hcju/proofwikips huggingface dataset:
Definition:Compact Space/Normed Vector Space/Subspace
Let $M = \struct{X, \norm {\,\cdot\,}}$ be a normed vector space. Let $K \subseteq X$ be a subset of $X$. The normed vector subspace $M_K = \struct {K, \norm {\,\cdot\,}_K}$ is '''compact in $M$'''  $M_K$ is itself a compact normed vector space.
Definition:Relation/Notation
If $\tuple {x, y}$ is an ordered pair such that $\tuple {x, y} \in \RR$, we use the notation: :$s \mathrel \RR t$ or: :$\map \RR {s, t}$ and can say: :'''$s$ bears $\RR$ to $t$''' :'''$s$ stands in the relation $\RR$ to $t$''' If $\tuple {s, t} \notin \RR$, we can write: $s \not \mathrel \RR t$, that is, by drawing a line through the relation symbol. See Complement of Relation.
Definition:Relation/Class Theory
Let $V$ be a basic universe. In the context of class theory, a '''relation''' $\RR$ is a subclass of the Cartesian product $V \times V$.
Definition:Braces
'''Braces''' is the name given to the pair of '''curly brackets''': :$\set {\cdots}$ These are conventionally used mostly for: :$(1): \quad$ Set delimiters, to membership of sets and classes. :$(2): \quad$ the fractional part of a real number: ::::$\fractpart x := x - \floor x$ :::where $\floor x$ denotes the floor of $x$. :$(3): \quad$ Karamata notation for the Stirling numbers of the second kind: ::::$\displaystyle {n \brace k}$ On , which implements the $\LaTeX$ mathematical markup language, '''braces''' are also used to delimit arguments to $\LaTeX$ commands.
Definition:Non-Limit Ordinal
An ordinal $\lambda$ is a '''non-limit ordinal'''  it is not a limit ordinal. That is, either the zero ordinal or a successor ordinal.
Definition:Natural Numbers/Inductive Set Definition
Let $x$ be a set which is an element of every inductive set. Then $x$ is a natural number.
Definition:Distinguished Element
Let $S$ be a set. A '''distinguished element of $S$''' is an element which is distinguished in some manner, by the imposition of some property or properties on $S$, from the other elements of $S$.
Definition:Inductive Set
Let $S$ be a set of sets. Then $S$ is '''inductive''' : {{begin-axiom}} {{axiom | n = 1         | lc= $S$ contains the empty set:         | q =          | m = \quad \O \in S }} {{axiom | n = 2         | lc= $S$ is closed under the successor mapping:         | q = \forall x         | m = \paren {x \in S \implies x^+ \in S}         | rc= where $x^+$ is the successor of $x$ }} {{axiom | rc= That is, where $x^+ = x \cup \set x$ }} {{end-axiom}}
Definition:Roster Notation
'''Roster notation''' is the technique of specifying the elements in a set by listing them between a pair of braces $\set{}$. === Explicit Definition === {{:Definition:Explicit Set Definition}} If there are many elements in a set, then it becomes tedious and impractical to list them all in one big long explicit definition. Fortunately, however, there are other techniques for listing sets. === Implicit Definition === {{:Definition:Implicit Set Definition}}
Definition:Inductive Class/General
Let $A$ be a class. Let $g: A \to A$ be a mapping on $A$. Then $A$ is '''inductive under $g$''' : {{begin-axiom}} {{axiom | n = 1         | lc= $A$ contains the empty set:         | q =          | m = \quad \O \in A }} {{axiom | n = 2         | lc= $A$ is closed under $g$:         | q = \forall x         | m = \paren {x \in A \implies \map g x \in A} }} {{end-axiom}}
Definition:Minimally Inductive Class under General Mapping/Definition 1
$A$ is '''minimally inductive under $g$''' : {{begin-axiom}} {{axiom | n = 1         | t = $A$ is inductive under $g$ }} {{axiom | n = 2         | t = No proper subclass of $A$ is inductive under $g$. }} {{end-axiom}}
Definition:Minimally Inductive Class under General Mapping/Definition 2
$A$ is '''minimally inductive under $g$''' : {{begin-axiom}} {{axiom | n = 1         | t = $A$ is inductive under $g$ }} {{axiom | n = 2         | t = Every subclass of $A$ which is inductive under $g$ contains all the elements of $A$. }} {{end-axiom}}
Definition:Minimally Inductive Class under General Mapping
Let $A$ be a class. Let $g: A \to A$ be a mapping. === Definition 1 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 1}} === Definition 2 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 2}} === Definition 3 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 3}}
Definition:Left Normal Element of Relation
Let $A$ be a class. Let $\RR$ be a relation on $A$. An element $x$ of $A$ is '''left normal''' with respect to $\RR$ : :$\forall y \in A: \map \RR {x, y}$ holds.
