hcju/proofwikips huggingface dataset:
Closed Class under Progressing Mapping Lemma
Let $N$ be a class which is closed under a progressing mapping $g$. Let $g$ be such that: :$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$ :if $\map g x = x$, then $x$ is the greatest element of $N$. Let the following hold: :$A \subseteq N$ is a subclass of $N$ :$x \in N$ is an element of $N$ Let $x$ be: :a proper subset of all elements of $A$ and: :the greatest element of $A$ with that property. Then $\map g x \in A$ and is the smallest element of $A$.
Minimally Inductive Class under Progressing Mapping is Well-Ordered under Inclusion
Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x$ be a fixed point of $g$. Then $M$ is well-ordered under the inclusion relation.
Minimally Closed Class under Progressing Mapping induces Nest
For all $x, y \in N$: :either $\map g x \subseteq y$ or $y \subseteq x$ and $N$ forms a nest: :$\forall x, y \in N: x \subseteq y$ or $y \subseteq x$
Minimally Closed Class under Progressing Mapping
Statement of Conditions: {{:Minimally Closed Class under Progressing Mapping/Statement}} Then the following results hold:
Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element
$g$ has no fixed point, unless possibly the greatest element, if there is one.
Smallest Element of Minimally Closed Class under Progressing Mapping
$b$ is the smallest element of $N$.
Equivalence of Definitions of Minimally Closed Class
Let $A$ be a class. Let $g$ be a mapping on $A$. {{TFAE|def = Minimally Closed Class|view = minimally closed class under $g$}}
Principle of General Induction/Minimally Closed Class
Let $M$ be a class. Let $g: M \to M$ be a mapping on $M$. Let $b \in M$ such that $M$ is minimally closed under $g$ with respect to $b$. Let $P: M \to \set {\T, \F}$ be a propositional function on $M$. Suppose that: :$(1): \quad \map P b = \T$ :$(2): \quad \forall x \in M: \map P x = \T \implies \map P {\map g x} = \T$ Then: :$\forall x \in M: \map P x = \T$
Double Induction Principle/Minimally Closed Class
Let $M$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $M$ such that $M$ is minimally closed under $g$ with respect to $b$. Let $\RR$ be a relation on $M$ which satisfies: {{begin-axiom}} {{axiom | n = \text D_1         | q = \forall x \in M         | m = \map \RR {x, b} }} {{axiom | n = \text D_2         | q = \forall x, y \in M         | m = \map \RR {x, y} \land \map \RR {y, x} \implies \map \RR {x, \map g y} }} {{end-axiom}} Then $\map \RR {x, y}$ holds for all $x, y \in M$.
Double Induction Principle/Minimally Closed Class/Lemma
Let $x$ be a right normal element of $M$ with respect to $\RR$. Then $x$ is also a left normal element of $M$ with respect to $\RR$.
Sandwich Principle for Minimally Closed Class
Let $N$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$. Then for all $x, y \in N$: :$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
Image of Proper Subset under Progressing Mapping on Minimally Closed Class
Let $N$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$. Then: :$x \subset y \implies \map g x \subseteq y$
Equation of Cornu Spiral/Parametric
Let $K$ be a Cornu spiral embedded in a Cartesian coordinate plane such that the origin coincides with the point at which $s = 0$. Then $K$ can be expressed by the parametric equations: :$\begin {cases} x = a \sqrt 2 \map {\operatorname C} {\dfrac s {a \sqrt 2} } \\ y = a \sqrt 2 \map {\operatorname S} {\dfrac s {a \sqrt 2} } \end {cases}$ where: :$\operatorname C$ denotes the Fresnel cosine integral function :$\operatorname S$ denotes the Fresnel sine integral function.
Sampling Function is its own Fourier Transform
Consider the sampling function $\operatorname {III}: \R \to \R$. Then: :$\map \FF {\operatorname {III} } = \operatorname {III}$ where $\FF$ denotes the Fourier transform.
Even Impulse Pair is Fourier Transform of Cosine Function
Consider the (real) cosine function $\cos: \R \to \R$. Then: :$\map \FF {\cos} = \operatorname {II}$ where: :$\FF$ denotes the Fourier transform :$\operatorname {II}$ denotes the even impulse pair function.
