nyuuzyou/wikis huggingface dataset:
Horizontal Section preserves Increasing Sequences of Sets

== Theorem ==
Let $X$ and $Y$ be sets.
Let $\sequence {A_n}_{n \mathop \in \N}$ be an increasing sequence in $X \times Y$. 
Let $y \in Y$. 
Then: 
:$\sequence {\paren {A_n}^y}_{n \mathop \in \N}$ is an increasing sequence.
== Proof ==
Since $\sequence {A_n}_{n \mathop \in \N}$ is increasing, we have: 
:$A_n \subseteq A_{n + 1}$
for each $n$. 
From Horizontal Section preserves Subsets, we have: 
:$\paren {A_n}^y \subseteq \paren {A_{n + 1} }^y$
for each $n$.
So:
:$\sequence {\paren {A_n}^y}_{n \mathop \in \N}$ is an increasing sequence.
Category:Horizontal Section of Sets
Category:Increasing Sequences of Sets
Horizontal Section preserves Pointwise Limits of Sequences of Functions

== Theorem ==
Let $X$ and $Y$ be sets. 
Let $f : X \times Y \to \overline \R$ be a function.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of functions converging pointwise to $f$. 
Let $y \in Y$. 
Then: 
:$\paren {f_n}^y \to f^y$
pointwise, where: 
:$\paren {f_n}^y$ denotes the $y$-horizontal section of $f_n$
:$f^y$ denotes the $y$-horizontal section of $f$. 
== Proof ==
From the definition of pointwise convergence, we have: 
:$\ds \map f {x, y} = \lim_{n \mathop \to \infty} \map {f_n} {x, y}$
for each $x \in X$ and $y \in Y$. 
Fix $y \in Y$. 
From the definition of the $y$-horizontal section, we have: 
:$\map {f_n} {x, y} = \map {\paren {f_n}^y} x$
and:
:$\map f {x, y} = \map {f^y} x$
So:
:$\ds \map {f^y} x = \lim_{n \mathop \to \infty} \map {\paren {f_n}^y} x$
for each $x \in X$. 
So:
:$\paren {f_n}^y \to f^y$
pointwise.
Category:Horizontal Section of Functions
Horizontal Section preserves Subsets

== Theorem ==
Let $X$ and $Y$ be sets.
Let $A \subseteq B$ be subsets of $X \times Y$. 
Let $y \in Y$. 
Then: 
:$A^y \subseteq B^y$
where $A^y$ is the $y$-horizontal section of $A$ and $B^y$ is the $y$-horizontal section of $B$. 
== Proof ==
Note that if: 
:$x \in A^y$
from the definition of $x$-vertical section, we have: 
:$\tuple {x, y} \in A$
so:
:$\tuple {x, y} \in B$
So, from the definition of $x$-vertical section, we have: 
:$x \in B^y$
So:
:if $x \in A^y$ then $x \in B^y$.
That is: 
:$A^y \subseteq B^y$
Category:Horizontal Section of Sets
Horner's Rule/Examples/Arbitrary Example 1

== Example of Use of Horner's Rule ==
The polynomial:
:$\map p x = 4 x^3 - 2 x^2 + 3 x - 1$
can be expressed using Horner's rule as:
:$\map p x = \paren {\paren {4 x - 2} x + 3} x - 1$
== Sources ==
* 
* 
Category:Examples of Use of Horner's Rule
Horses Born and Died at Same Time but with Different Ages

== Problem ==
$2$ horses were born at the same time, travelled the world, and died at the same time, but did not live to the same age.
How is this possible?
== Solution ==
This puzzle relies on the definition of age as:
:the number of days seen
which in turn is defined as:
:the number of times the sun has risen over the horse.
One horse travels east for its whole life.
Hence, travelling continually towards the sunrise, the time elapsed between consecutive sunrises is shorter than $24$ hours.
The other horse travels west for its whole life.
Hence, travelling continually away from the sunrise, the time elapsed between consecutive sunrises is longer than $24$ hours.
Hence, while being born and dying at the same time, the number of elapsed days they have seen will be different.
== Also see ==
* Three Consecutive Sabbath Days on Same Day, a variant of this
== Historical Note ==
== Sources ==
* 
* : Henry van Etten: $122$
Category:Geography Puzzles
Category:Classic Problems
Hurwitz's Theorem