Definition:Right Normal Element of Relation
Let $A$ be a class. Let $\RR$ be a relation on $A$. An element $x$ of $A$ is '''right normal''' with respect to $\RR$ : :$\forall y \in A: \map \RR {y, x}$ holds.
Definition:Comparable Sets
=== Comparable in Size === {{:Definition:Comparable Sets/Cardinality}} === Comparable by Subset Ordering === {{:Definition:Comparable Sets/Subset Ordering}}
Definition:Comparable Sets/Subset Ordering
Let $S$ and $T$ be sets. Then $S$ and $T$ are '''comparable (with respect to the subset ordering)'''  either: :$S \subseteq T$ or: :$T \subseteq S$ or both.
Definition:Nest/Class Theory
Let $C$ be a class. $C$ is a '''nest''' : :$\forall x, y \in C: x \subseteq y$ or $y \subseteq x$
Definition:Well-Ordered Class under Inclusion
Let $A$ be a class which is also a nest. Let $A$ have the property that every non-empty subclass of $A$ has a smallest element under the inclusion relation. Then $A$ is said to be '''well-ordered under inclusion'''.
Definition:Bounded Class
Let $B$ be a class. === Bounded by Set === {{:Definition:Bounded Class/Bounded by Set}} === Bounded Subset of Class === {{:Definition:Bounded Class/Bounded Subset of Class}}
Definition:Bounded Class/Bounded Subset of Class
Let $B$ be a subclass of a class $A$. Then $B$ is a '''bounded subset of $A$''' : :there exists a set $x \in A$ such that $B$ is '''bounded by $x$'''
Definition:Absolute Difference
Let $a$ and $b$ be real numbers. The '''absolute difference''' between $a$ and $b$ is defined and denoted as: :$\size {a - b}$ where $\size {\, \cdot \,}$ is the absolute value function.
Definition:Arbitrary Constant
An '''arbitrary constant''' is a symbol used to represent an object which is neither a specific number nor a variable. It is used to represent a general object (usually a number, but not necessarily) whose value can be assigned when the expression is instantiated. === In the context of Calculus === {{:Definition:Arbitrary Constant (Calculus)}}
Definition:Congruence (Number Theory)/Modulus
The number $m$ in this congruence is known as the '''modulus''' of the congruence.
Definition:Cubic (Geometry)
'''Cubic''' is an adjective which means '''in the shape of a cube'''.
Definition:Interior (Topology)/Definition 3
The '''interior''' of $H$ is the set of all interior points of $H$.
Definition:Interior Point (Topology)/Definition 3
Let $h \in H$. $h$ is an '''interior point''' of $H$  $h$ is an element of an open set of the subspace topology of $H$.
Definition:Weighted Mean/Normalized
Let the weights be normalized. Then the '''weighted mean''' of $S$ can be expressed in the form: :$\displaystyle \bar x := \sum_{i \mathop = 1}^n \map W {x_i} x_i$ as by definition of normalized weight function all the weights add up to $1$.
Definition:Operation/Operator
An '''operator''' is a symbol used to identify an '''operation'''.
Definition:Rate of Change/Dimension
Let a physical quantity $P$ have dimension $\mathsf D$. Then the '''rate of change''' of $P$ has dimension $\mathsf {D T}^{-1}$.
