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

Definition:Prime number

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

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

Definition:Empty set

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

$\emptyset .$

Definition:Subset

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

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

Definition:Intersection

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

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

the intersection

of the two sets.

Definition:Union

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

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

the union

of the sets.

Definition:Cartesian product

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

${}L\times M={\left\{(x,y)\mid x\in L,\,y\in M\right\}}\,$

is called the product set

(or Cartesian product) of the sets.

Definition:Mapping

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

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

Definition:Injective

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

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

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

are different.

Definition:Surjective

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

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

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

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

Definition:Bijective

Let ${}M$ and ${}L$ denote sets, and suppose that

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

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

surjective.

Definition:Inverse mapping

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

$G\colon M\longrightarrow L$

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

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

Definition:Composition

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

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

and

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

be mappings. Then the mapping

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

is called the composition of the mappings

${}F$ and ${}G$.

Definition:Operation

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

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

Definition:Field

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

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

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

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

Definition:Factorial

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

${}n!:=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1\,,$

and calls this ${}n$ factorial.

Definition:Binomial coefficient

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

${}{\binom {n}{k}}:={\frac {n!}{k!(n-k)!}}\,$

is called the binomial coefficient ${}n$ choose ${}k$

Definition:Ordered field

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

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

Definition:Archimedean ordered field

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

${}n\geq x\,.$

Definition:Real intervals

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

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

Definition:Floor

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

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

Definition:Modulus of a real number

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

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

Definition:Increasing function

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

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

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

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

Definition:Decreasing function

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

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

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

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

Definition:Strictly increasing function

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

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

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

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

Definition:Strictly decreasing function

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

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

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

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

Definition:Complex numbers

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

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

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

$\mathbb {C} .$

Definition:Real part, imaginary part

For a complex number

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

we call

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

the real part of ${}z$ and

${}\operatorname {Im} \,{\left(z\right)}=b\,$

the imaginary part of ${}z$.

Definition:Complex conjugation

The mapping

$\mathbb {C} \longrightarrow \mathbb {C} ,z=a+b{\mathrm {i} }\longmapsto {\overline {z}}:=a-b{\mathrm {i} },$

is called complex conjugation.

Definition:Modulus of a complex number

For a complex number

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

the modulus is defined by

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

Definition:Polynomial in one variable

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

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

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

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

Definition:Degree of a polynomial

The degree of a nonzero polynomial

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

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

is ${}n$.

Definition:Rational function

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

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

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

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

Definition:Real sequence

A real sequence is a mapping

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

Definition:Heron sequence

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

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

Definition:Convergent sequence

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

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

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

holds.

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

${}\lim _{n\rightarrow \infty }x_{n}:=x\,.$

If the sequence converges to a limit, we just say that the sequence converges, otherwise, that the sequence diverges.

Definition:Bounded subset

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

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

Definition:Increasing sequence

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

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

Definition:Decreasing sequence

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

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

Definition:Cauchy sequence

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

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

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

holds.

Definition:Subsequence

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

$i\mapsto x_{n_{i}}$

is called a subsequence of the sequence.

Definition:Completely ordered field

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

converges.

Definition:Nested intervals

A sequence of closed intervals

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

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

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

converges

to ${}0$.

Definition:Tending to ${}+\infty$

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

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

Definition:Tending to ${}-\infty$

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

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

Definition:Series

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

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

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

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

for its limit,

and this limit is called the sum of the series.

Definition:Absolute convergence of a series

A series

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

of real numbers is called absolutely convergent, if the series

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

converges.

Definition:Geometric series

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

$\sum _{k=0}^{\infty }x^{k}$

is called the geometric series in ${}x$.

Definition:Continuous function

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

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

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

${}x\in D$

Definition:Limit of a function

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

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

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

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

the estimate

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

holds. In this case, we write

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

Definition:Maximum

Let ${}M$ denote a set, and

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

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

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

Definition:Minimum

Let ${}M$ denote a set, and

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

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

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

Definition:Power series

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

$\sum _{n=0}^{\infty }c_{n}x^{n}$

is called the power series in ${}x$ for the coefficients ${}{\left(c_{n}\right)}_{n\in \mathbb {N} }$.

Definition:Cauchy product

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

$\sum _{k=0}^{\infty }c_{k}{\text{ with }}c_{k}:=\sum _{i=0}^{k}a_{i}b_{k-i}$

is called the Cauchy-product of the series.

Definition:Exponential series

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

$\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$

is called the exponential series in ${}x$.

Definition:Exponential function

The function

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

is called the (real)

exponential function.

Definition:Euler's number

The real number

${}e:=\sum _{k=0}^{\infty }{\frac {1}{k!}}\,$

is called Euler's number.

Definition:Natural logarithm

The natural logarithm

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

is defined as the inverse function of the

real exponential function.

Definition:Exponential function to base

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

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

Definition:Logarithm to base

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

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

Definition:Hyperbolic sine

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

${}\sinh x:={\frac {1}{2}}{\left(e^{x}-e^{-x}\right)}\,,$

is called hyperbolic sine.

Definition:Hyperbolic cosine

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

${}\cosh x:={\frac {1}{2}}{\left(e^{x}+e^{-x}\right)}\,,$

is called hyperbolic cosine.

Definition:Hyperbolic tangent

The function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}},$

is called hyperbolic tangent.

Definition:Even function

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

${}f(x)=f(-x)\,$

holds.

Definition:Odd function

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

${}f(x)=-f(-x)\,$

holds.