := \left\{{w \in \C: \csc \left({w}\right) = z}\right\}$ where $\csc \left({w}\right)$ is the cosecant of $w$.


Definition:Inverse Cosecant/Complex/Definition 2

Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''inverse cosecant of $z$''' is the multifunction defined as: :$\csc^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({\dfrac {i + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)}} z}\right) + 2 k \pi: k \in \Z}\right\}$ where: : $\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$ : $\ln$ denotes the complex natural logarithm considered as a  multifunction.


Definition:Inverse Cosecant/Complex/Arccosecant

The principal branch of the complex inverse cosecant function is defined as: :$\forall z \in \C_{\ne 0}: \map \arccsc z := \dfrac 1 i \, \map \Ln {\dfrac {i + \sqrt {z^2 - 1} } z}$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.


Definition:Inverse Hyperbolic Sine/Complex/Arcsine

The principal branch of the complex inverse hyperbolic sine function is defined as: :$\forall z \in \C: \map {\Sinh^{-1} } z := \map \Ln {z + \sqrt {z^2 + 1} }$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.


Definition:Inverse Hyperbolic Sine/Arcsine

{{:Definition:Complex Hyperbolic Arcsine}}


Definition:Inverse Hyperbolic Sine/Real/Definition 2

The '''inverse hyperbolic sine''' $\sinh^{-1}: \R \to \R$ is a real function defined on $\R$ as: :$\forall x \in \R: \map {\sinh^{-1} } x := \map \ln {x + \sqrt {x^2 + 1} }$ where: :$\ln$ denotes the natural logarithm of a (strictly positive) real number :$\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$.


Definition:Inverse Hyperbolic Cosine/Real/Definition 1

The '''inverse hyperbolic cosine''' $\cosh^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \cosh^{-1} \left({x}\right) := y \in \R_{\ge 0}: x = \cosh \left({y}\right)$ where $\cosh \left({y}\right)$ denotes the hyperbolic cosine function.


Definition:Inverse Hyperbolic Cotangent/Real/Definition 1

The '''inverse hyperbolic cotangent''' $\coth^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \coth^{-1} x := y \in \R: x = \coth y$ where $\coth y$ denotes the hyperbolic cotangent function.


Definition:Inverse Hyperbolic Secant/Real/Definition 1

The '''inverse hyperbolic secant''' $\operatorname{sech}^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \operatorname{sech}^{-1} \left({x}\right) := y \in \R_{\ge 0}: x = \operatorname{sech} \left({y}\right)$ where $\operatorname{sech} \left({y}\right)$ denotes the hyperbolic secant function.


Definition:Inverse Hyperbolic Cosecant/Real/Definition 1

The '''inverse hyperbolic cosecant''' $\csch^{-1}: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as: :$\forall x \in \R_{\ne 0}: \map {\csch^{-1} } x := y \in \R: x = \map \csch y$ where $\map \csch y$ denotes the hyperbolic cosecant function of $y$.


Definition:Inverse Hyperbolic Cosine/Complex/Arccosine

The principal branch of the complex inverse hyperbolic cosine function is defined as: :$\forall z \in \C: \map {\Cosh^{-1} } z := \map \Ln {z + \sqrt {z^2 - 1} }$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.


Definition:Inverse Hyperbolic Cosecant/Complex/Arccosecant

The principal branch of the complex inverse hyperbolic cosecant function is defined as: :$\forall z \in \C_{\ne 0}: \map {\Csch^{-1} } z := \map \Ln {\dfrac {1 + \sqrt {z^2 + 1} } z}$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.


Definition:Inverse Hyperbolic Secant/Complex/Arcsecant

The principal branch of the complex inverse hyperbolic secant function is defined as: :$\forall z \in \C: \map {\Sech^{-1} } z := \map \Ln {\dfrac {1 + \sqrt{1 - z^2} } z}$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.


Definition:Inverse Hyperbolic Tangent/Complex/Arctangent

The principal branch of the complex inverse hyperbolic tangent function is defined as: :$\forall z \in \C: \map {\Tanh^{-1} } z := \dfrac 1 2 \, \map \Ln {\dfrac {1 + z} {1 - z} }$ where $\Ln$ denotes the principal branch of the complex natural logarithm.


Definition:Inverse Hyperbolic Cotangent/Complex/Arccotangent

The principal branch of the complex inverse hyperbolic cotangent function is defined as: :$\forall z \in \C: \map {\Coth^{-1} } z := \dfrac 1 2 \, \map \Ln {\dfrac {z + 1} {z - 1} }$ where $\Ln$ denotes the principal branch of the complex natural logarithm.


Definition:Inverse Hyperbolic Cosine/Real/Definition 2

The '''inverse hyperbolic cosine''' $\cosh^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \map {\cosh^{-1} } x := \map \ln {x + \sqrt {x^2 - 1} }$ 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.