Definition:Section (Geometry)
=== Section of Line by Line === {{:Definition:Section of Line by Line}} === Definition:Section of Solid by Plane === {{:Definition:Section of Solid by Plane}}
Definition:Absolute Number
An '''absolute number''' is a number in an expression which has a single value. It is either expressed using actual figures, in an agreed number system, or by a symbol which is understood to represent that specific number. === Examples === :$2$, $\sqrt 5$, $1.976$, $\dfrac 2 3$, $\pi$ (pi), $e$ (Euler's number)
Definition:Long Period Prime
A '''long period prime''' is a prime number $p$ whose reciprocal has the maximum period $p - 1$.
Definition:Minimally Closed Class
Let $A$ be a class. Let $g: A \to A$ be a mapping. === Definition 1 === {{:Definition:Minimally Closed Class/Definition 1}} === Definition 2 === {{:Definition:Minimally Closed Class/Definition 2}}
Definition:Minimally Closed Class/Definition 1
$A$ is '''minimally closed under $g$ with respect to $b$''' : {{begin-axiom}} {{axiom | n = 1         | t = $A$ is closed under $g$ }} {{axiom | n = 2         | t = There exists $b \in A$ such that no proper subclass of $A$ containing $b$ is closed under $g$. }} {{end-axiom}}
Definition:Minimally Closed Class/Definition 2
$A$ is '''minimally closed under $g$ with respect to $b$''' : {{begin-axiom}} {{axiom | n = 1         | t = $A$ is closed under $g$ }} {{axiom | n = 2         | t = There exists $b \in A$ such that every subclass of $A$ containing $b$ which is closed under $g$ contains all the elements of $A$. }} {{end-axiom}}
Definition:Minimally Inductive Class under General Mapping/Definition 3
$A$ is '''minimally inductive under $g$'''  $A$ is minimally closed under $g$ with respect to $\O$.
Definition:Abstract Space
An '''abstract space''' is: :a set of objects together with: :a set of axioms which define operations on and relations between those objects. === Metric Space === {{:Definition:Metric Space}} === Topological Space === {{:Definition:Topological Space}} === Vector Space === {{:Definition:Vector Space}}
Definition:Rectangle Function
The '''rectangle function''' is the real function $\Pi: \R \to \R$ defined as: :$\forall x \in \R: \map \Pi x := \begin {cases} 1 : & \size x \le \dfrac 1 2 \\ 0 : & \size x > \dfrac 1 2 \end {cases}$
Definition:Triangle Function
The '''triangle function''' is the real function $\Lambda: \R \to \R$ defined as: :$\forall x \in \R: \map \Lambda x := \begin {cases} 1 - \size x : & \size x \le 1 \\ 0 : & \size x > 1 \end {cases}$ where $\size x$ denotes the absolute value function.
Definition:Rectangle Function/Graph
The graph of the rectangle function is illustrated below: :400px
Definition:Triangle Function/Graph
The graph of the triangle function is illustrated below: :400px
Definition:Heaviside Step Function/Graph
The graph of the Heaviside step function is illustrated below: :600px
Definition:Signum Function/Graph
The graph of the signum function is illustrated below: :400px
Definition:Dirac Delta Function/Graph
The graph of the construction of the Dirac delta function is illustrated below: :400px In the limit, the graph can be approximated as follows, where it is understood that the blue arrow represents a ray from $0$ up the $y$-axis: :400px
Definition:Lituus
The '''lituus''' is the locus of the equation expressed in Polar coordinates as: :$r^2 \theta = a^2$ :600px
Definition:Reciprocal Spiral
The '''reciprocal spiral''' is the locus of the equation expressed in Polar coordinates as: :$r = \dfrac a \theta$ :600px
Definition:Fermat's Spiral
'''Fermat's spiral''' is the locus of the equation expressed in Polar coordinates as: :$r^2 = \dfrac a \theta$ :600px
Definition:Cornu Spiral
The '''Cornu spiral''' is the locus $C$ of the equation expressed in intrinsic coordinates as: :$s = a^2 \kappa$ where: :$s$ denotes the length of arc at a point of $C$ from the origin :$\kappa$ denotes the curvature of $C$ at that point. :600px
Definition:Sampling Function
The '''sampling function''' is the real function $\operatorname {III}: \R \to \R$ defined as: :$\forall x \in \R: \map {\operatorname {III} } x := \displaystyle \sum_{n \mathop \in \Z} \map \delta {x - n}$ where $\delta$ denotes the Dirac delta function.