Convolution of Real Function with Rectangle Function
Let $f: \R \to \R$ be a real function. Consider the rectangle function $\Pi: \R \to \R$. Then: :$\forall x \in \R: \map \Pi x * \map f x = \displaystyle \int_{x \mathop - \frac 1 2}^{x \mathop + \frac 1 2} \map f u \rd u$ where $*$ denotes the convolution integral.
Repeated Fourier Transform of Even Function
Let $f: \R \to \R$ be an even real function which is Lebesgue integrable. Let $\displaystyle \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$. Let $\displaystyle \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier transform of $F$. Then: :$\map g t = \map f t$
Repeated Fourier Transform of Odd Function
Let $f: \R \to \R$ be an odd real function which is Lebesgue integrable. Let $\displaystyle \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$. Let $\displaystyle \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier transform of $F$. Then: :$\map g t = -\map f t$
Repeated Fourier Transform of Real Function
Let $f: \R \to \R$ be a real function which is Lebesgue integrable. Let $\displaystyle \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$. Let $\displaystyle \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier transform of $F$. Then: :$\map g t = \map f {-t}$
Fourier's Theorem/Integral Form
Let $f: \R \to \R$ be a real function which satisfies the Dirichlet conditions on $\R$. Then: :$\dfrac {\map f {t^+} + \map f {t^-} } 2 = \displaystyle \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i d t} \map f t \rd t} \rd s$ where: :$\map f {t^+}$ and $\map f {t^-}$ denote the limit from above and the limit from below of $f$ at $t$.
Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Lemma 1
:$\map {p_i^\to} {\map{\pr_k^\gets } {V_k} \cap Y_i}$ is open in $\struct{X_i, \tau_i}$
Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Injection
Let $\family {X_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set. Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$. Let $z \in X$. Let $i \in I$.  Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$. Let $p_i = \pr_i {\restriction_{Y_i}}$, where $\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is an injection.
Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Surjection
Let $\family {X_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set. Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$. Let $z \in X$. Let $i \in I$.  Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$. Let $p_i = \pr_i {\restriction_{Y_i} }$, where $\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is a surjection.
Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Continuous Mapping
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Let $z \in X$. Let $i \in I$.  Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$. Let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.  Let $p_i = \pr_i {\restriction_{Y_i}}$, where $\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is continuous.
Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Open Mapping
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Let $z \in X$. Let $i \in I$.  Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$. Let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.  Let $p_i = \pr_i {\restriction_{Y_i}}$, where $\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is an open mapping.
Triple Angle Formulas/Sine/Historical Note
The Triple Angle Formula for Sine is often attributed to {{AuthorRef|François Viète}}, although it was in fact discovered by {{AuthorRef|Jamshīd al-Kāshī}} a century or more earlier.
Exponential Distribution in terms of Continuous Uniform Distribution
Let $X \sim \mathrm U \hointl 0 1$ where $\mathrm U \hointl 0 1$ is the continuous uniform distribution on $\hointl 0 1$. Let $\beta$ be a positive real number. Then:  :$-\beta \ln X \sim \Exponential \lambda$ where $\operatorname {Exp}$ is the exponential distribution.
Expectation of Non-Negative Random Variable is Non-Negative
Let $X$ be a random variable.  Let $\map \Pr {X \ge 0} = 1$.  Then $\expect X \ge 0$, where $\expect X$ denotes the expectation of $X$.
Expectation of Non-Negative Random Variable is Non-Negative/Discrete
Let $X$ be a discrete random variable.  Let $\map \Pr {X \ge 0} = 1$.  Then $\expect X \ge 0$, where $\expect X$ denotes the expectation of $X$.
Expectation of Non-Negative Random Variable is Non-Negative/Continuous
Let $X$ be a continuous random variable.  Let $\map \Pr {X \ge 0} = 1$.  Then $\expect X \ge 0$, where $\expect X$ denotes the expectation of $X$.