Definition:Inverse Hyperbolic Tangent/Real/Definition 1

The '''inverse hyperbolic tangent''' $\tanh^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \tanh^{-1} \left({x}\right) := y \in \R: x = \tanh \left({y}\right)$ where $\tanh \left({y}\right)$ denotes the hyperbolic tangent function.


Definition:Inverse Hyperbolic Sine/Real/Definition 1

The '''inverse hyperbolic sine''' $\sinh^{-1}: \R \to \R$ is a real function defined on $\R$ as: :$\forall x \in \R: \map {\sinh^{-1} } x := y \in \R: x = \map \sinh y$ where $\map \sinh y$ denotes the hyperbolic sine function.


=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosine/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosine/Real/Definition 2}}


Definition:Image (Set Theory)/Mapping/Mapping/Definition 1

The '''image''' of a mapping $f: S \to T$ is the set: :$\Img f = \set {t \in T: \exists s \in S: \map f s = t}$ That is, it is the set of values taken by $f$.


Definition:Image (Set Theory)/Mapping/Mapping/Definition 2

The '''image''' of a mapping $f: S \to T$ is the set: :$\Img f = f \sqbrk S$ where $f \sqbrk S$ is the image of $S$ under $f$.


Let $S$ denote the open real interval: : $S := \left({-1 \,.\,.\, 1}\right)$ === Definition 1 === {{:Definition:Inverse Hyperbolic Tangent/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Tangent/Real/Definition 2}}


=== Definition 1 === {{:Definition:Inverse Hyperbolic Cotangent/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cotangent/Real/Definition 2}}


=== Definition 1 === {{:Definition:Inverse Hyperbolic Secant/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Secant/Real/Definition 2}}


=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosecant/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosecant/Real/Definition 2}}


Definition:Dominate (Set Theory)/Definition 1

Then '''$S$ is dominated by $T$''' {{iff}} there exists an injection from $S$ to $T$.


Definition:Dominate (Set Theory)/Definition 2

Then '''$S$ is dominated by $T$''' {{iff}} $S$ is equivalent to some subset of $T$. That is, {{iff}} there exists a bijection $f: S \to T'$ for some $T' \subseteq T$.


Definition:Dominate (Set Theory)/Strictly Dominate

'''$S$ is strictly dominated by set $T$''' {{iff}} $S \preccurlyeq T$ but $\neg T \preccurlyeq S$. This can be written $S \prec T$ or $S < T$.


Definition:Exponentiation of Cardinals

Let $A$ and $B$ be sets. Let $\mathbf a$ and $\mathbf b$ be the cardinals associated respectively with $A$ and $B$. Then the '''exponentiation''' of $\mathbf a$ to $\mathbf b$ is defined as: :$\mathbf a^{\mathbf b} := \operatorname{Card} \left({A^B}\right)$ where: :$A^B$ denotes the set of all mappings from $B$ to $A$ :$\operatorname{Card} \left({A^B}\right)$ denotes the cardinal associated with $A^B$.


Definition:Inverse Hyperbolic Tangent/Real/Definition 2

The '''inverse hyperbolic tangent''' $\tanh^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \map {\tanh^{-1} } x := \dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$ where $\ln$ denotes the natural logarithm of a (strictly positive) real number.


Definition:Inverse Hyperbolic Cotangent/Real/Definition 2

The '''inverse hyperbolic cotangent''' $\coth^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \coth^{-1} x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$ where $\ln$ denotes the natural logarithm of a (strictly positive) real number.


Definition:Inverse Hyperbolic Secant/Real/Definition 2

The '''inverse hyperbolic secant''' $\sech^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \map {\sech^{-1} } x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$ where: :$\sqrt {1 - x^2}$ denotes the positive square root of $1 - x^2$ :$\ln$ denotes the natural logarithm of a (strictly positive) real number.


Definition:Inverse Hyperbolic Cosecant/Real/Definition 2

The '''inverse hyperbolic cosecant''' $\csch^{-1}: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as: :$\forall x \in \R_{\ne 0}: \map {\csch^{-1} } x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$ 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.


Definition:Real Number/Axiomatic Definition

Let $\struct {R, +, \times, \le}$ be a Dedekind complete ordered field. Then $R$ is called the '''(field of) real numbers'''. ==== Real Number Axioms ==== {{:Definition:Real Number/Axioms}}


Definition:Natural Numbers/Inductive Sets in Real Numbers

Let $\R$ be the set of real numbers. Let $\II$ be the set of all inductive sets defined as subsets of $\R$. Then the '''natural numbers''' $\N$ are defined as: :$\N := \displaystyle \bigcap \II$ where $\displaystyle \bigcap$ denotes intersection.


Definition:Square Root/Positive Real

Let $x \in \R_{\ge 0}$ be a positive real number. The '''square roots of $x$''' are the real numbers defined as: :$x^{\paren {1 / 2} } := \set {y \in \R: y^2 = x}$ where $x^{\paren {1 / 2} }$ is the $2$nd root of $x$. The notation: :$y = \pm \sqrt x$ is usually encountered.


