# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/List of definitions

Definition:Prime number

A natural number ${\displaystyle {}n\geq 2}$ is called a prime number if it is only divisible

by ${\displaystyle {}1}$ and by ${\displaystyle {}n}$.

Definition:Empty set

The set which does not contain any element is called the empty set, denoted by

${\displaystyle \emptyset .}$

Definition:Subset

Let ${\displaystyle {}T}$ and ${\displaystyle {}M}$ denote sets. ${\displaystyle {}T}$ is called a subset

of ${\displaystyle {}M}$ if every element of ${\displaystyle {}T}$ is also an element of ${\displaystyle {}M}$.

Definition:Intersection

For sets ${\displaystyle {}L}$ und ${\displaystyle {}M}$, we call

${\displaystyle {}L\cap M={\left\{x\mid x\in L{\text{ and }}x\in M\right\}}\,}$

the intersection

of the two sets.

Definition:Union

For sets ${\displaystyle {}L}$ und ${\displaystyle {}M}$, we call

${\displaystyle {}L\cup M={\left\{x\mid x\in L{\text{ or }}x\in M\right\}}\,}$

the union

of the sets.

Definition:Cartesian product

Suppose that two sets ${\displaystyle {}L}$ and ${\displaystyle {}M}$ are given. Then the set

${\displaystyle {}L\times M={\left\{(x,y)\mid x\in L,\,y\in M\right\}}\,}$
is called the product set of the sets.

Definition:Mapping

Let ${\displaystyle {}L}$ and ${\displaystyle {}M}$ denote sets. A mapping ${\displaystyle {}F}$ from ${\displaystyle {}L}$ to ${\displaystyle {}M}$ is given by assigning, to every element of the set ${\displaystyle {}L}$, exactly one element of the set ${\displaystyle {}M}$. The unique element which is assigned to ${\displaystyle {}x\in L}$, is denoted by ${\displaystyle {}F(x)}$. For the mapping as a whole, we write

${\displaystyle F\colon L\longrightarrow M,x\longmapsto F(x).}$

Definition:Injective

Let ${\displaystyle {}L}$ and ${\displaystyle {}M}$ denote sets, and let

${\displaystyle F\colon L\longrightarrow M,x\longmapsto F(x),}$

be a mapping. Then ${\displaystyle {}F}$ is called injective, if for two different elements ${\displaystyle {}x,x'\in L}$, also ${\displaystyle {}F(x)}$ and ${\displaystyle {}F(x')}$

are different.

Definition:Surjective

Let ${\displaystyle {}L}$ and ${\displaystyle {}M}$ denote sets, and let

${\displaystyle F\colon L\longrightarrow M,x\longmapsto F(x),}$

be a mapping. Then ${\displaystyle {}F}$ is called surjective, if for every ${\displaystyle {}y\in M}$, there exists at least one element ${\displaystyle {}x\in L}$, such that

${\displaystyle {}F(x)=y\,.}$

Definition:Bijective

Let ${\displaystyle {}M}$ and ${\displaystyle {}L}$ denote sets and suppose that

${\displaystyle F\colon M\longrightarrow L,x\longmapsto F(x),}$

is a mapping. Then ${\displaystyle {}F}$ is called bijective if ${\displaystyle {}F}$ is injective as well as

surjective.

Definition:Inverse mapping

Let ${\displaystyle {}F\colon L\rightarrow M}$ denote a bijective mapping. Then the mapping

${\displaystyle G\colon M\longrightarrow L}$

which sends every element ${\displaystyle {}y\in M}$ to the uniquely determined element ${\displaystyle {}x\in L}$ with ${\displaystyle {}F(x)=y}$,

is called the inverse mapping of ${\displaystyle {}F}$.

Definition:Composition

Let ${\displaystyle {}L,\,M}$ and ${\displaystyle {}N}$ denote sets, let

${\displaystyle F\colon L\longrightarrow M,x\longmapsto F(x),}$

and

${\displaystyle G\colon M\longrightarrow N,y\longmapsto G(y),}$

be mappings. Then the mapping

${\displaystyle G\circ F\colon L\longrightarrow N,x\longmapsto G(F(x)),}$

is called the composition of the mappings

${\displaystyle {}F}$ and ${\displaystyle {}G}$.

