nyuuzyou/wikis huggingface dataset:
Properties of Class of All Ordinals/Superinduction Principle

== Theorem ==
Let $\On$ denote the class of all ordinals.
Let $A$ be a class which satisfies the following $3$ conditions:
That is, let $A$ be a superinductive class under the successor mapping.
Then $A$ contains all ordinals:
:$\On \subseteq A$
== Proof ==
We note that the zero ordinal, denoted $0$, is identified as the empty set:
:$0 \:= \O$
Hence by definition $A$ is indeed a superinductive class under the successor mapping.
By the definition of ordinal:
:$\alpha$ is an ordinal $\alpha$ is an element of every superinductive class.
Hence $\On$ is a subclass of every superinductive class.
The result follows.
== Sources ==
* : Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.2 \ (4)$
Category:Properties of Class of All Ordinals
Properties of Class of All Ordinals/Union of Chain of Ordinals is Ordinal

== Theorem ==
Let $\On$ denote the class of all ordinals.
Let $C$ be a chain of elements of $\On$.
Then its union $\bigcup C$ is also an element of $\On$.
== Proof ==
We have the result that Class of All Ordinals is Minimally Superinductive over Successor Mapping.
Hence $\On$ is a superinductive class the successor mapping.
Hence, by definition of superinductive class:
:$\On$ is closed under chain unions.
That is:
:$\forall C \in \On: \bigcup C \in \On$
where:
:$C$ is a chain of elements of $\On$
:$\bigcup C$ is the union of $C$.
== Sources ==
* : Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.2 \ (3)$
Category:Properties of Class of All Ordinals
Properties of Class of All Ordinals/Zero is Ordinal

== Theorem ==
Let $\On$ denote the class of all ordinals.
The natural number $0$ is an element of $\On$.
== Proof ==
We have the result that Class of All Ordinals is Minimally Superinductive over Successor Mapping.
Hence $\On$ is a superinductive class the successor mapping.
Hence, by definition of superinductive class:
:$\O \in \On$
We identify the natural number $0$ via the von Neumann construction of the natural numbers as:
:$0 := \O$
and the result follows.
== Sources ==
* : Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.2 \ (1)$
Category:Properties of Class of All Ordinals
Properties of Complex Exponential Function

== Theorem ==
Let $z \in \C$ be a complex number.
Let $\exp z$ be the exponential of $z$.
Then:
=== Image of Complex Exponential Function ===
=== Exponential of Sum ===
=== Reciprocal of Complex Exponential ===
=== Complex Exponential Function has Imaginary Period ===
=== Exponential Function is Continuous ===
=== Derivative of Exponential Function ===
Category:Exponential Function
Properties of Complex Numbers

== Complex Numbers are Uncountable ==
== Complex Numbers under Addition form Infinite Abelian Group ==
=== Complex Addition is Closed ===
=== Complex Addition is Associative ===
=== Complex Addition Identity is Zero ===
=== Inverse for Complex Addition ===
=== Complex Addition is Commutative ===
== Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group ==
=== Complex Multiplication is Closed ===
=== Complex Multiplication is Associative ===
=== Complex Multiplication Identity is One ===
=== Inverse for Complex Multiplication ===
=== Complex Multiplication is Commutative ===
== Complex Numbers form Ring ==
=== Complex Multiplication Distributes over Addition ===
== Complex Numbers form Field ==
== Substructures and Superstructures ==
=== Additive Group of Integers is Normal Subgroup of Complex ===
=== Additive Group of Rationals is Normal Subgroup of Complex ===
=== Additive Group of Reals is Normal Subgroup of Complex ===
=== Multiplicative Group of Rationals is Normal Subgroup of Complex ===
=== Multiplicative Group of Reals is Normal Subgroup of Complex ===
=== Rational Numbers form Subfield of Complex Numbers ===
=== Real Numbers form Subfield of Complex Numbers ===
== Further Structural Properties ==
=== Complex Numbers form Vector Space over Reals ===
=== Complex Numbers form Algebra ===
== Also see ==
* Properties of Natural Numbers
* Properties of Integers
* Properties of Rational Numbers
* Properties of Real Numbers
* Properties of Hamiltonian Numbers
Category:Complex Numbers
Properties of Content

