hcju/proofwikips huggingface dataset:
De Moivre's Formula/Negative Integer Index
Let $z \in \C$ be a complex number expressed in complex form: :$z = r \left({\cos x + i \sin x}\right)$ Then: :$\forall n \in \Z_{\le 0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$
De Moivre's Formula/Rational Index
Let $z \in \C$ be a complex number expressed in complex form: :$z = r \paren {\cos x + i \sin x}$ Then: :$\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$
Dot Product Operator is Commutative
:$\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u$
Dot Product with Self is Non-Negative
:$\mathbf u \cdot \mathbf u \ge 0$
Dot Product with Self is Zero iff Zero Vector
:$\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$
Vector Cross Product is Anticommutative/Complex
:$\forall z_1, z_2 \in \C: z_1 \times z_2 = -\paren {z_2 \times z_1}$
Polar Form of Complex Conjugate
Let $z := r \left({\cos \theta + i \sin \theta}\right) \in \C$ be a complex number expressed in polar form. Then: :$\overline z = r \left({\cos \theta - i \sin \theta}\right)$ where $\overline z$ denotes the complex conjugate of $z$.
Exponential Form of Complex Conjugate
Let $z := r e^{i \theta} \in \C$ be a complex number expressed in exponential form. Then: :$\overline z = r e^{-i \theta}$ where $\overline z$ denotes the complex conjugate of $z$.
Complex Dot Product in Exponential Form
Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be complex numbers expressed in exponential form. Then: :$z_1 \circ z_2 = r_1 r_2 \, \map \cos {\theta_2 - \theta_1}$ where $z_1 \circ z_2$ denotes the dot product of $z_1$ and $z_2$.
Complex Cross Product in Exponential Form
Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be complex numbers expressed in exponential form. Then: :$z_1 \times z_2 = r_1 r_2 \map \sin {\theta_2 - \theta_1}$ where $z_1 \times z_2$ denotes the dot product of $z_1$ and $z_2$.
Dot Product Distributes over Addition
:$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$
Dot Product Operator is Bilinear
:$\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$
Dot Product Associates with Scalar Multiplication
:$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$
Semantic Tableau Algorithm Terminates
Let $\mathbf A$ be a WFF of propositional logic. Then the Semantic Tableau Algorithm for $\mathbf A$ terminates. Each leaf node of the resulting semantic tableau is marked.
Complement of Closed Set in Complex Plane is Open
Let $S \subseteq \C$ be a closed subset of the complex plane $\C$. Then the complement of $S$ in $\C$ is open.
Condition for Quartic with Real Coefficients to have Wholly Imaginary Root
Let $Q$ be the quartic equation: :$(1): \quad z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ such that all of $a_1, a_2, a_3, a_4$ are real numbers. Then $Q$ has a root which is wholly imaginary : :$\text {(a)}: \quad a_3^2 + a_1^2 a_4 = a_1 a_2 a_3$ :$\text {(b)}: \quad a_1 a_3 > 0$
Soundness and Completeness of Semantic Tableaus
Let $\mathbf A$ be a WFF of propositional logic. Let $T$ be a completed semantic tableau for $\mathbf A$. Then $\mathbf A$ is unsatisfiable  $T$ is closed.
Semantic Tableau Algorithm is Decision Procedure for Tautologies
The Semantic Tableau Algorithm is a decision procedure for tautologies.
Soundness Theorem for Semantic Tableaus
Let $\mathbf A$ be a WFF of propositional logic. Let $T$ be a completed tableau for $\mathbf A$. Suppose that $T$ is closed. Then $\mathbf A$ is unsatisfiable for boolean interpretations.
Completeness Theorem for Semantic Tableaus
Let $\mathbf A$ be a WFF of propositional logic. Let $\mathbf A$ be unsatisfiable for boolean interpretations. Then every completed tableau for $\mathbf A$ is closed.