Definition:Naturally Ordered Semigroup

The concept of a '''naturally ordered semigroup''' is intended to capture the behaviour of the natural numbers $\N$, addition $+$ and the ordering $\le$ as they pertain to $\N$. === Naturally Ordered Semigroup Axioms === {{:Definition:Naturally Ordered Semigroup/Axioms}}


Definition:Square Root/Positive

The '''positive square root of $x$''' is the number defined as: :$+ \sqrt x := y \in \R_{>0}: y^2 = x$


Definition:Square Root/Negative

The '''negative square root of $x$''' is the number defined as: :$- \sqrt x := y \in \R_{<0}: y^2 = x$


Definition:Square Root/Negative Real

Let $x \in \R_{< 0}$ be a (strictly) negative real number. Then the '''square root of $x$''' is defined as: :$\sqrt x = i \paren {\pm \sqrt {-x} }$ where $i$ is the imaginary unit: :$i^2 = -1$


Definition:Square Root/Complex Number

{{:Definition:Square Root/Complex Number/Definition 1}}


Definition:Square Root/Complex Number/Definition 1

Let $z \in \C$ be a complex number expressed in polar form as $\left \langle{r, \theta}\right\rangle = r \left({\cos \theta + i \sin \theta}\right)$. The '''square root of $z$''' is the $2$-valued multifunction: {{begin-eqn}} {{eqn | l = z^{1/2}       | r = \left\{ {\sqrt r \left({\cos \left({\frac {\theta + 2 k \pi} 2}\right) + i \sin \left({\frac {\theta + 2 k \pi} 2}\right) }\right): k \in \left\{ {0, 1}\right\} }\right\}       | c =  }} {{eqn | r = \left\{ {\sqrt r \left({\cos \left({\frac \theta 2 + k \pi}\right) + i \sin \left({\frac \theta 2 + k \pi}\right) }\right): k \in \left\{ {0, 1}\right\} }\right\}       | c =  }} {{end-eqn}} where $\sqrt r$ denotes the positive square root of $r$.


Definition:Square Root/Complex Number/Definition 2

Let $z \in \C$ be a complex number expressed in polar form as $\left \langle{r, \theta}\right\rangle = r \left({\cos \theta + i \sin \theta}\right)$. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \left\{ {\pm \sqrt r \left({\cos \left({\dfrac \theta 2}\right) + i \sin \left({\dfrac \theta 2}\right) }\right)}\right\}$ where $\pm \sqrt r$ denotes the positive and negative square roots of $r$.


Definition:Square Root/Complex Number/Definition 3

Let $z \in \C$ be a complex number. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \left\{ {\sqrt {\left\vert{z}\right\vert} e^{\left({i / 2}\right) \arg \left({z}\right)} }\right\}$ where: : $\sqrt {\left\vert{z}\right\vert}$ denotes the positive square root of the complex modulus of $z$ : $\arg \left({z}\right)$ denotes the argument of $z$ considered as a  multifunction.


Definition:Square Root/Complex Number/Definition 4

Let $z \in \C$ be a complex number. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \left\{ {w \in \C: w^2 = z}\right\}$


Definition:Gamma Function/Integral Form

The '''Gamma function''' $\Gamma: \C \to \C$ is defined, for the open right half-plane, as: :$\displaystyle \map \Gamma z = \map {\MM \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$ where $\MM$ is the Mellin transform. For all other values of $z$ except the non-positive integers, $\map \Gamma z$ is defined as: :$\map \Gamma {z + 1} = z \, \map \Gamma z$


Definition:Gamma Function/Weierstrass Form

The '''{{AuthorRef|Karl Theodor Wilhelm Weierstrass|Weierstrass}} form''' of the '''Gamma function''' is: :$\displaystyle \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }$ where $\gamma$ is the Euler-Mascheroni constant. The '''Weierstrass form''' is valid for all $\C$.


Definition:Gamma Function/Euler Form

The '''{{AuthorRef|Leonhard Paul Euler|Euler}} form''' of the '''Gamma function''' is: :$\displaystyle \Gamma \left({z}\right) = \frac 1 z \prod_{n \mathop = 1}^\infty \left({\left({1 + \frac 1 n}\right)^z \left({1 + \frac z n}\right)^{-1}}\right) = \lim_{m \mathop \to \infty} \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$ which is valid except for $z \in \left\{{0, -1, -2, \ldots}\right\}$.