Expectation of Almost Surely Constant Random Variable
Let $X$ be an almost surely constant random variable. That is, there exists some $c \in \R$ such that:  :$\map \Pr {X = c} = 1$ Then:  :$\expect X = c$
Expectation Preserves Inequality
Let $X$, $Y$ be random variables. Let $\map \Pr {X \ge Y} = 1$. Then:  :$\expect X \ge \expect Y$
Covariance of Random Variable with Itself
Let $X$ be a random variable. Then $\cov {X, X} = \var X$.
Covariance is Symmetric
Let $X$ and $Y$ be random variables. Then $\cov {X, Y} = \cov {Y, X}$.
Covariance of Sums of Random Variables
Let $n$ be a strictly positive integer. Let $\sequence {X_i}_{1 \le i \le n}$, $\sequence {Y_j}_{1 \le j \le n}$ be sequences of random variables. Then:  :$\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, \sum_{j \mathop = 1}^n Y_j} = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \cov {X_i, Y_j}$
Expectation of Linear Transformation of Random Variable/Discrete
Let $X$ be a discrete random variable.  Let $a, b$ be real numbers.  Then we have:  :$\expect {a X + b} = a \expect X + b$ where $\expect X$ denotes the expectation of $X$.
Expectation of Linear Transformation of Random Variable/Continuous
Let $X$ be a continuous random variable.  Let $a, b$ be real numbers.  Then we have:  :$\expect {a X + b} = a \expect X + b$ where $\expect X$ denotes the expectation of $X$.
Cauchy's Mean Theorem/Proof of Equality Condition
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive. Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$. Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$. Then: :$A_n = G_n$ : :$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$ That is,  all terms are equal. Then:
Fourier Series/Square Wave
600pxthumbrightSquare Wave and $9$th Approximation Let $\map S x$ be the square wave defined on the real numbers $\R$ as: :$\forall x \in \R: \map S x = \begin {cases} 1 & : x \in \openint 0 l \\ -1 & : x \in \openint {-l} 0 \\ \map S {x + 2 l} & : x < -l \\ \map S {x - 2 l} & : x > +l \end {cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map S x       | o = \sim       | r = \frac 4 \pi \sum_{r \mathop = 0}^\infty \frac 1 {2 r + 1} {\sin \frac {\pi x} l}       | c =  }} {{eqn | r = \frac 4 \pi \paren {\sin \frac {\pi x} l + \dfrac 1 3 \sin \frac {3 \pi x} l + \dfrac 1 5 \sin \frac {5 \pi x} l + \dotsb}       | c =  }} {{end-eqn}}
Fourier Series/Triangle Wave
600pxthumbrightTriangle Wave and $9$th Approximation  Let $\map T x$ be the triangle wave defined on the real numbers $\R$ as: :$\forall x \in \R: \map T x = \begin {cases} \size x & : x \in \closedint {-l} l \\ \map T {x + 2 l} & : x < -l \\ \map T {x - 2 l} & : x > +l \end {cases}$ where: :$l$ is a given real constant :$\size x$ denotes the absolute value of $x$. Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map T x       | o = \sim       | r = \frac l 2 - \frac {4 l} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi x} l       | c =  }} {{eqn | r = \frac l 2 - \frac {4 l} {\pi^2} \paren {\cos \dfrac {\pi x} l + \frac 1 {3^2} \cos \dfrac {3 \pi x} l + \frac 1 {5^2} \cos \dfrac {5 \pi x} l + \dotsb}       | c =  }} {{end-eqn}}
Subspace of Product Space Homeomorphic to Factor Space/Product with Singleton/Lemma
:$f$ is a bijection.
Standard Continuous Uniform Distribution in terms of Exponential Distribution
Let $X$ and $Y$ be independent random variables. Let $\beta$ be a strictly positive real number. Let $X$ and $Y$ be random samples from the exponential distribution with parameter $\beta$.  Then: :$\dfrac X {X + Y} \sim \operatorname U \openint 0 1$ where $\operatorname U \openint 0 1$ is the uniform distribution on $\openint 0 1$.
Identity Function is Odd Function
Let $I_\R: \R \to \R$ denote the identity function on $\R$. Then $I_\R$ is an odd function.