Definition:Operation

An operation (or binary operation) ${\displaystyle {}\circ }$ on a set ${\displaystyle {}M}$ is a mapping

${\displaystyle \circ \colon M\times M\longrightarrow M,(x,y)\longmapsto \circ (x,y)=x\circ y.}$

Definition:Field

A set ${\displaystyle {}K}$ is called a field if there are two binary operations (called addition and multiplication)

${\displaystyle +:K\times K\longrightarrow K{\text{ and }}\cdot :K\times K\longrightarrow K}$

and two different elements ${\displaystyle {}0,1\in K}$, which fulfill the following properties.

1. Law of associativity: ${\displaystyle {}(a+b)+c=a+(b+c)}$ holds for all ${\displaystyle {}a,b,c\in K}$.
2. Law of commutativity: ${\displaystyle {}a+b=b+a}$ holds for all ${\displaystyle {}a,b\in K}$.
3. ${\displaystyle {}0}$ is the neutral element of the addition, i.e. ${\displaystyle {}a+0=a}$ holds for all ${\displaystyle {}a\in K}$.
4. Existence of the negative: For every ${\displaystyle {}a\in K}$, there exists an element ${\displaystyle {}b\in K}$ with ${\displaystyle {}a+b=0}$.
2. Axioms of the multiplication:
1. Law of associativity: ${\displaystyle {}(a\cdot b)\cdot c=a\cdot (b\cdot c)}$ holds for all ${\displaystyle {}a,b,c\in K}$.
2. Law of commutativity: ${\displaystyle {}a\cdot b=b\cdot a}$ holds for all ${\displaystyle {}a,b\in K}$.
3. ${\displaystyle {}1}$ is the neutral element for the multiplication, i.e. ${\displaystyle {}a\cdot 1=a}$ holds for all ${\displaystyle {}a\in K}$.
4. Existence of the inverse: For every ${\displaystyle {}a\in K}$ with ${\displaystyle {}a\neq 0}$, there exists an element ${\displaystyle {}c\in K}$ such that ${\displaystyle {}a\cdot c=1}$.
3. Law of distributivity: ${\displaystyle {}a\cdot (b+c)=(a\cdot b)+(a\cdot c)}$ holds for all ${\displaystyle {}a,b,c\in K}$.

Definition:Factorial

For a natural number ${\displaystyle {}n}$, one puts

${\displaystyle {}n!:=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1\,,}$
and calls this ${\displaystyle {}n}$ factorial.

Definition:Binomial coefficient

Let ${\displaystyle {}k}$ and ${\displaystyle {}n}$ denote natural numbers with ${\displaystyle {}k\leq n}$. Then

${\displaystyle {}{\binom {n}{k}}:={\frac {n!}{k!(n-k)!}}\,}$
is called the binomial coefficient ${\displaystyle {}n}$ choose ${\displaystyle {}k}$

Definition:Ordered field

A field ${\displaystyle {}K}$ is called an ordered field, if there is a relation ${\displaystyle {}>}$ (larger than) between the elements of ${\displaystyle {}K}$, fulfilling the following properties (${\displaystyle {}a\geq b}$ means ${\displaystyle {}a>b}$ or ${\displaystyle {}a=b}$).

1. For two elements ${\displaystyle {}a,b\in K}$, we have either ${\displaystyle {}a>b}$ or ${\displaystyle {}a=b}$ or ${\displaystyle {}b>a}$.
2. From ${\displaystyle {}a\geq b}$ and ${\displaystyle {}b\geq c}$, one may deduce ${\displaystyle {}a\geq c}$ (for any ${\displaystyle {}a,b,c\in K}$).
3. ${\displaystyle {}a\geq b}$ implies ${\displaystyle {}a+c\geq b+c}$ (for any ${\displaystyle {}a,b,c\in K}$).
4. From ${\displaystyle {}a\geq 0}$ and ${\displaystyle {}b\geq 0}$, one may deduce ${\displaystyle {}ab\geq 0}$ (for any ${\displaystyle {}a,b\in K}$).