Cosine to Power of Even Integer/Proof 2
Let $n \in \Z$ be an even integer. Then: :$\displaystyle \cos^n \theta = \frac 1 {2^{n - 1} } \sum_{k \mathop = 0}^{n / 2} \paren {\binom n k \cos \paren {n - 2 k} \theta}$ That is: :$\cos^n \theta = \dfrac 1 {2^{n - 1} } \paren {\cos n \theta + n \cos \paren {n - 2} \theta + \dfrac {n \paren {n - 1} } 2 \cos \paren {n - 4} \theta + \cdots + \dfrac {n!} {\paren {\paren {\frac n 2}!}^2} \cos \theta}$
Conjugate of Real Polynomial is Polynomial in Conjugate
Let $\map P z$ be a polynomial in a complex number $z$. Let the coefficients of $P$ all be real. Then: :$\overline {\map P z} = \map P {\overline z}$ where $\overline z$ denotes the complex conjugate of $z$.
Equation for Line through Two Points in Complex Plane
Let $z_1, z_2 \in \C$ be complex numbers. Let $L$ be a straight line through $z_1$ and $z_2$ in the complex plane. === Formulation 1 === {{:Equation for Line through Two Points in Complex Plane/Formulation 1}} === Parametric Form $1$ === {{:Equation for Line through Two Points in Complex Plane/Parametric Form 1}} === Parametric Form $2$ === {{:Equation for Line through Two Points in Complex Plane/Parametric Form 2}} === Symmetric Form === {{:Equation for Line through Two Points in Complex Plane/Symmetric Form}}
Absolute Value of Components of Complex Number no greater than Root 2 of Modulus
Let $z = x + i y \in \C$ be a complex number. Then: :$\size x + \size y \le \sqrt 2 \cmod z$ where: :$\size x$ and $\size y$ denote the absolute value of $x$ and $y$ :$\cmod z$ denotes the complex modulus of $z$.
Exclusive Or with Factor of Exclusive Or
:$\left({p \oplus q}\right) \oplus q \dashv \vdash p$
Biconditional with Factor of Biconditional
:$\paren {p \iff q} \iff q \dashv \vdash p$
Conditional iff Biconditional of Consequent with Disjunction
:$p \implies q \dashv \vdash q \iff \left({p \lor q}\right)$
Functionally Incomplete Logical Connectives
These sets of logical connectives are ''not'' functionally complete.
Functionally Incomplete Logical Connectives/Conjunction and Disjunction
:$\set {\land, \lor}$: And and Or
Functionally Incomplete Logical Connectives/Negation and Biconditional
:$\set {\neg, \iff}$: Not and Iff
Theory of Set of Formulas is 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. Let $T \left({\mathcal F}\right)$ be the $\mathcal L$-theory of $\mathcal F$. Then $T \left({\mathcal F}\right)$ is a theory.
Semantic Consequence is Transitive
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F, \mathcal G$ and $\mathcal H$ be sets of $\mathcal L$-formulas. Suppose that: :$\mathcal F \models_{\mathscr M} \mathcal G$ :$\mathcal G \models_{\mathscr M} \mathcal H$ Then $\mathcal F \models_{\mathscr M} \mathcal H$.
Lemniscate of Bernoulli as Locus in Complex Plane
The locus of $z$ on the complex plane such that: :$\cmod {z - a} \cmod {z - a} = a^2$ is a lemniscate of Bernoulli.
Sum of Cosines of Arithmetic Sequence of Angles
Let $\alpha \in \R$ be a real number such that $\alpha \ne 2 \pi k$ for $k \in \Z$. Then:
Sum of Sines of Arithmetic Sequence of Angles
Let $\alpha \in \R$ be a real number such that $\alpha \ne 2 \pi k$ for $k \in \Z$. Then:
Product of Tangent and Cotangent
:$\tan \theta \cot \theta = 1$
Product of Cotangents of Fractions of Pi
Let $m \in \Z$ such that $m > 1$. Then: :$\displaystyle \prod_{k \mathop = 1}^{m - 1} \cot \frac {k \pi} {2 m} = 1$
Uncountable Set less Countable Set is Uncountable
Let $S$ be an uncountable set. Let $T \subseteq S$ be a countable subset of $S$. Then: :$S \setminus T$ is uncountable where $\setminus$ denotes set difference.
Irrational Numbers are Uncountably Infinite
The set $\R \setminus \Q$ of irrational numbers is uncountable.