Half-Range Fourier Series/Identity Function
Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $\map f x: \openint 0 \lambda \to \R$ be the identity function on the open real interval $\openint 0 \lambda$: :$\forall x \in \openint 0 \lambda: \map f x = x$ The half-range Fourier series of $f$ over $\openint 0 \lambda$ can be given in the following forms:
Half-Range Fourier Series/Identity Function/Cosine
The half-range Fourier cosine series for $\map f x$ can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi x} \lambda       | c =  }} {{eqn | r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \paren {\cos \dfrac {\pi x} \lambda + \frac 1 {3^2} \cos \dfrac {3 \pi x} \lambda + \frac 1 {5^2} \cos \dfrac {5 \pi x} \lambda + \dotsb}       | c =  }} {{end-eqn}}
Half-Range Fourier Series/Identity Function/Sine
The half-range Fourier sine series for $\map f x$ can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \dfrac {2 \lambda} \pi \displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \frac {n \pi x} \lambda       | c =  }} {{eqn | r = \dfrac {2 \lambda} \pi \paren {\sin \dfrac {\pi x} \lambda - \frac 1 2 \sin \dfrac {2 \pi x} \lambda + \frac 1 3 \sin \dfrac {3 \pi x} \lambda - \dotsb}       | c =  }} {{end-eqn}}
Absolute Value Function is Even Function
Let $\size {\, \cdot \,} : \R \to \R$ denote the absolute value function on $\R$: Then $\size {\, \cdot \,}$ is an even function.
Fourier Series/Absolute Value Function over Symmetric Range
Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the absolute value function on the open real interval $\openint {-\lambda} \lambda$: :$\forall x \in \openint {-\lambda} \lambda: \map f x = \size x$ The Fourier series of $f$ over $\openint {-\lambda} \lambda$ can be given as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi x} \lambda       | c =  }} {{eqn | r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \paren {\cos \dfrac {\pi x} \lambda + \frac 1 {3^2} \cos \dfrac {3 \pi x} \lambda + \frac 1 {5^2} \dfrac {5 \pi x} \lambda + \dotsb}       | c =  }} {{end-eqn}}
Factor Spaces of Hausdorff Product Space are Hausdorff
Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$. Let $T$ be a $T_2$ (Hausdorff) space. Then: :for each $\alpha \in I$, $\struct {S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space.
Cartesian Product is Empty iff Factor is Empty/Family of Sets
Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Then: :$S = \O$   $S_i = \O$ for some $i \in I$
Derivative of P-Norm wrt P
Let $p \ge 1$ be a real number. Let $\ell^p$ denote the $p$-sequence space. Let $\mathbf x = \sequence {x_n} \in \ell^p$. Let $\norm {\mathbf x}_p$ be a p-norm. Suppose, $\norm {\mathbf x}_p \ne 0$. Then: :$\displaystyle \dfrac \d {\d p} \norm {\mathbf x}_p = \frac {\norm {\mathbf x}_p} p \paren { \frac {\sum_{n \mathop = 0}^\infty \size {x_n}^p \map \ln {\size {x_n} } } {\norm {\bf x}_p^p} - \map \ln {\norm {\bf x}_p} }$
Extension of Half-Range Fourier Sine Function to Symmetric Range
Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Let $\map f x$ be represented by the half-range Fourier sine series $\map S x$: :$\map f x \sim \map S x = \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$ Then $\map S x$ also represents the extension $g: \openint {-\lambda} \lambda \to \R$ of $f$, defined as: :$\forall x \in \openint {-\lambda} \lambda: \map g x = \begin {cases} \map f x & : x > 0 \\ -\map f {-x} & : x < 0 \\ 0 & : x = 0 \end {cases}$
Primitive of x by Logarithm of x squared plus a squared
:$\displaystyle \int x \map \ln {x^2 + a^2} \rd x = \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C$
Equivalence of Definitions of Sets Separated by Neighborhoods
Let $T = \struct {S, \tau}$ be a topological space. {{TFAE|def = Separated by Neighborhoods/Sets|view = Sets Separated by Neighborhoods}}
Equivalence of Definitions of Points Separated by Neighborhoods
Let $T = \struct {S, \tau}$ be a topological space. {{TFAE|def = Points Separated by Neighborhoods}}
Product Space is T3 iff Factor Spaces are T3/Product Space is T3 implies Factor Spaces are T3
Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a $T_3$ space. Then for each $\alpha \in I$, $\struct {S_\alpha, \tau_\alpha}$ is a $T_3$ space.