Definition:Archimedean ordered field

Let ${\displaystyle {}K}$ be an ordered field. ${\displaystyle {}K}$ is called Archimedean, if the following Archimedean axiom holds, i.e. if for every ${\displaystyle {}x\in K}$ there exists a natural number ${\displaystyle {}n}$ such that

${\displaystyle {}n\geq x\,.}$

Definition:Real intervals

For real numbers ${\displaystyle {}a,b}$, ${\displaystyle {}a\leq b}$, we call

1. ${\displaystyle {}[a,b]={\left\{x\in \mathbb {R} \mid x\geq a{\text{ and }}x\leq b\right\}}}$ the closed interval.
2. ${\displaystyle {}]a,b[={\left\{x\in \mathbb {R} \mid x>a{\text{ and }}x the open interval.
3. ${\displaystyle {}]a,b]={\left\{x\in \mathbb {R} \mid x>a{\text{ and }}x\leq b\right\}}}$ the half-open interval (closed on the right).
4. ${\displaystyle {}[a,b[={\left\{x\in \mathbb {R} \mid x\geq a{\text{ and }}x the half-open interval (closed on the left).

Definition:Floor

For a real number ${\displaystyle {}x}$, the floor ${\displaystyle {}\left\lfloor x\right\rfloor }$ is defined as

${\displaystyle \left\lfloor x\right\rfloor =n,{\text{ if }}x\in [n,n+1[{\text{ and }}n\in \mathbb {Z} .}$

Definition:Modulus of a real number

For a real number ${\displaystyle {}x\in \mathbb {R} }$, the modulus is defined in the following way.

${\displaystyle {}\vert {x}\vert ={\begin{cases}x\,,{\text{ if }}x\geq 0\,,\\-x,\,{\text{ if }}x<0\,.\end{cases}}\,}$

Definition:Increasing function

Let ${\displaystyle {}I\subseteq \mathbb {R} }$ denote an interval and let

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

denote a function. Then ${\displaystyle {}f}$ is called increasing, if

${\displaystyle f(x')\geq f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'\geq x.}$

Definition:Decreasing function

Let ${\displaystyle {}I\subseteq \mathbb {R} }$ denote an interval and let

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

denote a function. Then ${\displaystyle {}f}$ is called decreasing if

${\displaystyle f(x')\leq f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'\geq x.}$

Definition:Strictly increasing function

Let ${\displaystyle {}I\subseteq \mathbb {R} }$ denote an interval and let

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

denote a function. Then ${\displaystyle {}f}$ is called strictly increasing if

${\displaystyle f(x')>f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'>x.}$

Definition:Strictly decreasing function

Let ${\displaystyle {}I\subseteq \mathbb {R} }$ denote an interval and let

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

denote a function. Then ${\displaystyle {}f}$ is called strictly decreasing if

${\displaystyle f(x')x.}$

Definition:Complex numbers

The set ${\displaystyle {}\mathbb {R} ^{2}}$ with ${\displaystyle {}0:=(0,0)}$ and ${\displaystyle {}1:=(1,0)}$, with componentwise addition and the multiplication defined by

${\displaystyle {}(a,b)\cdot (c,d):=(ac-bd,ad+bc)\,,}$

is called the field of complex numbers. We denote it by

${\displaystyle \mathbb {C} .}$

Definition:Real part, imaginary part

For a complex number

${\displaystyle {}z=a+b{\mathrm {i} }\,,}$

we call

${\displaystyle {}\operatorname {Re} \,{\left(z\right)}=a\,}$

the real part of ${\displaystyle {}z}$ and

${\displaystyle {}\operatorname {Im} \,{\left(z\right)}=b\,}$
the imaginary part of ${\displaystyle {}z}$.

Definition:Complex conjugation

The mapping

${\displaystyle \mathbb {C} \longrightarrow \mathbb {C} ,z=a+b{\mathrm {i} }\longmapsto {\overline {z}}:=a-b{\mathrm {i} },}$
is called complex conjugation.