Product of Diagonals from Point of Regular Polygon
Let $A_0, A_1, \ldots, A_{n - 1}$ be the vertices of a regular $n$-gon $P = A_0 A_1 \cdots A_{n - 1}$ which is circumscribed by a unit circle. Then: :$\displaystyle \prod_{k \mathop = 2}^{n - 2} A_0 A_k = \frac {n \csc^2 \frac \pi n} 4$ where $A_0 A_k$ is the length of the line joining $A_0$ to $A_k$.
Condition on Conjugate from Real Product of Complex Numbers
Let $z_1, z_2 \in \C$ be complex numbers such that $z_1 z_2 \in \R_{\ne 0}$. Then: :$\exists p \in \R: z_1 = p \overline {z_2}$ where $\overline {z_2}$ denotes the complex conjugate of $z_2$.
Exponential of Sum
=== Real Numbers === {{:Exponential of Sum/Real Numbers}} === Complex Numbers === {{:Exponential of Sum/Complex Numbers}} == Corollary == {{:Exponential of Sum/Corollary}}
Cayley-Menger Determinant
Let $S$ be a $j$-simplex in $\R^n$. Let the vertices of $S$ be $v_1, v_2, \ldots, v_{j+1}$. Let $B = \left[{\beta_{ij} }\right]$ denote the $\left({j + 1}\right) \times \left({j + 1}\right)$ matrix given by: :$\beta_{ij} = \left\vert{v_i - v_j}\right\vert_2^2$ where $\left\vert{v_i - v_j}\right\vert_2$ is the vector 2-norm of the vector $v_i - v_j$. Then the content $V_j$ is given by: {{begin-eqn}} {{eqn | l = V_j^2 \left({S}\right)         | r = \frac {\left({-1}\right)^{j+1} } {2^j \left({j!}\right)^2} \det C  }} {{end-eqn}} where $C$ is the $\left({j + 2}\right) \times \left({j + 2}\right)$ matrix obtained from $B$ by bordering $B$ with a top row $\left({0, 1, \ldots, 1}\right)$ and a left column $\left({0, 1, \ldots, 1}\right)^\intercal$.
Rule of Commutation/Disjunction/Formulation 2/Forward Implication
:$\vdash \left({p \lor q}\right) \implies \left({q \lor p}\right)$
Provable by Gentzen Proof System iff Negation has Closed Tableau
Let $\mathscr G$ be instance 1 of a Gentzen proof system.
Provable by Gentzen Proof System iff Negation has Closed Tableau/Formula
Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is a $\mathscr G$-theorem iff: :$\neg \mathbf A$ has a closed semantic tableau where $\neg \mathbf A$ is the negation of $\mathbf A$.
Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas
Let $U$ be a set of WFFs of propositional logic. Then $U$ is a $\mathscr G$-theorem iff: :$\bar U$ has a closed semantic tableau where $\bar U$ is the set comprising the logical complements of all WFFs in $U$.
Domain of Real Natural Logarithm
Let $\ln$ be the natural logarithm function on the real numbers. Then the domain of $\ln$ is the set of strictly positive real numbers: :$\Dom \ln = \R_{>0}$
Image of Real Natural Logarithm
Let $\ln$ be the natural logarithm function on the real numbers. Then the image of $\ln$ is the set of real numbers: :$\Img \ln = \R$
Soundness and Completeness of Gentzen Proof System
Let $\mathscr G$ be instance 1 of a Gentzen proof system. Let $\mathrm{BI}$ be the formal semantics of boolean interpretations. Then $\mathscr G$ is a sound and complete proof system for $\mathrm{BI}$.
Definition:Deduction Rule
Let $\mathcal L$ be the language of propositional logic. The '''Deduction Rule''' is the rule of inference: :From $U, \mathbf A \vdash \mathbf B$, one may infer $U \vdash \mathbf A \implies \mathbf B$ where: :$\mathbf A, \mathbf B$ are WFFs of propositional logic :$U$ is a set of WFFs For a given proof system, the '''Deduction Rule''' can be either a basic rule of inference, or a derived rule.