Product Space is T3 iff Factor Spaces are T3/Factor Spaces are T3 implies Product Space is T3
Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. For each $\alpha \in I$, let $\struct{S_\alpha, \tau_\alpha}$ be a $T_3$ space. Then $T$ is a $T_3$ space.
Mittag-Leffler Expansion for Cosecant Function/Real Domain
:$\pi \cosec \pi \alpha = \dfrac 1 \alpha + \displaystyle 2 \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac {\alpha} {\alpha^2 - n^2}$
Sum from -m to m of 1 minus Cosine of n + alpha of theta over n + alpha
For $0 < \theta < 2 \pi$: :$\displaystyle \sum_{n \mathop = -m}^m \dfrac {1 - \cos \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \sin {\alpha u} \dfrac {\sin \paren {m + \frac 1 2} u \rd u} {\sin \frac 1 2 u}$
Sum over Integers of Cosine of n + alpha of theta over n + alpha
Let $\alpha \in \R$ be a real number which is specifically not an integer. For $0 < \theta < 2 \pi$: :$\displaystyle \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \sum_{n \mathop \in \Z} \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$
Sum of Complex Exponentials of i times 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 Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1
:$\displaystyle \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \frac {n \alpha} 2} } \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} }$
Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2
:$\displaystyle \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$
Sum of Sines of Arithmetic Sequence of Angles/Formulation 1
{{begin-eqn}} {{eqn | l = \sum_{k \mathop = 0}^n \map \sin {\theta + k \alpha}       | r = \sin \theta + \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb }} {{eqn | r = \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \sin {\theta + \frac {n \alpha} 2} }} {{end-eqn}}
Sum of Sines of Arithmetic Sequence of Angles/Formulation 2
{{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n \map \sin {\theta + k \alpha}       | r = \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb }} {{eqn | r = \map \sin {\theta + \frac {n + 1} 2 \alpha}\frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} } }} {{end-eqn}}
Sum of Cosines of Arithmetic Sequence of Angles/Formulation 2
{{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n \map \cos {\theta + k \alpha}       | r = \map \cos {\theta + \alpha} + \map \cos {\theta + 2 \alpha} + \map \cos {\theta + 3 \alpha} + \dotsb }} {{eqn | r = \map \cos {\theta + \frac {n + 1} 2 \alpha} \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} } }} {{end-eqn}}
Weierstrass Approximation Theorem/Lemma 1
:$\displaystyle \sum_{k \mathop = 0}^n k \map {p_{n,k} } t = n t$
Weierstrass Approximation Theorem/Lemma 2
:$\displaystyle \sum_{k \mathop = 0}^n \paren {k - nt}^2 \map {p_{n,k} } t = n t \paren {1 - t}$
Convergent Sequence in Normed Vector Space has Unique Limit
Let $\struct {X, \norm {\,\cdot\,}}$ be a normed vector space. Let $\sequence {x_n}$ be a sequence in $\struct {X, \norm {\,\cdot\,} }$. Then $\sequence {x_n}$ can have at most one limit.
Product Space is T3 1/2 iff Factor Spaces are T3 1/2/Product Space is T3 1/2 implies Factor Spaces are T3 1/2
Let $\mathbb S = \family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a $T_{3 \frac 1 2}$ space. Then for each $\alpha \in I$, $\struct{S_\alpha, \tau_\alpha}$ is a $T_{3 \frac 1 2}$ space.
Product Space is T3 1/2 iff Factor Spaces are T3 1/2/Factor Spaces are T3 1/2 implies Product Space is T3 1/2
Let $\mathbb S = \family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$ with $S_\alpha \neq \O$ for every $\alpha \in I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. For each $\alpha \in I$, let $\struct{S_\alpha, \tau_\alpha}$ be a $T_{3 \frac 1 2}$ space. Then $T$ is a $T_{3 \frac 1 2}$ space.