== Theorem ==
Let $f, g \in \Q \sqbrk X$ and $q \in \Q$.
The content of a polynomial satisfies the following:
=== Content of Scalar Multiple ===
=== Content of Monic Polynomial ===
=== Polynomial has Integer Coefficients iff Content is Integer ===
=== Content of Rational Polynomial is Multiplicative ===
=== Content of Polynomial in Dedekind Domain is Multiplicative ===
Category:Content of Polynomial
Properties of Convergents of Continued Fractions

== Theorem ==
Let $n \in \N \cup \set \infty$ be an extended natural number.
Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be a continued fraction in $\R$ of length $n$.
Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators.
Let $C_0, C_1, C_2, \ldots$ be the convergents of $\sqbrk {a_0, a_1, a_2, \ldots}$.
Then the following results apply:
=== Difference between Adjacent Convergents ===
=== Difference between Adjacent Convergents But One ===
== Simple continued fractions ==
Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be a simple continued fraction in $\R$ of length $n$.
=== Convergents are Rationals in Canonical Form ===
=== Even convergents are strictly increasing ===
=== Odd convergents are strictly decreasing ===
=== Even convergents are smaller than odd convergents ===
== Also see ==
* Accuracy of Convergents of Continued Fraction
Category:Continued Fractions
Properties of Cumulative Distribution Function

== Theorem ==
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $F_X$ be the cumulative distribution function.
That is: 
:$\map {F_X} x = \map \Pr {X \le x}$
for each $x \in \R$.
Then $F_X$ has the following properties:
=== Bounds for Cumulative Distribution Function ===
=== Cumulative Distribution Function is Increasing ===
=== Cumulative Distribution Function is Right-Continuous ===
=== Limit of Cumulative Distribution Function at Positive Infinity ===
=== Limit of Cumulative Distribution Function at Negative Infinity ===
Category:Cumulative Distribution Functions
Properties of Degree

== Theorem ==
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$.
Let $R \sqbrk X$ denote the ring of polynomial forms over $R$ in the indeterminate $X$.
For $f \in R \sqbrk X$ let $\map \deg f$ denote the degree of $f$.
Then the following hold:
=== Degree of Sum of Polynomials ===
=== Degree of Product of Polynomials over Ring ===
=== Degree of Product of Polynomials over Integral Domain not Less than Degree of Factors ===
Category:Polynomial Theory
Properties of Digamma Function

== Theorem ==
This page gathers together some of the properties of the digamma function:
=== Digamma Function in terms of General Harmonic Number ===
=== Recurrence Relation for Digamma Function ===
=== Digamma Reflection Formula ===
=== Digamma Additive Formula ===
=== Gauss's Digamma Theorem ===
=== Ramanujan Phi Function in terms of Digamma Function ===
=== Gauss's Integral Form of Digamma Function ===
=== Dirichlet's Integral Form of Digamma Function ===
Category:Digamma Function
Properties of Dirichlet Convolution

== Theorem ==
Let $f, g, h$ be arithmetic functions.
Let $*$ denote Dirichlet convolution.
Let $\iota$ be the identity arithmetic function.
Then the following properties hold:
=== Dirichlet Convolution is Commutative ===
=== Dirichlet Convolution is Associative ===
=== Identity Element for Dirichlet Convolution ===
=== Dirichlet Convolution Preserves Multiplicativity ===
== Also see ==
* Definition:Ring of Arithmetic Functions
Category:Dirichlet Convolution
Properties of Discrete Topology

== Properties of Discrete Topology ==
Let $T = \struct {S, \tau}$ be a discrete topological space.
This page gathers up some results about $\struct {S, \tau}$.
=== Set in Discrete Topology is Clopen ===
=== Discrete Topology is Finest Topology ===
=== Topological Space is Discrete iff All Points are Isolated ===
Category:Discrete Topology
Properties of Domain of Exponential Map