* Hurwitz's Theorem (Complex Analysis)
* Hurwitz's Theorem (Number Theory)
* Hurwitz's Theorem (Normed Division Algebras)
* Hurwitz's Automorphisms Theorem
* Riemann-Hurwitz Formula‎ (with )
* Routh-Hurwitz Theorem (with )
Category:Hurwitz's Theorem
Hurwitz's Theorem (Normed Division Algebras)

== Theorem ==
The only normed division algebras over the real numbers are:
:the real numbers $\R$ themselves
:the complex numbers $\C$
:the quaternions $\H$
:the octonions $\mathbb O$
== Proof ==
== Sources ==
* 
Category:Hurwitz's Theorem
Category:Normed Division Algebras
Hurwitz's Theorem (Number Theory)

== Theorem ==
Let $\xi$ be an irrational number.
Then there are infinitely many relatively prime integers $p, q \in \Z$ such that:
:$\size {\xi - \dfrac p q} < \dfrac 1 {\sqrt 5 \, q^2}$
== Proof ==
=== Lemma 1 ===
=== Lemma 2 ===
There are an infinite number of convergents to $\xi$.
Taking these in sets of $3$ at a time, it can be seen from Lemma 2 that at least one of them satisfies the given inequality.
From Lemma 1 it is seen that this inequality is the best possible.
== Also see ==
* Liouville's Theorem (Number Theory)
== Sources ==
* : $11.8$: The measure of the closest approximation to an arbitrary irrational
* 
Category:Hurwitz's Theorem (Number Theory)
Category:Hurwitz's Theorem
Category:Number Theory
Hurwitz's Theorem (Number Theory)/Lemma 1

== Lemma ==
Let $\xi$ be an irrational number.
Let $A \in \R$ be a real number strictly greater than $\sqrt 5$.
Then there may exist at most a finite number of relatively prime integers $p, q \in \Z$ such that:
:$\size {\xi - \dfrac p q} < \dfrac 1 {A \, q^2}$
== Proof ==
We will take as our example of such an irrational number:
:$\xi = \dfrac {\sqrt 5 - 1} 2$
This is equal to $\phi - 1$, where $\phi$ is the Golden mean.
 that there exist an infinite number of $p, q$ with $p \perp q$ such that:
:$\size {\xi - \dfrac p q} < \dfrac 1 {A \, q^2}$
Then there exist an infinite number of $p, q$ with $p \perp q$ such that:
:$\xi = \dfrac p q + \dfrac \delta {q^2}$
where:
:$A > \sqrt 5$
Therefore:
:$\dfrac 1 A < \dfrac 1 {\sqrt 5}$
and:
:$0 < \size \delta < \dfrac 1 A < \dfrac 1 {\sqrt 5}$
Hence:
Over the interval ${-\dfrac 1 {\sqrt 5} } < \delta < \dfrac 1 {\sqrt 5}$, the of $(1)$ takes on values:
:$\dfrac 1 {5 q^2} - 1 < \dfrac {\delta^2} {q^2} - \delta \sqrt 5 < \dfrac 1 {5 q^2} + 1$ 
At the same time, the of $(1)$ is always an integer.
Thus, for the equality to hold infinitely many times, it must hold at zero:
Hence $\dfrac {-1 \pm \sqrt 5} 2$, is a rational number.
Hence $\sqrt 5$ is also rational.
But by Square Root of Prime is Irrational, $\sqrt 5$ is irrational.
This is a contradiction.
Hence by Proof by Contradiction there cannot be an infinite number of such $p, q$.
Hence the result.
== Sources ==
* : $11.8$: The measure of the closest approximation to an arbitrary irrational: Theorem $194$
Category:Hurwitz's Theorem (Number Theory)
Hurwitz's Theorem (Number Theory)/Lemma 2