Definition:Modulus of a complex number

For a complex number

${\displaystyle {}z=a+b{\mathrm {i} }\,,}$

the modulus is defined by

${\displaystyle {}\vert {z}\vert ={\sqrt {a^{2}+b^{2}}}\,.}$

Definition:Polynomial in one variable

Let ${\displaystyle {}K}$ be a field. An expression of the form

${\displaystyle {}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,,}$

with ${\displaystyle {}a_{i}\in K}$ and ${\displaystyle {}n\in \mathbb {N} }$,

is called a polynomial in one variable over ${\displaystyle {}K}$.

Definition:Degree of a polynomial

The degree of a nonzero polynomial

${\displaystyle {}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,}$

with ${\displaystyle {}a_{n}\neq 0}$

is ${\displaystyle {}n}$.

Definition:Rational function

For polynomials ${\displaystyle {}P,Q\in \mathbb {R} [X]}$, ${\displaystyle {}Q\neq 0}$, the function

${\displaystyle D\longrightarrow \mathbb {R} ,z\longmapsto {\frac {P(z)}{Q(z)}},}$

where ${\displaystyle {}D}$ is the complement of the zeroes

of ${\displaystyle {}Q}$, is called a rational function.

Definition:Real sequence

A real sequence is a mapping

${\displaystyle \mathbb {N} \longrightarrow \mathbb {R} ,n\longmapsto x_{n}.}$

Definition:Heron sequence

Let ${\displaystyle {}c\in \mathbb {R} _{+}}$ denote a positive real number. The Heron-sequence, with the positive initial value ${\displaystyle {}x_{0}}$, is defined recursively by

${\displaystyle {}x_{n+1}:={\frac {x_{n}+{\frac {c}{x_{n}}}}{2}}\,}$

Definition:Convergent sequence

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ denote a real sequence, and let ${\displaystyle {}x\in \mathbb {R} }$. We say that the sequence converges to ${\displaystyle {}x}$, if the following property holds.

For every positive ${\displaystyle {}\epsilon >0}$, ${\displaystyle {}\epsilon \in \mathbb {R} }$, there exists some ${\displaystyle {}n_{0}\in \mathbb {N} }$, such that for all ${\displaystyle {}n\geq n_{0}}$, the estimate

${\displaystyle {}\vert {x_{n}-x}\vert \leq \epsilon \,}$

holds.

If this condition is fulfilled, then ${\displaystyle {}x}$ is called the limit of the sequence. For this we write

${\displaystyle {}\lim _{n\rightarrow \infty }x_{n}:=x\,.}$
If the sequence converges to a limit, we just say that the sequence converges, otherwise, that the sequence diverges.

Definition:Bounded subset

A subset ${\displaystyle {}M\subseteq \mathbb {R} }$ of the real numbers is called bounded, if there exist real numbers ${\displaystyle {}s\leq S}$ such that

${\displaystyle {}M\subseteq [s,S]}$.

Definition:Increasing sequence

A real sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ is called increasing, if ${\displaystyle {}x_{n+1}\geq x_{n}}$ holds for all

${\displaystyle {}n\in \mathbb {N} }$.

Definition:Decreasing sequence

A real sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ is called decreasing, if ${\displaystyle {}x_{n+1}\leq x_{n}}$ holds for all

${\displaystyle {}n\in \mathbb {N} }$.

Definition:Cauchy sequence

A real sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ is called a Cauchy sequence, if the following condition holds.

For every ${\displaystyle {}\epsilon >0}$, there exists an ${\displaystyle {}n_{0}\in \mathbb {N} }$, such that for all ${\displaystyle {}n,m\geq n_{0}}$, the estimate

${\displaystyle {}\vert {x_{n}-x_{m}}\vert \leq \epsilon \,}$
holds.

Definition:Subsequence

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be a real sequence. For any strictly increasing mapping ${\displaystyle {}\mathbb {N} \rightarrow \mathbb {N} ,i\mapsto n_{i}}$, the sequence

${\displaystyle i\mapsto x_{n_{i}}}$
is called a subsequence of the sequence.

Definition:Completely ordered field

An ordered field ${\displaystyle {}K}$ is called complete or completely ordered, if every Cauchy sequence in ${\displaystyle {}K}$

converges.