== Theorem ==
Let $\struct {M, g, \nabla}$ be a Riemannian or pseudo-Riemannian manifold endowed with the Levi-Civita connection.
Let $T_p M$ be the tangent space of $M$ at $p \in M$.
Let $v \in T_p M$.
Let $I \subseteq \R$ be an open real interval.
Let $\gamma_v : I \to M$ be the unique maximal geodesic such that:
:$\map {\gamma '} 0 = v$
where $\gamma'$ is the velocity of $\gamma$.
Let $TM$ be the tangent bundle of $M$.
Let $\exp : \EE \to M$ be the exponential map where:
:$\EE = \set {v \in TM : \text{$\gamma_v$ is defined on $I : \closedint 0 1 \subseteq I$}}$
Then $\EE$ is an open subset of $TM$ containing the image of the zero section.
Furthermore, $\forall p \in M$ the set $\EE_p \subseteq T_p M$ is star-shaped with respect to $0$.
== Proof ==
== Sources ==
* : $\S 5$: The Levi-Civita Connection. The Exponential Map
Category:Exponential Maps
Properties of Dot Product

== Theorem ==
Let $\mathbf u, \mathbf v, \mathbf w$ be vectors in the real Euclidean space $\R^n$.
Let $c$ be a real scalar.
The dot product has the following properties:
=== Dot Product with Self is Non-Negative ===
=== Dot Product with Self is Zero iff Zero Vector ===
=== Dot Product Operator is Commutative ===
=== Dot Product Operator is Bilinear ===
That last result is often broken down into two less powerful ones:
=== Dot Product Distributes over Addition ===
=== Dot Product Associates with Scalar Multiplication ===
Category:Dot Product
Properties of Exponential Function

== Theorem ==
Let $x \in \R$ be a real number.
Let $\exp x$ be the exponential of $x$.
Then:
=== Exponential of Zero ===
=== Exponential of One ===
=== Exponential is Strictly Increasing ===
=== Exponential is Strictly Convex ===
=== Exponential Tends to Zero and Infinity ===
=== Exponential of Natural Logarithm ===
=== Exponential Function is Continuous ===
:$\forall c \in \R: \ds \lim_{x \mathop \to c} \exp x = \exp c$
Category:Exponential Function
Properties of Falling Factorial

== Theorem ==
Let $x^{\underline n}$ denote the $n$th falling factorial power of $x$.
This page gathers together some of the properties of the falling factorial.
=== Rising Factorial in terms of Falling Factorial ===
=== Rising Factorial in terms of Falling Factorial of Negative ===
=== Falling Factorial as Quotient of Factorials ===
=== Integer to Power of Itself Falling is Factorial ===
=== Integer to Power of Itself Less One Falling is Factorial ===
=== Number to Power of One Falling is Itself ===
=== Number to Power of Zero Falling is One ===
=== Sum of Indices of Falling Factorial ===
== Also see ==
* Properties of Rising Factorial
Category:Falling Factorials
Properties of Family of 333,667 and Related Numbers/Mistake

== Source Work ==
:
:The Dictionary
::$333,667$
:
:The Dictionary
::$333,667$
== Mistake ==
:The same author shows other patterns involving the same number: ... 
::$3,333,366,667 \times 1,111,333 = 371,113,711,137,111$ and so on.
== Correction ==
That multiplication should read:
::$3,333,366,667 \times 111,333 = 371,113,711,137,111$
Whether this mistake originates in the source where this result comes from, or whether it was a copying error in , has not been determined.
The original author was , writing in volumes $18$ and $21$ of , which have not been inspected by .
== Sources ==
* : $333,667$
* : $333,667$
Category:Mistakes/Curious and Interesting Numbers
Properties of Family of 333,667 and Related Numbers/Product with Certain Repetitive Numbers/Historical Note

== Historical Note on Properties of Family of 333,667 and Related Numbers: Product with Certain Repetitive Numbers ==
This result was reported by in his of $1986$ as having appeared in articles by in Volumes $18$ and $21$ of .
However, this has not been corroborated.
== Sources ==
* : $333,667$
Category:Historical Notes
Properties of Family of 333,667 and Related Numbers/Squares

== Theorem ==
This page reports on certain properties, difficult to classify, of the number $333 \, 667$, and patterns arising.
The square of any number consisting of:
:a string of $3$s
followed by:
:a string of $6$s
followed by:
:a single $7$
has its digits all in an increasing sequence:
The same is true of numbers of the following form:
== Proof ==
== Sources ==
* : $333,667$
* 
* : $333,667$
Category:Recreational Mathematics
Category:333,667
Category:Square Numbers
Properties of Fermi Coordinates