== Lemma ==
Let $\xi$ be an irrational number.
Let there be $3$ consecutive convergents of the continued fraction to $\xi$.
Then at least one of them, $\dfrac p q$ say, satisfies:
:$\size {\xi - \dfrac p q} < \dfrac 1 {\sqrt 5 \, q^2}$
== Proof ==
By definition of simple infinite continued fraction, the partial denominators are strictly positive integers:
:$\forall n \in \N_{>0}: a_n > 0$
Let $\dfrac {p_k} {q_k}$ be an arbitrary convergent to $\xi$.
Let:
:$\dfrac {q_{n - 1} } {q_n} = b_n$
By definition of numerators and denominators of continued fraction:
:$q_{n + 1} = a_{n + 1} q_n + q_{n - 1}$
Then:
From Difference between Adjacent Convergents of Simple Continued Fraction:
Therefore:
We aim to show that given $3$ consecutive convergents of the continued fraction to $\xi$, at least one of them satisfies:
:$\size {\xi - \dfrac p q} < \dfrac 1 {\sqrt 5 \, q^2}$
 that $3$ consecutive convergents do NOT satisfy the inequality.
Thus for arbitrary $n$:
Therefore:
Using the Quadratic Formula, we arrive at:
Therefore:
:$\phi - 1 < b_{n + 1} < \phi$
Taking reciprocals:
:$\phi > \dfrac 1 {b_{n + 1} } > \phi - 1$
Starting with $(4)$ above and replacing $n$ with $n - 1$, and replacing $n + 1$ with $n$, an identical argument can be made for $b_n$.
We now have:
:$\phi - 1 < \dfrac 1 {b_{n + 1} } < \phi$
and
:$\phi - 1 < b_n < \phi$
And from $(2)$ above:
:$a_{n + 1} = \dfrac 1 {b_{n + 1} } - b_n$
Therefore, from $(2)$ and the two inequalities immediately above:
:$a_{n + 1} = \dfrac 1 {b_{n + 1} } - b_n < 1$
But this contradicts our premise that the partial denominators are strictly positive integers.
Hence the result by Proof by Contradiction.
== Sources ==
* : $11.8$: The measure of the closest approximation to an arbitrary irrational: Theorem $195$
Category:Hurwitz's Theorem (Number Theory)
Category:Continued Fractions
Huygens-Steiner Theorem

== Theorem ==
Let $B$ be a body of mass $M$.
Let $I_0$ be the moment of inertia of $B$ about some axis $A$ through the centre of mass of $B$.
Let $I$ be the moment of inertia of $B$ about another axis $A'$ parallel to $A$.
Then $I_0$ and $I$ are related by:
:$I = I_0 + M l^2$
where $l$ is the distance between $A$ and $A'$.
== Proof ==
, suppose $I$ is oriented along the $z$-axis.
By definition of moment of inertia:
:$\ds I = \sum m_j \lambda_j^2$
:$\ds I_0 = \sum m_j \lambda_j'^2$
where:
:$\lambda_j$ is the position vector to the $j$th particle from the $z$-axis
:$\lambda_j'$ is related to $\lambda_j$ by:
::$\lambda_j = \lambda_j' + R_\perp$
:$R_\perp$ is the perpendicular distance from $A$ to the center of mass of $B$.
Therefore:
:$\ds I = \sum m_j \lambda_j^2 = \sum m_j \paren {\lambda_j'^2 + 2 \lambda_j' \cdot R_\perp + R_\perp^2}$
The middle term is:
:$\ds 2 R_\perp \cdot \sum m_j \lambda_j' = 2 R_\perp \cdot \sum m_j \paren {\lambda_j - R_\perp} = 2 R_\perp \cdot M \paren {R_\perp - R_\perp} = 0$
Thus:
:$\ds I = \sum m_j \lambda_j^2 = \sum m_j \paren {\lambda_j'^2 + R_\perp^2} = I_0 + M l^2$
== Also known as ==
== Also see ==
* Perpendicular Axes Theorem
== Sources ==
* 
* 
Category:Huygens-Steiner Theorem
Category:Moments of Inertia
Hyperbola can be Drawn through Four Non-Collinear Points

== Theorem ==
Let $A, B, C, D$ be points in the plane of which no $3$ are collinear.
Then a hyperbola can be drawn so that it passes through all points $A, B, C, D$.
== Proof ==
== Sources ==
* : $4$
* : $4$
Category:Hyperbolas
Hyperbolic Cosecant Function is Odd