Definition:Sampling Function/Graph
The graph of the sampling function is illustrated below: :600px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \Z$.
Definition:Even Impulse Pair Function
The '''even impulse pair function''' is the real function $\operatorname {II}: \R \to \R$ defined as: :$\forall x \in \R: \map {\operatorname {II} } x := \dfrac 1 2 \map \delta {x + \dfrac 1 2} + \dfrac 1 2 \map \delta {x - \dfrac 1 2}$ where $\delta$ denotes the Dirac delta function.
Definition:Even Impulse Pair Function/Graph
The graph of the even impulse pair function is illustrated below: :400px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \set {-\dfrac 1 2, \dfrac 1 2}$.
Definition:Odd Impulse Pair Function
The '''odd impulse pair function''' is the real function $\operatorname {I_I}: \R \to \R$ defined as: :$\forall x \in \R: \map {\operatorname {I_I} } x := \dfrac 1 2 \map \delta {x + \dfrac 1 2} - \dfrac 1 2 \map \delta {x - \dfrac 1 2}$ where $\delta$ denotes the Dirac delta function.
Definition:Odd Impulse Pair Function/Graph
The graph of the odd impulse pair function is illustrated below: :300px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \set {-\dfrac 1 2, \dfrac 1 2}$.
Definition:Filtering Function
The '''filtering function''' is the real function $\operatorname {sinc}: \R \to \R$ defined as: :$\forall x \in \R: \map {\operatorname {sinc} } x := \dfrac {\sin \pi x} {\pi x}$ where $\sin$ denotes the (real) sine function.
Definition:Filtering Function/Graph
The graph of the filtering function is illustrated below: :500px
Definition:Convolution Integral/Positive Real Domain
Let $f$ and $g$ be supported on the positive real numbers $\R_{\ge 0}$ only. The '''convolution integral''' of $f$ and $g$ may be defined as: :$\displaystyle \map f t * \map g t := \int_0^t \map f u \map g {t - u} \rd u$
Definition:Convolution of Real Sequences
Let $\sequence f$ and $\sequence g$ be real sequences. The '''convolution''' of $f$ and $g$ is defined as: :$\displaystyle \sequence {f_i} * \sequence {g_i} := \sum_{j \mathop \in \Z_{\ge 0} } f_i g_{i - j}$
Definition:Even Impulse Pair Function/2 Dimensional
Let $\operatorname {II}: \R \to \R$ denote the '''even impulse pair function'''. The $2$-dimensional form of $\operatorname {II}$ is defined and denoted: :$\forall x, y \in \R: \map {\operatorname { {}^2 II} } {x, y} := \map {\operatorname {II} } x \map {\operatorname {II} } y$
Definition:Dirac Delta Function/2 Dimensional
Let $\delta: \R \to \R$ denote the '''Dirac delta function'''. The $2$-dimensional form of $\delta$ is defined and denoted: :$\forall x, y \in \R: \map { {}^2 \delta} {x, y} := \map \delta x \map \delta y$
Definition:Sampling Function/2 Dimensional
Let $\operatorname {III}: \R \to \R$ denote the '''sampling function'''. The $2$-dimensional form of $\operatorname {III}$ is defined and denoted: :$\forall x, y \in \R: \map {\operatorname { {}^2 III} } {x, y} := \map {\operatorname {III} } x \map {\operatorname {III} } y$
Definition:Filtering Function/2 Dimensional
Let $\operatorname {sinc}: \R \to \R$ denote the '''filtering function'''. The $2$-dimensional form of $\operatorname {sinc}$ is defined and denoted: :$\forall x, y \in \R: \map {\operatorname { {}^2 sinc} } {x, y} := \map {\operatorname {sinc} } x \map {\operatorname {sinc} } y$
Definition:Sub-Gaussian Distribution
The distribution of a random variable $X$ with expectation $\mu = \expect X$ is called '''sub-Gaussian''' if there exists a $\sigma \in \R_{>0}$ such that: :$\expect {e^{\lambda \paren {X - \mu} } } \le e^{\sigma^2 \lambda^2 / 2}$ for all $\lambda \in \R$.