Definition:Square Root/Complex Number/Principal Square Root

Let $z \in \C$ be a complex number. Let $z^{1/2} = \set {w \in \C: w^2 = z}$ be the square root of $z$. The '''principal square root''' of $z$ is the element $w$ of $z^{1/2}$ such that: :$\begin{cases} \map \Im w > 0 : & \map \Im z \ne 0 \\ \map \Re w \ge 0 : & \map \Im z = 0 \end{cases}$


Definition:Reduction Formula

Let $f: \R \times \Z_{\ge 0}: \R$ be a Darboux integrable real-valued function on $\R \times \N$. Let: :$\displaystyle I_n = \int \map f {x, n} \rd x$ for some $n \in \Z_{\ge 0}$. A '''reduction formula''' is a recurrence relation of the form: :$\displaystyle I_k = \map g {\displaystyle I_n}$ such that $k < n$. That is, it is a technique to '''reduce''' the integer parameter in the integrand in order to allow evaluation of the integral at, usually, $n = 0$ or $n = 1$. This technique relies upon the supposition that $\map f {x, 0}$ or $\map f {x, 1}$ can be integrated directly. Category:Definitions/Proof Techniques Category:Definitions/Integral Calculus 1kwzi5xzaqj9e47w5qfe5ui8vd20c7g


Definition:Zero Complement

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $0$ be the zero of $S$. Let $S^* := \relcomp S {\set 0} = S \setminus \set 0$ be the complement of $\set 0$ in $S$. Then $S^*$ is called the '''zero complement''' of $S$.


Definition:Beta Function

{{:Definition:Beta Function/Definition 1}}


Definition:Subtraction/Naturally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $m, n \in S$ such that $m \preceq n$. By axiom $(NO3)$, there exists a $p \in S$ such that: :$m + p = n$ This $p$ is the '''difference between $m$ and $n$''', and denoted $n - m$. The operation $-$, assigning to $m, n \in S$ with $m \preceq n$ their '''difference''' $n - m$ is called '''subtraction'''.


Definition:Bernoulli Numbers

The '''Bernoulli numbers''' $B_n$ are a sequence of rational numbers defined by:  === Generating Function === {{:Definition:Bernoulli Numbers/Generating Function}} === Recurrence Relation === {{:Definition:Bernoulli Numbers/Recurrence Relation}}


Definition:Convex Real Function/Definition 1

$f$ is '''convex on $I$''' {{iff}}: :$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \le \alpha f \left({x}\right) + \beta f \left({y}\right)$


Definition:Convex Real Function/Definition 2

$f$ is '''convex on $I$''' {{iff}}: :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \le \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$


Definition:Convex Real Function/Definition 3

$f$ is '''convex on $I$''' {{iff}}: :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \le \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$


Definition:Concave Real Function/Definition 1

$f$ is '''concave on $I$''' {{iff}}: :$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \ge \alpha f \left({x}\right) + \beta f \left({y}\right)$


Definition:Concave Real Function/Definition 2

$f$ is '''concave on $I$''' {{iff}}: :$\displaystyle \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \frac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$


Definition:Concave Real Function/Definition 3

$f$ is '''concave on $I$''' {{iff}}: :$\forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$


Definition:Convex Real Function/Definition 1/Strictly

$f$ is '''strictly convex on $I$''' {{iff}}: :$\forall x, y \in I, x \ne y: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) < \alpha f \left({x}\right) + \beta f \left({y}\right)$


Definition:Convex Real Function/Definition 2/Strictly

$f$ is '''strictly convex on $I$''' {{iff}}: :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} < \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$


Definition:Convex Real Function/Definition 3/Strictly

$f$ is '''strictly convex on $I$''' {{iff}}: :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} < \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$


Definition:Concave Real Function/Definition 1/Strictly

$f$ is '''strictly concave on $I$''' {{iff}}: :$\forall x, y \in I, x \ne y: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) > \alpha f \left({x}\right) + \beta f \left({y}\right)$


Definition:Concave Real Function/Definition 2/Strictly

$f$ is '''strictly concave on $I$''' {{iff}}: :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} > \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$


Definition:Concave Real Function/Definition 3/Strictly

$f$ is '''strictly concave on $I$''' {{iff}}: :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} > \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$


Definition:Beta Function/Definition 1

:$\displaystyle \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$


Definition:Beta Function/Definition 2

:$\displaystyle \Beta \left({x, y}\right) := 2 \int_0^{\pi / 2} \left({\sin \theta}\right)^{2 x - 1} \left({\cos \theta}\right)^{2 y - 1} \rd \theta$


Definition:Beta Function/Definition 3

:$\map \Beta {x, y} := \dfrac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }$ where $\Gamma$ is the Gamma function.


==== Open Above ==== {{:Definition:Improper Integral/Half Open Interval/Open Above}} ==== Open Below ==== {{:Definition:Improper Integral/Half Open Interval/Open Below}}


Definition:Improper Integral/Open Interval

Let $f$ be a real function which is continuous on the open interval $\openint a b$. Then the improper integral of $f$ over $\openint a b$ is defined as: :$\displaystyle \int_{\mathop \to a}^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to b} \int_c^\gamma \map f t \rd t$ for some $c \in \openint a b$.