== Theorem ==
Let $\csch: \C \to \C$ be the hyperbolic cosecant function on the set of complex numbers.
Then $\csch$ is odd:
:$\map \csch {-x} = -\csch x$
== Proof ==
== Also see ==
* Hyperbolic Sine Function is Odd
* Hyperbolic Cosine Function is Even
* Hyperbolic Tangent Function is Odd
* Hyperbolic Cotangent Function is Odd
* Hyperbolic Secant Function is Even
* Cosecant Function is Odd
== Sources ==
* : $8.17$: Functions of Negative Arguments
Category:Hyperbolic Cosecant Function
Category:Examples of Odd Functions
Hyperbolic Cosecant in terms of Cosecant

== Theorem ==
Let $z \in \C$ be a complex number.
Then:
:$i \csch z = -\csc \paren {i z}$
where:
: $\csc$ denotes the cosecant function
: $\csch$ denotes the hyperbolic cosecant
: $i$ is the imaginary unit: $i^2 = -1$.
== Proof ==
 
== Also see ==
* Hyperbolic Sine in terms of Sine
* Hyperbolic Cosine in terms of Cosine
* Hyperbolic Tangent in terms of Tangent
* Hyperbolic Cotangent in terms of Cotangent
* Hyperbolic Secant in terms of Secant
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.77$: Relationship between Hyperbolic and Trigonometric Functions
Category:Cosecant Function
Category:Hyperbolic Cosecant Function
Hyperbolic Cosecant of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \csch {a + b i} = \dfrac {\sinh a \cos b - i \cosh a \sin b} {\sinh^2 a \cos^2 b + \cosh^2 a \sin^2 b}$
where:
:$\csch$ denotes the hyperbolic cosecant function.
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Cosecant Function
Category:Complex Numbers
Hyperbolic Cosine Function in terms of Hypergeometric Function

== Theorem ==
:$\cosh x = \map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {x^2} 4}$
where:
:$x$ is a real number with $x \in \R$
:$\cosh$ denotes the hyperbolic cosine function
:$\map { {}_0 \operatorname F_1} {a; x}$ is the generalized hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac 1 { a^{\overline k} } \dfrac {x^k} {k!}$
== Proof ==
== Also see ==
* Hyperbolic Sine Function in terms of Hypergeometric Function
* Cosine Function in terms of Hypergeometric Function
* Sine Function in terms of Hypergeometric Function
Category:Hypergeometric Functions
Category:Hyperbolic Cosine Function
Hyperbolic Cosine Function is Even

== Theorem ==
Let $\cosh: \C \to \C$ be the hyperbolic cosine function on the set of complex numbers.
Then $\cosh$ is even:
:$\map \cosh {-x} = \cosh x$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Hyperbolic Sine Function is Odd
* Hyperbolic Tangent Function is Odd
* Hyperbolic Cotangent Function is Odd
* Hyperbolic Secant Function is Even
* Hyperbolic Cosecant Function is Odd
* Cosine Function is Even
== Sources ==
* : $8.15$: Functions of Negative Arguments
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
* 
* 
* 
* 
Category:Hyperbolic Cosine Function
Category:Hyperbolic Cosine Function is Even
Category:Examples of Even Functions
Hyperbolic Cosine in terms of Cosine

== Theorem ==
Let $z \in \C$ be a complex number.
Then:
:$\cosh z = \map \cos {i z}$
where:
:$\cos$ denotes the complex cosine
:$\cosh$ denotes the hyperbolic cosine
:$i$ is the imaginary unit: $i^2 = -1$.
== Proof ==
 
== Also see ==
* Hyperbolic Sine in terms of Sine
* Hyperbolic Tangent in terms of Tangent
* Hyperbolic Cotangent in terms of Cotangent
* Hyperbolic Secant in terms of Secant
* Hyperbolic Cosecant in terms of Cosecant
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $(4.22)$
* : $\S 8$: Hyperbolic Functions: $8.75$: Relationship between Hyperbolic and Trigonometric Functions
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
* 
* 
Category:Cosine Function
Category:Hyperbolic Cosine Function
Hyperbolic Cosine of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \cosh {a + b i} = \cosh a \cos b + i \sinh a \sin b$
where:
:$\cos$ denotes the real cosine function
:$\sin$ denotes the real sine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Cosine Function
Category:Complex Numbers
Category:Hyperbolic Cosine of Complex Number
Hyperbolic Cosine of Complex Number/Proof 1