Definition:Nested intervals

A sequence of closed intervals

${\displaystyle I_{n}=[a_{n},b_{n}],\,n\in \mathbb {N} ,}$

in ${\displaystyle {}\mathbb {R} }$ is called (a sequence of) nested intervals, if ${\displaystyle {}I_{n+1}\subseteq I_{n}}$ holds for all ${\displaystyle {}n\in \mathbb {N} }$, and if the sequence of the lengths of the intervals, i.e.

${\displaystyle {\left(b_{n}-a_{n}\right)}_{n\in \mathbb {N} },}$
to ${\displaystyle {}0}$.

Definition:Tending to ${\displaystyle {}+\infty }$

A real sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ is said to tend to ${\displaystyle {}+\infty }$, if for every ${\displaystyle {}s\in \mathbb {R} }$, there exists some ${\displaystyle {}N\in \mathbb {N} }$, such that

${\displaystyle x_{n}\geq s{\text{ holds for all }}n\geq N.}$

Definition:Tending to ${\displaystyle {}-\infty }$

A real sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ is said to tend to ${\displaystyle {}-\infty }$, if for every ${\displaystyle {}s\in \mathbb {R} }$, there exists some ${\displaystyle {}N\in \mathbb {N} }$. such that

${\displaystyle x_{n}\leq s{\text{ holds for all }}n\geq N.}$

Definition:Series

Let ${\displaystyle {}{\left(a_{k}\right)}_{k\in \mathbb {N} }}$ be a sequence of real numbers. The series ${\displaystyle {}\sum _{k=0}^{\infty }a_{k}}$ is the sequence ${\displaystyle {}{\left(s_{n}\right)}_{n\in \mathbb {N} }}$ of the partial sums

${\displaystyle {}s_{n}:=\sum _{k=0}^{n}a_{k}\,.}$

If the sequence ${\displaystyle {}{\left(s_{n}\right)}_{n\in \mathbb {N} }}$ converges, then we say that the series converges. In this case, we write also

${\displaystyle \sum _{k=0}^{\infty }a_{k}}$

for its limit,

and this limit is called the sum of the series.

Definition:Absolute convergence of a series

${\displaystyle \sum _{k=0}^{\infty }a_{k}}$

of real numbers is called absolutely convergent, if the series

${\displaystyle \sum _{k=0}^{\infty }\vert {a_{k}}\vert }$
converges.

Definition:Geometric series

For every ${\displaystyle {}x\in \mathbb {R} }$, the series

${\displaystyle \sum _{k=0}^{\infty }x^{k}}$
is called the geometric series in ${\displaystyle {}x}$.

Definition:Continuous function

Let ${\displaystyle {}D\subseteq \mathbb {R} }$ be a subset,

${\displaystyle f\colon D\longrightarrow \mathbb {R} }$

a function, and ${\displaystyle {}x\in D}$ a point. We say that ${\displaystyle {}f}$ is continuous in the point ${\displaystyle {}x}$, if for every ${\displaystyle {}\epsilon >0}$, there exists a ${\displaystyle {}\delta >0}$, such that for all ${\displaystyle {}x'\in D}$ fulfilling ${\displaystyle {}\vert {x-x'}\vert \leq \delta }$, the estimate ${\displaystyle {}\vert {f(x)-f(x')}\vert \leq \epsilon }$ holds. We say that ${\displaystyle {}f}$ continuous, if it is continuous in every point

${\displaystyle {}x\in D}$

Definition:Limit of a function

Let ${\displaystyle {}T\subseteq \mathbb {R} }$ denote a subset and ${\displaystyle {}a\in \mathbb {R} }$ a point. Let

${\displaystyle f\colon T\longrightarrow \mathbb {R} }$

be a function. Then ${\displaystyle {}b\in \mathbb {R} }$ is called limit of ${\displaystyle {}f}$ in ${\displaystyle {}a}$, if for every ${\displaystyle {}\epsilon >0}$ there exists some ${\displaystyle {}\delta >0}$ such that for all ${\displaystyle {}x\in T}$ fulfilling