== Theorem ==
Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.
Let $P$ be an embedded $p$-dimensional submanifold. 
Let $U$ be a normal neighborhood of $P$ in $M$.
Let $U_0 \subseteq U$ be an open subset.
Let $\tuple{x^1, \ldots, x^p; v^1, \ldots, v^{n-p} }$ be Fermi coordinates on $U_0$. 
Let $x^{p + j} = v^j$ for $j = 1, \ldots, n - p$.
Let $\pi : NP \to P$ be the normal bundle of $P$ in $M$.
Let $\tuple {E_1, \ldots, E_{n - p}}$ be a local orthonormal frame for $NP$.
Let $\Gamma^k_{ij}$ be the Christoffel symbol.
Let $\gamma_v : \R \to M$ be a geodesic such that:
:$\map {\gamma_v} 0 = q$
:$\map {\gamma'_v} 0 = v$
where $\gamma'_v$ denotes the velocity of $\gamma_v$.
Then $\forall q \in P \cap U_0$:
:$(1): \quad \map {x^{p + 1} } q = \ldots = \map {x^n} q = 0$
:$(2): \quad g_{ij} = g_{ji} = \begin{cases}
0 & : 1 \le i \le p \text{ and } p + 1 \le j \le n\\
\delta_{ij} & : p + 1 \le i, j \le n
\end{cases}$
:$(3): \quad \forall v = v^1 \valueat {E_1} q + \ldots + v^{n - p} \valueat {E_{n - p} } q \in N_q P : \map {\gamma_v} t = \tuple {\map {x^1} q, \ldots, \map {x^p} q, t v^1, \ldots, t v^{n - p} }$
:$(4): \quad \forall i, j \in \N : p + 1 \le i, j \le n : \map {\Gamma^k_{ij} } q = 0$
:$(5): \quad \forall i, j, k \in \N : p + 1 \le i, j, k \le n : \partial_i \map {g_{jk} } q = 0$
== Proof ==
== Sources ==
* : $\S 5$: The Levi-Civita Connection. Tubular Neighborhoods and Fermi Coordinates
Category:Fermi Coordinates
Properties of Fibonacci Numbers

== Theorem ==
Let $F_n$ denote the $n$th Fibonacci number:
=== Fibonacci Number with Negative Index ===
=== Euler-Binet Formula ===
=== Fibonacci Number as Sum of Binomial Coefficients ===
=== Zeckendorf's Theorem ===
=== Sum of Non-Consecutive Fibonacci Numbers ===
=== Sum of Sequence of Fibonacci Numbers ===
=== Sum of Sequence of Even Index Fibonacci Numbers ===
=== Sum of Sequence of Odd Index Fibonacci Numbers ===
=== Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers ===
=== Sum of Even Sequence of Products of Consecutive Fibonacci Numbers ===
=== Lucas Number as Sum of Fibonacci Numbers ===
=== Millin Series ===
=== Recurrence Formula for Sum of Sequence of Fibonacci Numbers ===
=== Cassini's Identity ===
=== Catalan's Identity ===
=== Vajda's Identity ===
=== d'Ocagne's Identity ===
=== Honsberger's Identity ===
=== Fibonacci Number in terms of Larger Fibonacci Numbers ===
=== Divisibility of Fibonacci Number ===
=== GCD of Fibonacci Numbers ===
Category:Fibonacci Numbers
Properties of Floor Function

== Theorem ==
This page gathers together some basic propeties of the floor function.
=== Floor is between Number and One Less ===
=== Floor of Number plus Integer ===
=== Real Number minus Floor ===
=== Real Number is between Floor Functions ===
=== Real Number is Floor plus Difference ===
=== Floor Function is Idempotent ===
== Range of Values of Floor Function ==
== Also see ==
* Properties of Ceiling Function
Category:Floor Function
Properties of Fourier Transform

== Theorem ==
In the following:
:$\map f x$ and $\map g x$ denote Lebesgue integrable functions
:for a given $\map f x$, $\map {\hat f} \zeta$ denotes its Fourier transform.
=== Linearity ===
=== Translation ===
=== Modulation ===
=== Time Scaling ===
Category:Fourier Transforms
Properties of Fourier Transform/Linearity