== Theorem ==
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Cosine of Complex Number
Hyperbolic Cosine of Complex Number/Proof 2

== Theorem ==
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Cosine of Complex Number
Hyperbolic Cosine of Sum

== Theorem ==
:$\map \cosh {a + b} = \cosh a \cosh b + \sinh a \sinh b$
where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine.
=== Corollary ===
== Proof ==
== Also see ==
* Hyperbolic Sine of Sum
* Hyperbolic Tangent of Sum
* Hyperbolic Cotangent of Sum
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.21$: Addition Formulas
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
* 
Category:Hyperbolic Cosine Function
Category:Addition Formulas for Hyperbolic Functions
Hyperbolic Cosine of Sum/Corollary

== Corollary of Hyperbolic Cosine of Sum ==
:$\map \cosh {a - b} = \cosh a \cosh b - \sinh a \sinh b$
where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine.
== Proof ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.21$: Addition Formulas
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
Category:Subtraction Formulas for Hyperbolic Functions
Category:Hyperbolic Cosine Function
Hyperbolic Cotangent Function is Odd

== Theorem ==
Let $\coth: \C \to \C$ be the hyperbolic cotangent function on the set of complex numbers.
Then $\coth$ is odd:
:$\map \coth {-x} = -\coth x$
== Proof ==
== Also see ==
* Hyperbolic Sine Function is Odd
* Hyperbolic Cosine Function is Even
* Hyperbolic Tangent Function is Odd
* Hyperbolic Secant Function is Even
* Hyperbolic Cosecant Function is Odd
* Cotangent Function is Odd
== Sources ==
* : $8.19$: Functions of Negative Arguments
Category:Hyperbolic Cotangent Function
Category:Examples of Odd Functions
Hyperbolic Cotangent in terms of Cotangent

== Theorem ==
Let $z \in \C$ be a complex number.
Then:
:$\coth z = -\cot \paren {i z}$
where:
: $\cot$ denotes the cotangent function
: $\coth$ denotes the hyperbolic cotangent
: $i$ is the imaginary unit: $i^2 = -1$.
== Proof ==
 
== Also see ==
* Hyperbolic Sine in terms of Sine
* Hyperbolic Cosine in terms of Cosine
* Hyperbolic Tangent in terms of Tangent
* Hyperbolic Secant in terms of Secant
* Hyperbolic Cosecant in terms of Cosecant
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.79$: Relationship between Hyperbolic and Trigonometric Functions
Category:Cotangent Function
Category:Hyperbolic Cotangent Function
Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent

== Theorem ==
:$\coth x = \dfrac 1 {\tanh x}$
where $\tanh$ and $\coth$ denote hyperbolic tangent and hyperbolic cotangent respectively.
== Proof ==
== Sources ==
* : $8.8$: Relationships among Hyperbolic Functions
* 
* 
Category:Hyperbolic Cotangent Function
Category:Hyperbolic Tangent Function
Hyperbolic Cotangent of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
=== Formulation 1 ===
=== Formulation 2 ===
=== Formulation 3 ===
where:
:$\cot$ denotes the real cotangent function
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function
:$\coth$ denotes the hyperbolic cotangent function.
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cosecant of Complex Number
Category:Hyperbolic Cotangent Function
Category:Complex Numbers
Category:Hyperbolic Cotangent of Complex Number
Hyperbolic Cotangent of Complex Number/Formulation 1

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\coth \paren {a + b i} = \dfrac {\cosh a \cos b + i \sinh a \sin b} {\sinh a \cos b + i \cosh a \sin b}$
where:
:$\coth$ denotes the hyperbolic cotangent function
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function.
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
Category:Hyperbolic Cotangent of Complex Number
Hyperbolic Cotangent of Complex Number/Formulation 2