==== Unbounded Above ==== {{:Definition:Improper Integral/Unbounded Closed Interval/Unbounded Above}} ==== Unbounded Below ==== {{:Definition:Improper Integral/Unbounded Closed Interval/Unbounded Below}}


Definition:Improper Integral/Unbounded Open Interval/Unbounded Above

Let $f$ be a real function which is continuous on the unbounded open interval $\openint a {+\infty}$. Then the improper integral of $f$ over $\openint a {+\infty}$ is defined as: :$\displaystyle \int_{\mathop \to a}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$ for some $c \in \openint a {+\infty}$.


Definition:Improper Integral/Unbounded Open Interval/Unbounded Below

Let $f$ be a real function which is continuous on the unbounded open interval $\openint {-\infty} b$. Then the improper integral of $f$ over $\openint {-\infty} b$ is defined as: :$\displaystyle \int_{\mathop \to -\infty}^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to b} \int_c^\gamma \map f t \rd t$ for some $c \in \openint {-\infty} b$.


Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below

Let $f$ be a real function which is continuous everywhere. Then the improper integral of $f$ over $\R$ is defined as: :$\displaystyle \int_{\mathop \to -\infty}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$ for some $c \in \R$. Usually $c$ is taken to be $0$ as this usually simplifies the evaluation of the expressions.


==== Unbounded Above ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Above}} ==== Unbounded Below ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Below}} A specific and important instance of this occurs when the interval in question is the set of all real numbers: ==== Unbounded Above and Below ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below}}


Definition:Improper Integral/Half Open Interval/Open Above

Let $f$ be a real function which is continuous on the half open interval $\hointr a b$. Then the improper integral of $f$ over $\hointr a b$ is defined as: :$\displaystyle \int_a^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to b} \int_a^\gamma \map f t \rd t$


Definition:Improper Integral/Half Open Interval/Open Below

Let $f$ be a real function which is continuous on the half open interval $\hointl a b$. Then the improper integral of $f$ over $\hointl a b$ is defined as: :$\displaystyle \int_{\mathop \to a}^b \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^b \map f t \rd t$


Definition:Improper Integral/Unbounded Closed Interval/Unbounded Above

Let $f$ be a real function which is continuous on the unbounded closed interval $\hointr a {+\infty}$. Then the improper integral of $f$ over $\hointr a {+\infty}$ is defined as: :$\displaystyle \int_a^{\mathop \to + \infty} \map f t \rd t := \lim_{\gamma \mathop \to +\infty} \int_a^\gamma \map f t \rd t$


Definition:Improper Integral/Unbounded Closed Interval/Unbounded Below

Let $f$ be a real function which is continuous on the unbounded closed interval $\hointl {-\infty} b$. Then the improper integral of $f$ over $\hointl {-\infty} b$ is defined as: :$\displaystyle \int_{\mathop \to -\infty}^b \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^b \map f t \rd t$


Definition:Positive Real Function

Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''positive (on $I$)''' {{iff}}: :$\forall x \in I: f \left({x}\right) \ge 0$


Definition:Negative Real Function

Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''negative (on $I$)''' {{iff}}: :$\forall x \in I: \map f x \le 0$


Definition:Strictly Positive Real Function

Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''strictly positive (on $I$)''' iff: :$\forall x \in I: f \left({x}\right) > 0$


Definition:Strictly Negative Real Function

Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''strictly negative (on $I$)''' {{iff}}: :$\forall x \in I: \map f x < 0$


Definition:Horizontal Line

=== Definition 1 === {{Definition:Horizontal Line/Definition 1}} === Definition 2 === {{Definition:Horizontal Line/Definition 2}} === Definition 3 === {{Definition:Horizontal Line/Definition 3}} File:Horizontal.png


Definition:Vertical Line

=== Definition 1 === {{:Definition:Vertical Line/Definition 1}} === Definition 2 === {{:Definition:Vertical Line/Definition 2}} :::File:Vertical.png


Definition:Laplace Transform

Let $f: \R_{\ge 0} \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \set {\R, \C}$. The '''Laplace transform''' of $f$, denoted $\laptrans f$ or $F$, is defined as: :$\laptrans {\map f t} = \map F s = \displaystyle \int_0^{\to +\infty} e^{-s t} \map f t \rd t$ whenever this improper integral converges. If this improper integral does not converge, then $\laptrans {\map f t}$ does not exist.


Definition:Euler Numbers

The '''Euler Numbers''' $E_n$ are a sequence of integers defined by the exponential generating function: :$\displaystyle \sech x = \frac {2 e^x} {e^{2 x} + 1} = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$ where $\size x < \dfrac \pi 2$.


Definition:Catalan Number

The '''Catalan Numbers''' $C_n$ are a sequence of natural numbers defined by: :$C_n = \dfrac 1 {n + 1} \dbinom {2 n} n $


Definition:Population/Infinite

An '''infinite population''' is a population which is infinite.