${\displaystyle {}\vert {x-a}\vert \leq \delta \,,}$

the estimate

${\displaystyle {}\vert {f(x)-b}\vert \leq \epsilon \,}$

holds. In this case, we write

${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.}$

Definition:Maximum

Let ${\displaystyle {}M}$ denote a set, and

${\displaystyle f\colon M\longrightarrow \mathbb {R} }$

a function. We say that ${\displaystyle {}f}$ attains in a point ${\displaystyle {}x\in M}$ its maximum, if

${\displaystyle f(x)\geq f(x'){\text{ holds for all }}x'\in M.}$

Definition:Minimum

Let ${\displaystyle {}M}$ denote a set, and

${\displaystyle f\colon M\longrightarrow \mathbb {R} }$

a function. We say that ${\displaystyle {}f}$ attains in a point ${\displaystyle {}x\in M}$ its minimum, if

${\displaystyle f(x)\leq f(x'){\text{ holds for all }}x'\in M.}$

Definition:Power series

Let ${\displaystyle {}{\left(c_{n}\right)}_{n\in \mathbb {N} }}$ be a sequence of real numbers and ${\displaystyle {}x}$ another real number. Then the series

${\displaystyle \sum _{n=0}^{\infty }c_{n}x^{n}}$
is called the power series in ${\displaystyle {}x}$ for the coefficients ${\displaystyle {}{\left(c_{n}\right)}_{n\in \mathbb {N} }}$.

Definition:Cauchy product

For two series ${\displaystyle {}\sum _{i=0}^{\infty }a_{i}}$ and ${\displaystyle {}\sum _{j=0}^{\infty }b_{j}}$ of real numbers, the series

${\displaystyle \sum _{k=0}^{\infty }c_{k}{\text{ with }}c_{k}:=\sum _{i=0}^{k}a_{i}b_{k-i}}$
is called the Cauchy-product of the series.

Definition:Exponential series

For every ${\displaystyle {}x\in \mathbb {R} }$, the series

${\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$
is called the exponential series in ${\displaystyle {}x}$.

Definition:Exponential function

The function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \exp x:=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}},}$

is called the (real)

exponential function.

Definition:Euler's number

The real number

${\displaystyle {}e:=\sum _{k=0}^{\infty }{\frac {1}{k!}}\,}$
is called Euler's number.

Definition:Natural logarithm

The natural logarithm

${\displaystyle \ln \colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto \ln x,}$

is defined as the inverse function of the

real exponential function.

Definition:Exponential function to base

For a positive real number ${\displaystyle {}b>0}$, the exponential function for the base ${\displaystyle {}b}$ is defined as

${\displaystyle {}b^{x}:=\exp(x\ln b)\,.}$

Definition:Logarithm to base

For a positive real number ${\displaystyle {}b>0}$, ${\displaystyle {}b\neq 1}$, the logarithm to base ${\displaystyle {}b}$ of ${\displaystyle {}x\in \mathbb {R} _{+}}$ is defined by

${\displaystyle {}\log _{b}x:={\frac {\ln x}{\ln b}}\,.}$

Definition:Hyperbolic sine

The function defined for ${\displaystyle {}x\in \mathbb {R} }$ by

${\displaystyle {}\sinh x:={\frac {1}{2}}{\left(e^{x}-e^{-x}\right)}\,,}$
is called hyperbolic sine.

Definition:Hyperbolic cosine

The function defined for ${\displaystyle {}x\in \mathbb {R} }$ by

${\displaystyle {}\cosh x:={\frac {1}{2}}{\left(e^{x}+e^{-x}\right)}\,,}$
is called hyperbolic cosine.

Definition:Hyperbolic tangent

The function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}},}$
is called hyperbolic tangent.

Definition:Even function

A function ${\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }$ is called even, if for all ${\displaystyle {}x\in \mathbb {R} }$, the identity

${\displaystyle {}f(x)=f(-x)\,}$
holds.

Definition:Odd function

A function ${\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }$ is called odd, if for all ${\displaystyle {}x\in \mathbb {R} }$, the identity

${\displaystyle {}f(x)=-f(-x)\,}$
holds.