== Theorem ==
Let $\map f x$ and $\map g x$ be Lebesgue integrable functions.
Let $a$ and $b$ be complex numbers.
Let $\map h x$ be a Lebesgue integrable function such that:
:$\map h x = a \map f x + b \map g x$
Then:
:$\map {\hat h} s = a \map {\hat f} s + b \map {\hat g} s$
where $\map {\hat h} s$, $\map {\hat f} s$ and $\map {\hat g} s$ are the Fourier transforms of $\map h x$, $\map f x$ and $\map g x$ respectively.
== Proof ==
== Sources == 
* 
Category:Fourier Transforms
Properties of Fourier Transform/Modulation

== Theorem ==
Let $\map f x$ be a Lebesgue integrable function.
Let $s_0$ be a real number.
Let $\map h x$ be a Lebesgue integrable function such that:
:$\map h x = e^{2 \pi i x s_0} \map f x$
Then:
:$\map {\hat h} s = \map {\hat f} {s - s_0}$
where $\map {\hat h} s$ and $\map {\hat f} s$ are the Fourier transforms of $\map h x$ and $\map f x$ respectively.
== Proof ==
== Sources == 
* 
Category:Fourier Transforms
Properties of Fourier Transform/Scaling

== Theorem ==
Let $\map f x$ be a Lebesgue integrable function.
Let $a$ be a non-zero real number.
Let $\map h x$ be a Lebesgue integrable function such that:
:$\map h x = \map f {a x}$
Then:
:$\map {\hat h} s = \dfrac 1 {\size a} \map {\hat f} {\dfrac s a}$
where $\map {\hat h} s$ and $\map {\hat f} s$ are the Fourier transforms of $\map h x$ and $\map f x$ respectively.
== Proof ==
== Sources == 
* 
Category:Fourier Transforms
Properties of Fourier Transform/Translation

== Theorem ==
Let $\map f x$ be a Lebesgue integrable function.
Let $x_0$ be a real number.
Let $\map h x$ be a Lebesgue integrable function such that:
:$\map h x = \map f {x - x_0}$
Then:
:$\map {\hat h} s = e^{-2 \pi i x_0 s} \map {\hat f} s$
where $\map {\hat h} s$ and $\map {\hat f} s$ are the Fourier transforms of $\map h x$ and $\map f x$ respectively.
== Proof ==
== Sources == 
* 
Category:Fourier Transforms
Properties of Gamma Function

== Theorem ==
The gamma function $\Gamma \left({z}\right)$ has the following properties:
=== Gamma Difference Equation ===
=== Euler's Reflection Formula ===
=== Legendre's Duplication Formula ===
=== Reciprocal times Derivative ===
=== Complex Conjugate ===
=== Gamma Function of One Half ===
Category:Gamma Function
Properties of General Harmonic Numbers

== Theorem ==
This page gathers together some of the properties of the general harmonic numbers:
=== Digamma Function in terms of General Harmonic Number ===
=== Recurrence Relation for General Harmonic Numbers ===
=== General Harmonic Number Reflection Formula ===
=== Nth Derivative of General Harmonic Number Order One ===
=== General Harmonic Numbers in terms of Riemann Zeta and Hurwitz Zeta Functions ===
=== Sum of General Harmonic Numbers in terms of Riemann Zeta Function ===
=== General Harmonic Number Additive Formula ===
Category:General Harmonic Numbers
Properties of Generalized Hypergeometric Function

== Theorem ==
From the , we have:
:$\ds \map { {}_m \operatorname F_n} { { {a_1, \ldots, a_m} \atop {b_1, \ldots, b_n} } \, \middle \vert \, z} = \sum_{k \mathop = 0}^\infty \dfrac { {a_1}^{\overline k} \cdots {a_m}^{\overline k} } { {b_1}^{\overline k} \cdots {b_n}^{\overline k} } \dfrac {z^k} {k!}$
where $x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
This page gathers together some of the properties of hypergeometric functions.
=== Gauss's Hypergeometric Theorem ===
=== Kummer's Hypergeometric Theorem ===
=== Dixon's Hypergeometric Theorem ===
=== Watson's Hypergeometric Theorem ===
=== Pfaff-Saalschütz Theorem ===
=== Dougall's Hypergeometric Theorem ===
=== Dougall-Ramanujan Identity ===
Category:Hypergeometric Functions
Properties of Hadamard Product