Definition:Population/Finite

A '''finite population''' is a population which is finite.


Definition:Statistics

'''Statistics''' is the branch of science which studies the collection and analysis of data, and the presentation of organized inferences from that analysis. It is also concerned with the design of experiments and sample surveys, data reduction, data processing, and various other things. As such, it can be considered as a form of applied probability theory. It has been defined as '''the technology of the scientific method'''.


Definition:Inductive Statistics

'''Inductive statistics''' is the phase of statistics which is concerned with the conditions under which conclusions about populations can be drawn from analysis of particular samples.


Definition:Descriptive Statistics

'''Descriptive statistics''' is the phase of statistics whose task is to collect, analyze and summarize data about statistical samples, without drawing conclusions or inferences about the populations from which those samples are taken.


Definition:Variable/Domain

The collection of all possible objects that a variable may refer to has to be specified. This collection is the '''domain''' of the variable.


Definition:Variable/Continuous

A '''continuous variable''' is a variable which can take any value between two given values.


Definition:Variable/Discrete

A '''discrete variable''' is a variable which is not continuous.


Definition:Sample Statistic/Discrete

Data which can be described with a discrete variable are known as '''discrete data'''.


Definition:Sample Statistic/Continuous

Data which can be described with a continuous variable are known as '''continuous data'''.


Definition:Measurable Property/Measurement

'''Measurement''' is the process of determining the quantity of a measurable property. A '''measurement''' is reported as a (real) number multiplied by a unit of measurement for that quantity.


Definition:Rounding

'''Rounding''' is the process of approximation of a value of a variable to a multiple of a given power of whatever number base one is working in (usually decimal). Let $n \in \Z$ be an integer. Let $x \in \R$ be a real number. Let $y \in \R$ such that: :$y = 10^n \floor {\dfrac x {10^n} + \dfrac 1 2}$ or: :$y = 10^n \ceiling {\dfrac x {10^n} - \dfrac 1 2}$ where $\floor {\, \cdot \,}$ denotes the floor function and $\ceiling {\, \cdot \,}$ denotes the ceiling function. Then $y$ is defined as '''$x$ rounded to the nearest $n$th power of $10$'''. Both of these definitions amount to the same thing, except for when $\dfrac x {10^n}$ is exactly halfway between $\floor {\dfrac x {10^n} }$ and $\ceiling {\dfrac x {10^n} }$. How these instances is treated is known as the '''treatment of the half'''.


Definition:Derivative/Higher Derivatives/Second Derivative

Let $f$ be a real function which is differentiable on an open interval $I$. Hence $f'$ is defined on $I$ as the derivative of $f$. Let $\xi \in I$ be a point in $I$. Let $f'$ be differentiable at the point $\xi$. Then the '''second derivative''' $\map {f''} \xi$ is defined as: :$\displaystyle f'' := \lim_{x \mathop \to \xi} \dfrac {\map {f'} x - \map {f'} \xi} {x - \xi}$


Definition:Derivative/Higher Derivatives/Third Derivative

Let $f$ be a real function which is twice differentiable on an open interval $I$. Let $f''$ denote the second derivate. Then the '''third derivative''' $f'''$ is defined as: :$f''' := \dfrac {\d} {\d x} f'' = \map {\dfrac {\d} {\d x} } {\dfrac {\d^2} {\d x^2} f}$


Definition:Derivative/Higher Derivatives/Higher Order

The $n$th derivative of a function $y = \map f x$ is defined as: :$\map {f^{\paren n} } x = \dfrac {\d^n y} {\d x^n} := \begin {cases} \map {\dfrac \d {\d x} } {\dfrac {\d^{n - 1} y} {\d x^{n - 1} } } & : n > 0 \\ y & : n = 0 \end {cases}$ assuming appropriate differentiability for a given $f^{\paren {n - 1} }$.


Definition:Derivative/Higher Derivatives/Zeroth Derivative

The '''zeroth derivative''' of a real function $f$ is defined as $f$ itself: :$f^{\paren 0} := f$ where $f^{\paren n}$ denotes the $n$th derivative of $f$.


Definition:Dirichlet Eta Function

The '''Dirichlet $\eta$ (eta) function''' $\eta$ is the complex function defined on the half-plane $\map \Re s > 0$ as the series: :$\displaystyle \map \eta s = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} n^{-s}$


Definition:Partial Derivative/Second Derivative

Let $\map f {x, y}$ be a function of the two independent variables $x$ and $y$. The '''second partial derivatives of $f$ with respect to $x$ and $y$''' are defined and denoted by: {{begin-eqn}} {{eqn | n = 1       | l = \dfrac {\partial^2 f} {\partial x^2}       | r = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial x} }       | rr= =: \map {f_{1 1} } {x, y}       | c =  }} {{eqn | n = 2       | l = \dfrac {\partial^2 f} {\partial y^2}       | r = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial y} }       | rr= =: \map {f_{2 2} } {x, y}       | c =  }} {{eqn | n = 3       | l = \quad \dfrac {\partial^2 f} {\partial x \partial y}       | r = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial y} }       | rr= =: \map {f_{2 1} } {x, y}       | c =  }} {{eqn | n = 4       | l = \dfrac {\partial^2 f} {\partial y \partial x}       | r = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial x} }       | rr= =: \map {f_{1 2} } {x, y}       | c =  }} {{end-eqn}}


Definition:Primitive (Calculus)/Integration

The process of finding a primitive for a function is known as '''integration'''.