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \coth {a + b i} = \dfrac {1 - i \coth a \cot b} {\coth a - i \cot b}$
where:
:$\cot$ denotes the real cotangent function
:$\coth$ denotes the hyperbolic cotangent function.
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
Category:Hyperbolic Cotangent of Complex Number
Hyperbolic Cotangent of Complex Number/Formulation 3

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \coth {a + b i} = \dfrac {\coth a + \coth a \cot^2 b} {\coth^2 a + \cot^2 b} + \dfrac {\cot b - \coth^2 a \cot b} {\coth^2 a + \cot^2 b} i$
where:
:$\cot$ denotes the real cotangent function
:$\coth$ denotes the hyperbolic cotangent function.
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
Category:Hyperbolic Cotangent of Complex Number
Hyperbolic Cotangent of Sum

== Theorem ==
:$\map \coth {a + b} = \dfrac {\coth a \coth b + 1} {\coth b + \coth a}$
where $\coth$ denotes the hyperbolic cotangent.
=== Corollary ===
== Proof ==
== Also see ==
* Hyperbolic Sine of Sum
* Hyperbolic Cosine of Sum
* Hyperbolic Tangent of Sum
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.23$: Addition Formulas
Category:Hyperbolic Cotangent Function
Category:Addition Formulas for Hyperbolic Functions
Hyperbolic Cotangent of Sum/Corollary

== Corollary to Hyperbolic Cotangent of Sum ==
:$\map \coth {a - b} = \dfrac {\coth a \coth b - 1} {\coth b - \coth a}$
where $\coth$ denotes the hyperbolic cotangent.
== Proof ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.23$: Addition Formulas
Category:Subtraction Formulas for Hyperbolic Functions
Category:Hyperbolic Cotangent Function
Hyperbolic Secant Function is Even

== Theorem ==
Let $\sech: \C \to \C$ be the hyperbolic secant function on the set of complex numbers.
Then $\sech$ is even:
:$\map \sech {-x} = \sech x$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Hyperbolic Sine Function is Odd
* Hyperbolic Cosine Function is Even
* Hyperbolic Tangent Function is Odd
* Hyperbolic Cotangent Function is Odd
* Hyperbolic Cosecant Function is Odd
* Secant Function is Even
== Sources ==
* : $8.18$: Functions of Negative Arguments
Category:Hyperbolic Secant Function
Category:Hyperbolic Secant Function is Even
Category:Examples of Even Functions
Hyperbolic Secant in terms of Secant

== Theorem ==
Let $z \in \C$ be a complex number.
Then:
:$\sech z = \sec \paren {i z}$
where:
: $\sec$ denotes the secant function
: $\sech$ denotes the hyperbolic secant
: $i$ is the imaginary unit: $i^2 = -1$.
== Proof ==
 
== Also see ==
* Hyperbolic Sine in terms of Sine
* Hyperbolic Cosine in terms of Cosine
* Hyperbolic Tangent in terms of Tangent
* Hyperbolic Cotangent in terms of Cotangent
* Hyperbolic Cosecant in terms of Cosecant
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.78$: Relationship between Hyperbolic and Trigonometric Functions
Category:Secant Function
Category:Hyperbolic Secant Function
Hyperbolic Secant of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\sech \paren {a + b i} = \dfrac {\cosh a \cos b - i \sinh a \sin b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}$
where:
:$\sech$ denotes the hyperbolic secant function.
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Secant Function
Category:Complex Numbers
Hyperbolic Sine Function in terms of Hypergeometric Function

== Theorem ==
:$\sinh x = x \paren {\map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {x^2} 4} }$
where:
:$x$ is a real number with $x \in \R$
:$\sinh$ denotes the hyperbolic sine function
:$\map { {}_0 \operatorname F_1} {a; x}$ is the generalized hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac 1 { a^{\overline k} } \dfrac {x^k} {k!}$
== Proof ==
== Also see ==
* Cosine Function in terms of Hypergeometric Function
* Sine Function in terms of Hypergeometric Function
* Hyperbolic Cosine Function in terms of Hypergeometric Function
Category:Hypergeometric Functions
Category:Hyperbolic Sine Function