Definition:Sampling Function/Linguistic Note
The name '''shah''' for the '''sampling function''' derives from its similarity in shape and appearance to the Russian '''Ш''', whose name is itself pronounced '''shah'''.
Definition:Fourier Transform/Real Function
{{:Definition:Fourier Transform/Real Function/Formulation 1}}
Definition:Sub-Exponential Distribution
The distribution of a random variable $X$ with expectation $\mu = \expect X$ is called '''sub-exponential''' if there exists $\nu\in \R_{> 0}, \alpha \in \R_{\ge 0}$ such that: :$\expect {e^{\lambda \paren {X - \mu} } } \le e^{\nu^2 \lambda^2 / 2}$ for all $\size \lambda < \dfrac 1 \alpha$. {{explain|Where does $\alpha$ come into it?}}
Definition:Progression
The word '''progression''' is a term used for either '''sequence''' or '''series''' in the following specific contexts. Because its usage is in general ambiguous, it is deprecated on .
Definition:Arithmetic Progression
The term '''arithmetic progression''' is used to mean one of the following: === Arithmetic Sequence === {{:Definition:Arithmetic Sequence}} === Arithmetic Series === {{:Definition:Arithmetic Series}}
Definition:Geometric Progression
The term '''geometric progression''' is used to mean one of the following: === Geometric Sequence === {{:Definition:Geometric Sequence}} === Geometric Series === {{:Definition:Geometric Series}}
Definition:Arithmetic-Geometric Progression
The term '''arithmetic-geometric progression''' is used to mean one of the following: === Arithmetic-Geometric Sequence === {{:Definition:Arithmetic-Geometric Sequence}} === Arithmetic-Geometric Series === {{:Definition:Arithmetic-Geometric Series}}
Definition:Harmonic Progression
The term '''harmonic progression''' is used to mean one of the following: === Harmonic Sequence === {{:Definition:Harmonic Sequence}} === Harmonic Series === {{:Definition:Harmonic Series}}
Definition:Harmonic Series/General
Let $\sequence {x_n}$ be a sequence of numbers such that $\sequence {\size {x_n} }$ is a harmonic sequence. Then the series defined as: :$\displaystyle \sum_{n \mathop = 1}^\infty x_n$ is a '''harmonic series'''.
Definition:Square Wave/Points of Discontinuity
The points $\alpha + n \lambda$, for $n \in \Z$, are jump discontinuities. The values $\map S {\alpha + n \lambda}$ may or may not be explicitly defined. It is a common approach to include either endpoint of the intervals from $\alpha$ to $\alpha + \lambda$, and from $\alpha - \lambda$ to $\alpha$, in order to ensure that the domain of $S$ is simply defined, for example: :$\forall x \in \R: \map S x = \begin {cases} \delta + \gamma & : x \in \hointr \alpha {\alpha + \lambda} \\ \delta - \gamma & : x \in \hointr {\alpha - \lambda} \alpha \\ \map S {x + 2 \lambda} & : x < \alpha - \lambda \\ \map S {x - \lambda l} & : x \ge \alpha + \lambda \end {cases}$ Another approach is to make $\map S {\alpha + n \lambda} = \delta$ for all $n \in \Z$. The precise treatment of the discontinuities is often irrelevant or immaterial.
Definition:Square Wave
A '''square wave''' is a periodic real function $S: \R \to \R$ defined as follows: :$\forall x \in \R: \map S x = \begin {cases} \delta + \gamma & : x \in \openint \alpha {\alpha + \lambda} \\ \delta - \gamma & : x \in \openint {\alpha - \lambda} \alpha \\ \map S {x + 2 \lambda} & : x < \alpha - \lambda \\ \map S {x - 2 \lambda} & : x > \alpha + \lambda \end {cases}$ where: :$\alpha$, $\lambda$, $\gamma$ and $\delta$ are given real constants.
Definition:Square Wave/Graph
The graph of $S$ is given below: :800px
Definition:Given
A '''given''' is an object whose properties and existence are stipulated at the beginning of an exposition.