Definition:Bernoulli Numbers/Generating Function

:$\displaystyle \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$


Definition:Bernoulli Numbers/Recurrence Relation

:$B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n + 1 - k} & : n > 0 \end{cases}$ or equivalently: :$B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \frac 1 {n+1} \sum_{k \mathop = 0}^{n - 1} \binom {n+1} k B_k & : n > 0 \end{cases}$


Definition:Logarithmic Integral/Eulerian

The '''Eulerian logarithmic integral''' is defined as: :$\displaystyle \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$


Definition:Heaviside Step Function

Let $c \ge 0$ be a constant real number. The '''Heaviside step function on $c$''' is the real function $u_c: \R \to \R$ defined as: :$\map {u_c} t := \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$ If $c = 0$, the subscript is often omitted: :$\map u t := \begin{cases} 1 & : t > 0 \\ 0 & : t < 0 \end{cases}$


Definition:Doubly Periodic Function

Let $f: \C \to \C$ be a complex function.  Then $f \left({z}\right)$ is a '''doubly-periodic function''' if there exist $\omega_1, \omega_2 \in \C$ such that: : $(1): \quad \omega_1, \omega_2 \ne 0$ : $(2): \quad \dfrac {\omega_1} {\omega_2} \notin \R$ : $(3): \quad \forall z \in \C: f \left({z}\right) = f \left({z + \omega_1}\right) = f \left({z + \omega_2}\right)$


Definition:Elliptic Function

Let $\displaystyle y \left({x}\right) = \int_0^x \dfrac {\d t} {\sqrt {P \left({t}\right)} }$ be an elliptic integral, where $P \left({t}\right)$ is a polynomial of degree $3$ or $4$. Consider the inverse of $y \left({x}\right)$: :$x = \phi \left({y}\right)$ Then $\phi$ is an '''elliptic function'''.


Definition:Quadratic Function

A '''quadratic function''' is an expression of the form: :$\map Q x := a_0 + a_1 x + a_2 x^2$ where $a_0, a_1, a_2$ are constants.


Definition:Exponential Order/Real Index

Let $e^{a t}$ be the exponential function, where $a \in \R$ is constant. Then $\map f t$ is said to be of '''exponential order''' $a$, denoted $f \in \mathcal E_a$, {{iff}} there exist strictly positive real numbers $M, K$ such that: :$\forall t \ge M: \size {\map f t} < K e^{a t}$


=== Definition 1 === {{:Definition:Minimal Infinite Successor Set/Definition 1}} === Definition 2 === {{:Definition:Minimal Infinite Successor Set/Definition 2}} === Definition 3 === {{:Definition:Minimal Infinite Successor Set/Definition 3}}


Definition:Step Function

A real function $f: \R \to \R$ is a '''step function''' {{iff}} it can be expressed as a finite linear combination of the form: :$\map f x = \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + \cdots + \lambda_n \chi_{\mathbb I_n}$ where: :$\lambda_1, \lambda_2, \ldots, \lambda_n$ are real constants :$\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$ are open intervals, where these intervals partition $\R$ (except for the endpoints) :$\chi_{\mathbb I_1}, \chi_{\mathbb I_2}, \ldots, \chi_{\mathbb I_n}$ are characteristic functions of $\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$.


Definition:Limit Inferior of Sequence of Sets/Notation

The limit inferior of $E_n$ can also be seen denoted as: :$\displaystyle {\underline {\lim} }_{n \mathop \to \infty} \ E_n$ but this notation is not used on {{ProofWiki}} because it does not render well. Some sources merely present this as: :$\ds \underline \lim E_n$


Definition:Limit Superior of Sequence of Sets/Notation

The limit superior of $E_n$ can also be seen denoted as: :$\ds \overline {\lim}_{n \mathop \to \infty} \ E_n$ but this notation is not used on {{ProofWiki}} because it does not render well. Some sources merely present this as: :$\ds \overline \lim E_n$


Definition:Gamma Function/Graph

The graph of the Gamma function is illustrated here for real arguments. <!--thumbright400pxThe Gamma function: $\Gamma \left({z}\right)$ (red solid line) and $\dfrac 1 {\Gamma\left({z}\right)}$ (blue broken line)--> :800pxThe Gamma function: $\Gamma \left({z}\right)$ (red solid line) and $\dfrac 1 {\Gamma\left({z}\right)}$ (blue broken line) The Gamma function: :$\Gamma \left({z}\right)$ (red solid line) :$\dfrac 1 {\Gamma \left({z}\right)}$ (blue broken line)


Definition:O Notation/Big-O Notation/Implied Constant

From the definition of the limit of a function, it can be seen that this is also equivalent to: :$\exists c \in \R: c > 0, k \ge 0: \forall n > k, f \left({n}\right) \le c g \left({n}\right)$ For some fixed $k$ (appropriate to the function under consideration) the infimum of such $c$ is called the '''implied constant'''.