== Theorem ==
Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {S, \cdot}$.
Let $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$.
Let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$.
The operation $\circ$ of Hadamard product satisfies the following properties:
:$\circ$ is closed on $\map {\MM_S} {m, n}$ $\cdot$ is closed on $\struct {S, \cdot}$
:$\circ$ is associative on $\map {\MM_S} {m, n}$ $\cdot$ is associative on $\struct {S, \cdot}$
:$\circ$ is commutative on $\map {\MM_S} {m, n}$ $\cdot$ is commutative on $\struct {S, \cdot}$.
== Proof ==
=== Closure of Hadamard Product ===
=== Associativity of Hadamard Product ===
=== Commutativity of Hadamard Product ===
Category:Hadamard Product
Properties of Hausdorff Space

== Theorem ==
=== Subspace of Hausdorff Space is Hausdorff ===
=== Product of Hausdorff Factor Spaces is Hausdorff ===
=== Domain of Continuous Injection to Hausdorff Space is Hausdorff ===
=== Hausdorff Condition is Preserved under Homeomorphism ===
Let $T_A = \struct {S_A, \tau_A}, T_B = \struct {S_B, \tau_B}$ be topological spaces.
Let $\phi: T_A \to T_B$ be a homeomorphism.
Category:Hausdorff Spaces
Properties of Idempotent Semigroup/1

== Property of Idempotent Semigroup ==
Let $\struct {S, \circ}$ be an idempotent semigroup.
Let $x \circ y = y$ and $y \circ x = x$.
Then for all $z \in S$:
:$z \circ x \circ z \circ y = z \circ y$
and:
:$z \circ y \circ z \circ x = z \circ x$
== Proof ==
From we take it for granted that $\circ$ is associative.
== Sources ==
* : Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.19 \ \text {(a)}$
Category:Idempotent Semigroups
Properties of Idempotent Semigroup/2

== Property of Idempotent Semigroup ==
Let $\struct {S, \circ}$ be an idempotent semigroup.
Let $x \circ y = x$ and $y \circ x = y$.
Then for all $z \in S$:
:$x \circ z \circ y \circ z = x \circ z$
and:
:$y \circ z \circ x \circ z = y \circ z$
== Proof ==
From we take it for granted that $\circ$ is associative.
== Sources ==
* : Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.19 \ \text {(c)}$
Category:Idempotent Semigroups
Properties of Indiscrete Topology

== Properties of Indiscrete Topology ==
Let $S$ be a set.
Let $\tau$ be the indiscrete topology on $S$.
This page gathers up some results about $\left({S, \tau}\right)$.
=== Indiscrete Topology is Topology ===
=== Indiscrete Topology is Coarsest Topology ===
Category:Indiscrete Topology
Properties of Infinite Products

== Theorem ==
From the , we have:
:$\ds \prod_{\map R j} a_j = \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ -n \mathop \le j \mathop < 0} } a_j} \times \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ 0 \mathop \le j \mathop \le n} } a_j}$
This page gathers together some of the properties of infinite products.
=== Golden Mean as Infinite Product ===
=== Square Root of 2 as Infinite Product ===
=== Square Root of 3 as Infinite Product ===
=== Multiple of Pi as Infinite Product ===
=== Vieta's Formula for Pi ===
=== Wallis's Product ===
=== Binomial Coefficient as Infinite Product ===
=== Infinite Product of One Plus Reciprocals of Squares ===
=== Infinite Product of One Plus Reciprocals of Even Squares ===
=== Infinite Product of One Minus Reciprocals of Squares ===
=== Infinite Product of One Minus Reciprocals of Even Squares ===
=== Infinite Product of Ratio of One Less Than Square Over One More Than Square ===
=== Infinite Product of One Plus Reciprocals of Cubes ===
=== Infinite Product of One Minus Reciprocals of Cubes ===
=== Infinite Product of Ratio of One Less Than Cube Over One More Than Cube ===
Category:Examples of Infinite Products