Definition:Finite Discontinuity
Let $A \subseteq \R$ be a subset of the real numbers. Let $f: A \to \R$ be a real function. Let $f$ be discontinuous at $a \in A$. The point $a$ is a '''finite discontinuity''' of $f$  either: :$(1): \quad a$ is a removable discontinuity or: :$(2): \quad a$ is a jump discontinuity.
Definition:Triangle Wave
A '''triangle wave''' is a periodic real function $T: \R \to \R$ defined as follows: :$\forall x \in \R: \map T x = \begin {cases} \size x & : x \in \closedint {-\lambda} \lambda \\ \map T {x + 2 \lambda} & : x < -\lambda \\ \map T {x - 2 \lambda} & : x > +\lambda \end {cases}$ where: :$\lambda$ is a given real constant :$\size x$ denotes the absolute value of $x$.
Definition:Triangle Wave/Graph
The graph of $T$ is given below: :1000px
Definition:Sawtooth Wave
A '''sawtooth wave''' is a periodic real function $S: \R \to \R$ defined as follows: :$\forall x \in \R: \map S x = \begin {cases} x & : x \in \openint {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x > +\lambda \end {cases}$ where $\lambda$ is a given real constant
Definition:Sawtooth Wave/Points of Discontinuity
The points $\paren {2 r + 1} \lambda$, for $r \in \Z$, are jump discontinuities. The values $\map S {\paren {2 r + 1} \lambda}$ may or may not be explicitly defined. It is a common approach to include one of the endpoints of the interval from $-\lambda$ to $\lambda$, in order to ensure that the domain of $S$ is simply defined. For the sawtooth wave for example: :$\forall x \in \R: \map S x = \begin {cases} x & : x \in \hointr {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x \ge +\lambda \end {cases}$ and, for the inverse sawtooth wave: :$\forall x \in \R: \map S x = \begin {cases} -x & : x \in \hointr {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x \ge +\lambda \end {cases}$ Another approach is to make $\map S {\paren {2 r + 1} \lambda} = 0$ for all $r \in \Z$. The precise treatment of the discontinuities is often irrelevant or immaterial.
Definition:Sawtooth Wave/Graph
The graph of $S$ is given below: :800px
Definition:Sawtooth Wave/Inverse/Graph
The graph of $S$ is given below: :800px
Definition:Set/Point Set
A '''set''' whose elements are all (geometric) points is often called a '''point set'''. In particular, the Cartesian plane and complex plane can each be seen referred to as a '''two-dimensional point set'''.
Definition:Definite Integral/Limits of Integration/Upper Limit
The limit $b$ is referred to as the '''upper limit''' of the integral'''.
Definition:Definite Integral/Limits of Integration/Lower Limit
The limit $a$ is referred to as the '''lower limit''' of the integral'''.
Definition:Fourier Series/Range 2 Pi
Let $\alpha \in \R$ be a real number. Let $f: \R \to \R$ be a function such that $\displaystyle \int_\alpha^{\alpha + 2 \pi} \map f x \rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = a_n       | r = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x }} {{eqn | l = b_n       | r = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x }} {{end-eqn}} Then: :$\dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ is called the '''Fourier Series''' for $f$.
Definition:Half-Range Fourier Cosine Series/Formulation 1
Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Then the ''' half-range Fourier cosine series''' of $\map f x$ over $\openint 0 \lambda$ is the series: :$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$ where for all $n \in \Z_{\ge 0}$: :$a_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$
Definition:Half-Range Fourier Sine Series/Formulation 1
Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Then the '''half-range Fourier sine series''' of $\map f x$ over $\openint 0 \lambda$ is the series: :$\map f x \sim \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$
Definition:Half-Range Fourier Series/Range Pi
=== Half-Range Fourier Cosine Series === {{:Definition:Half-Range Fourier Cosine Series/Range Pi}} === Half-Range Fourier Sine Series === {{:Definition:Half-Range Fourier Sine Series/Range Pi}}
Definition:Half-Range Fourier Cosine Series/Range Pi
Let $\map f x$ be a real function defined on the interval $\openint 0 \pi$. Then the ''' half-range Fourier cosine series''' of $\map f x$ over $\openint 0 \pi$ is the series: :$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos n x$ where for all $n \in \Z_{\ge 0}$: :$a_n = \displaystyle \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$