Definition:Arithmetic Series

An '''arithmetic series''' is a series whose underlying sequence is an arithmetic sequence: {{begin-eqn}} {{eqn | l = S_n       | r = \sum_{k \mathop = 0}^{n - 1} a + k d       | c =  }} {{eqn | r = a + \paren {a + d} + \paren {a + 2 d} + \cdots + \paren {a + \paren {n - 1} d}       | c =  }} {{end-eqn}}


Definition:Arithmetic Sequence/Common Difference

The term $d$ is the '''common difference''' of $\sequence {a_k}$.


Definition:Arithmetic-Geometric Sequence

An '''arithmetic-geometric sequence''' is a sequence $\sequence {a_k}$ in $\R$ defined as: :$a_k = \paren {a_0 + k d} r^k$ for $k = 0, 1, 2, \ldots$ Thus its general form is: :$a_0, \paren {a_0 + d} r, \paren {a_0 + 2 d} r^2, \paren {a_0 + 3 d} r^3, \ldots$


Definition:Arithmetic-Geometric Series

An '''arithmetic-geometric series''' is a series whose underlying sequence is an arithmetic-geometric sequence: {{begin-eqn}} {{eqn | l = S_n       | r = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k       | c =  }} {{eqn | r = a + \paren {a + d} r + \paren {a + 2 d} r^2 + \cdots + \paren {a + \paren {n - 1} d}r^{n-1}       | c =  }} {{end-eqn}}


Definition:Arithmetic Sequence/Last Term

The term $a_{n-1} = a_0 + \paren {n - 1} d$ is the '''last term''' of $\sequence {a_k}$.


Definition:Geometric Mean/Mean Proportional

In the language of {{AuthorRef|Euclid}}, the geometric mean of two magnitudes is called the '''mean proportional'''. Thus the '''mean proportional''' of $a$ and $b$ is defined as that magnitude $c$ such that: :$a : c = c : b$ where $a : c$ denotes the ratio between $a$ and $c$.


Definition:SI Units/Base Units

{| class="wikitable" style="margin:1em auto 1em auto" |+ SI base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! metre | $\mathrm m$ | $\mathsf L$: Length | $l$ |- ! kilogram | $\mathrm {kg}$ | $\mathsf M$: Mass | $m$ |- ! second | $\mathrm s$ | $\mathsf T$: Time | $t$ |- ! ampere | $\mathrm A$ | $\mathsf I$: Electric Current | $I$ |- ! kelvin | $\mathrm K$ | $\Theta$: Temperature | $T$ |- ! candela | $\mathrm {cd}$ | $\mathsf J$: Luminous Intensity | $I_v$ |- ! mole | $\mathrm {mol}$ | $\mathsf N$: Amount of Substance | $n$ |- |}


A different definition of the Bernoulli numbers can be found in older literature. Usually denoted with the symbol ${B_n}^*$, they are considered archaic, and will not be used on {{ProofWiki}}. === Definition 1 === {{:Definition:Bernoulli Numbers/Archaic Form/Definition 1}} === Definition 2 === {{:Definition:Bernoulli Numbers/Archaic Form/Definition 2}}


Definition:Planar Graph/Face

The '''faces''' of a planar graph are the areas which are surrounded by edges. In the above, the '''faces''' are $ABHC$, $CEGH$, $ACD$, $CDFE$ and $ADFEGHIHB$. === Incident === {{:Definition:Incident (Graph Theory)/Planar Graph}} === Adjacent === {{:Definition:Adjacent (Graph Theory)/Faces}} In the above diagram, $ABHC$ and $ACD$ are adjacent, but $ABHC$ and $CDFE$ are ''not'' adjacent.


Definition:Tangent Map/Affine Transformation

Let $\mathcal E$ and $\mathcal F$ be affine spaces with difference spaces $E$ and $F$ respectively. Let $\mathcal L : \mathcal E \to \mathcal F$ be an affine transformation. The associated linear transformation $L: E \to F$ of difference spaces is called the '''tangent map''' of $\mathcal L$.


Definition:Tangent Map/Differentable Mappings

{{stub}}


Definition:Cube

=== Geometry === {{:Definition:Cube/Geometry}} === Algebra === {{:Definition:Cube/Algebra}}