Deduction Theorem
Let $\mathscr H$ be instance 1 of a Hilbert proof system. Then the deduction rule: ::$\dfrac{U,\mathbf A \vdash \mathbf B}{U \vdash \mathbf A \implies \mathbf B}$ is a derived rule for $\mathscr H$.
Soundness Theorem for Hilbert Proof System
Let $\mathscr H$ be instance 1 of a Hilbert proof system. Let $\mathrm{BI}$ be the formal semantics of boolean interpretations. Then $\mathscr H$ is a sound proof system for $\mathrm{BI}$: :Every $\mathscr H$-theorem is a tautology.
Equivalence of Definitions of Dominate (Set Theory)
Let $S, T$ be sets. {{TFAE|def = Dominate (Set Theory)|view = Dominate|context = Set Theory}}
Element of Toset has at most One Immediate Successor
Let $\struct {S, \preceq}$ be a toset. Let $a \in S$. Then $a$ has at most one immediate successor.
Element of Toset has at most One Immediate Predecessor
Let $\struct {S, \preceq}$ be a toset. Let $a \in S$. Then $a$ has at most one immediate predecessor.
Equivalence of Definitions of Complex Inverse Tangent Function
{{TFAE|def = Complex Inverse Tangent}} Let $S$ be the subset of the complex plane: :$S = \C \setminus \set {0 + i, 0 - i}$
Equivalence of Definitions of Complex Inverse Cotangent Function
{{TFAE|def = Complex Inverse Cotangent}} Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{0 + i, 0 - i}\right\}$
Square Root of Complex Number in Cartesian Form
Let $z \in \C$ be a complex number. Let $z = x + i y$ where $x, y \in \R$ are real numbers. Let $z$ not be wholly real, that is, such that $y \ne 0$. Then the square root of $z$ is given by: :$z^{1/2} = \pm \paren {a + i b}$ where: {{begin-eqn}} {{eqn | l = a       | r = \sqrt {\frac {x + \sqrt {x^2 + y^2} } 2}       | c =  }} {{eqn | l = b       | r = \frac y {\cmod y} \sqrt {\frac {-x + \sqrt {x^2 + y^2} } 2}       | c =  }} {{end-eqn}}
Zero is Identity in Naturally Ordered Semigroup
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $0$ be the zero of $\struct {S, \circ, \preceq}$. Then $0$ is the identity for $\circ$. That is: :$\forall n \in S: n \circ 0 = n = 0 \circ n$
Reduction Formula for Integral of Power of Sine
Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Then: :$\displaystyle \int \sin^n x \rd x = \dfrac {n - 1} n \int \sin^{n - 2} x \rd x - \dfrac {\sin^{n - 1} x \cos x} n$ is a reduction formula for $\displaystyle \int \sin^n x \rd x$.
Reduction Formula for Definite Integral of Power of Sine
Let $n \in \Z_{> 0}$ be a positive integer. Let $I_n$ be defined as: :$\displaystyle I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$ Then $\left\langle{I_n}\right\rangle$ is a decreasing sequence of real numbers which satisfies: :$n I_n = \left({n - 1}\right) I_{n - 2}$ Thus: :$I_n = \dfrac {n - 1} n  I_{n - 2}$ is a reduction formula for $I_n$.
Stirling's Formula/Proof 2/Lemma 4
Let $I_n$ be defined as: :$\displaystyle I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$ Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac {I_{2 n} } {I_{2 n + 1} } = 1$
Definite Integral from 0 to Half Pi of Even Power of Sine x
:$\displaystyle \int_0^{\frac \pi 2} \sin^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$
Definite Integral from 0 to Half Pi of Odd Power of Sine x
:$\displaystyle \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$
Zero Complement is Not Empty
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $S^*$ be the zero complement of $S$. Then $S^*$ is not empty.
Stirling's Formula/Proof 2/Lemma 5
:$\dfrac {n!} {n^n \sqrt n e^{-n} } \to \sqrt {2 \pi}$ as $n \to \infty$
Limit of Error in Stirling's Formula
Consider Stirling's Formula: :$n! \sim \sqrt {2 \pi n} \left({\dfrac n e}\right)^n$ The ratio of $n!$ to its approximation $\sqrt {2 \pi n} \left({\dfrac n e}\right)^n$ is bounded as follows: :$e^{1 / \left({12 n + 1}\right)} \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$
Zero Strictly Precedes One
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $0$ be the zero of $S$. Let $1$ be the one of $S$. Then: :$0 \prec 1$
Stirling's Formula/Proof 2/Lemma 2
The sequence $\left \langle {d_n} \right \rangle$ defined as: : $\displaystyle d_n = \ln \left({n!}\right) - \left({n + \frac 1 2}\right) \ln n + n$ is decreasing.
Power Series Expansion for Real Arctangent Function
The arctangent function has a Taylor series expansion: :$\arctan x = \begin {cases} \displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \displaystyle \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \ge 1 \\ \displaystyle -\frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \le -1 \end {cases}$ That is: :$\arctan x = \begin {cases} \displaystyle x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots & : -1 \le x \le 1 \\ \displaystyle \frac \pi 2 - \frac 1 x + \frac 1 {3 x^3} - \frac 1 {5 x^5} + \cdots & : x \ge 1 \\ \displaystyle -\frac \pi 2 - \frac 1 x + \frac 1 {3 x^3} - \frac 1 {5 x^5} + \cdots & : x \le -1 \end {cases}$
Ordering in terms of Addition
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Then $\forall m, n \in S$:  :$m \preceq n \iff \exists p \in S: m \circ p = n$
Difference in Naturally Ordered Semigroup is Unique
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $n, m \in S$ such that $m \preceq n$. Then there exists a unique difference $n - m$ of $m$ and $n$.
Equivalence of Definitions of Beta Function
{{TFAE|def = Beta Function}} For $\map \Re x, \map \Re y > 0$:
Ordering of Naturally Ordered Semigroup is Strongly Compatible
:$\forall m, n, p \in S: m \preceq n \iff m \circ p \preceq n \circ p$
Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible
:$\forall m, n, p \in S: m \prec n \iff m \circ p \prec n \circ p$
Sum with One is Immediate Successor in Naturally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $1$ be the one of $S$. Let $n \in S$. Then $n \circ 1$ is the immediate successor of $n$. That is, for all $m \in S$: :$n \prec m \iff n \circ 1 \preceq m$
Riemann Zeta Function at Even Integers
The Riemann $\zeta$ function can be calculated for even integers as follows: {{begin-eqn}} {{eqn | l = \map \zeta {2 n}       | r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = \frac 1 {1^{2 n} } + \frac 1 {2^{2 n} } + \frac 1 {3^{2 n} } + \frac 1 {4^{2 n} } + \cdots       | c =  }} {{end-eqn}} where: :$B_n$ are the Bernoulli numbers :$n$ is a positive integer.
Tangent Inequality
:$x < \tan x$ for all $x$ in the interval $\left({0 \,.\,.\, \dfrac {\pi} 2}\right)$.
Integral Representation of Riemann Zeta Function in terms of Gamma Function
For $\Re \paren s > 1$, the Riemann Zeta function is given by: :$\displaystyle \map \zeta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {t^{s - 1}} {e^t - 1} \rd t$ where $\Gamma$ is the Gamma function.
Gamma Function is Unique Extension of Factorial
Let $f: \R_{>0} \to \R$ be a real function which is positive and continuous. Let $\ln \mathop \circ f$ be convex on $\R_{>0}$. Let $f$ satisfy the conditions: :$f \left({x + 1}\right) = \begin{cases} 1 & : x = 0 \\ x f \left({x}\right) & : x > 0 \end{cases}$ Then: : $\forall x \in \R_{>0}: f \left({x}\right) = \Gamma \left({x}\right)$ where $\Gamma \left({x}\right)$ is the Gamma function.
Equivalence of Definitions of Convex Real Function
Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Convex Real Function}}
Power Series Expansion for Real Arcsine Function
The (real) arcsine function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \arcsin x       | r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = x + \frac 1 2 \frac {x^3} 3 + \frac {1 \times 3} {2 \times 4} \frac {x^5} 5 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} \frac {x^7} 7 + \cdots       | c =  }} {{end-eqn}} which converges for $-1 \le x \le 1$.
Power Series Expansion for Real Arccosine Function
The arccosine function has a Taylor Series expansion: {{begin-eqn}} {{eqn | l = \arccos x       | r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = \frac \pi 2 - \paren {x + \frac 1 2 \frac {x^3} 3 + \frac {1 \times 3} {2 \times 4} \frac {x^5} 5 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} \frac {x^7} 7 + \cdots}       | c =  }} {{end-eqn}} which converges for $-1 \le x \le 1$.
Real Function is Concave iff its Negative is Convex
Let $f$ be a real function. Let $I \subseteq \R$ be an interval of $\R$. Then $f$ is concave on $I$  $-f$ is convex on $\R$.
Concave Real Function is Left-Hand and Right-Hand Differentiable
Let $f$ be a real function which is concave on the open interval $\left({a \,.\,.\, b}\right)$. Then the left-hand derivative $f'_- \left({x}\right)$ and right-hand derivative $f'_+ \left({x}\right)$ both exist for all $x \in \left({a \,.\,.\, b}\right)$.
Absolute Value Function is Convex
Let $f: \R \to \R$ be the absolute value function on the real numbers. Then $f$ is convex.
Real Function is Concave iff Derivative is Decreasing
Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Then $f$ is concave on $\openint a b$  its derivative $f'$ is decreasing on $\openint a b$. Thus the intuitive result that a concave function "gets less steep".
Real Function is Strictly Convex iff Derivative is Strictly Increasing
Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Then $f$ is strictly convex on $\openint a b$  its derivative $f'$ is strictly increasing on $\openint a b$.
Real Function is Strictly Concave iff Derivative is Strictly Decreasing
Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Then $f$ is strictly concave on $\openint a b$  its derivative $f'$ is strictly decreasing on $\openint a b$.
Inverse of Strictly Decreasing Convex Real Function is Convex
Let $f$ be a real function which is convex on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is convex on $J$.
Inverse of Strictly Increasing Concave Real Function is Convex
Let $f$ be a real function which is concave on the open interval $I$. Let $J = f \sqbrk I$. If $f$ is strictly increasing on $I$, then $f^{-1}$ is convex on $J$.
Inverse of Strictly Decreasing Concave Real Function is Concave
Let $f$ be a real function which is concave on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is concave on $J$.
Mean Value of Concave Real Function
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Let $f$ be concave on $\left({a \,.\,.\, b}\right)$. Then: : $\forall \xi \in \left({a \,.\,.\, b}\right): f \left({x}\right) - f \left({\xi}\right) \le f^{\prime} \left({\xi}\right) \left({x - \xi}\right)$
Differentiable Bounded Concave Real Function is Constant
Let $f$ be a real function which is: :$(1): \quad$ Differentiable on $\R$ :$(2): \quad$ Bounded on $\R$ :$(3): \quad$ Concave on $\R$. Then $f$ is constant.
Second Derivative of Strictly Concave Real Function is Strictly Negative
Let $f$ be a real function which is twice differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Then $f$ is strictly concave on $\left({a \,.\,.\, b}\right)$  its second derivative $f'' < 0$ on $\left({a \,.\,.\, b}\right)$.
Second Derivative of Strictly Convex Real Function is Strictly Positive
Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$. Then $f$ is strictly convex on $\openint a b$  its second derivative $f'' > 0$ on $\openint a b$.
Inverse of Strictly Decreasing Strictly Concave Real Function is Strictly Concave
Let $f$ be a real function which is strictly concave on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is strictly concave on $J$.
Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave
Let $f$ be a real function which is strictly convex on the open interval $I$. Let $J = f \sqbrk I$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is strictly concave on $J$.
Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex
Let $f$ be a real function which is strictly concave on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is strictly convex on $J$.
Gamma Function as Integral of Natural Logarithm
Let $x \in \R_{>0}$ be a strictly positive real number. Then: :$\displaystyle \map \Gamma x = \int_{\to 0}^1 \paren {\ln \frac 1 t}^{x - 1} \rd t$ where $\Gamma$ denotes the Gamma function.
Reversal of Limits of Definite Integral
Let $a \le b$. Then: :$\displaystyle \int_a^b \map f x \rd x = - \int_b^a \map f x \rd x$