Operators

Symbol: Supremum —

\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right)

— Supremum of a set or function

This operator can be called with 1 or 3 arguments.

Called with 1 argument, Supremum(S), rendered

\operatorname{sup}\left(S\right)

, represents the supremum of the set

S

. This operator is only defined if

S

is a subset of

\mathbb{R} \cup \left\{-\infty, +\infty\right\}

. The supremum does not need to be an element of

S

itself; in particular, for an open interval

S = \left(a, b\right)

, we have

\operatorname{sup}\left(S\right) = b

.

Called with 3 arguments, Supremum(f(x), x, P(x)), rendered

\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right)

, represents

\operatorname{sup}\left(\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}\right)

where

P\!\left(x\right)

is a predicate defining the range of

x

.

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`Supremum`	$\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right)$	Supremum of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension

Source code for this entry:

Entry(ID("6ec976"),
    SymbolDefinition(Supremum, Supremum(f(x), x, P(x)), "Supremum of a set or function"),
    Description("This operator can be called with 1 or 3 arguments."),
    Description("Called with 1 argument, ", SourceForm(Supremum(S)), ", rendered", Supremum(S), ", represents the supremum of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), ".", "The supremum does not need to be an element of", S, "itself; in particular, for an open interval", Equal(S, OpenInterval(a, b)), ", we have", Equal(Supremum(S), b), "."),
    Description("Called with 3 arguments, ", SourceForm(Supremum(f(x), x, P(x))), ", rendered", Supremum(f(x), x, P(x)), ", represents", Supremum(SetBuilder(f(x), x, P(x))), "where", P(x), "is a predicate defining the range of", x, "."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

bbeb35

Symbol: Infimum —

\mathop{\operatorname{inf}}\limits_{P\left(x\right)} f\!\left(x\right)

— Infimum of a set or function

This operator can be called with 1 or 3 arguments.

Called with 1 argument, Infimum(S), rendered

\operatorname{inf}\left(S\right)

, represents the infimum of the set

S

. This operator is only defined if

S

is a subset of

\mathbb{R} \cup \left\{-\infty, +\infty\right\}

. The infimum does not need to be an element of

S

itself; in particular, for an open interval

S = \left(a, b\right)

, we have

\operatorname{inf}\left(S\right) = a

.

Called with 3 arguments, Infimum(f(x), x, P(x)), rendered

\mathop{\operatorname{inf}}\limits_{P\left(x\right)} f\!\left(x\right)

, represents

\operatorname{inf}\left(\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}\right)

where

P\!\left(x\right)

is a predicate defining the range of

x

.

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`Infimum`	$\mathop{\operatorname{inf}}\limits_{P\left(x\right)} f\!\left(x\right)$	Infimum of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension

Source code for this entry:

Entry(ID("bbeb35"),
    SymbolDefinition(Infimum, Infimum(f(x), x, P(x)), "Infimum of a set or function"),
    Description("This operator can be called with 1 or 3 arguments."),
    Description("Called with 1 argument, ", SourceForm(Infimum(S)), ", rendered", Infimum(S), ", represents the infimum of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), ".", "The infimum does not need to be an element of", S, "itself; in particular, for an open interval", Equal(S, OpenInterval(a, b)), ", we have", Equal(Infimum(S), a), "."),
    Description("Called with 3 arguments, ", SourceForm(Infimum(f(x), x, P(x))), ", rendered", Infimum(f(x), x, P(x)), ", represents", Infimum(SetBuilder(f(x), x, P(x))), "where", P(x), "is a predicate defining the range of", x, "."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

d0cb24

Symbol: Minimum —

\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)

— Minimum value of a set or function

This operator can be called with 1 or 3 arguments.

Called with 1 argument, Minimum(S), rendered

\min\left(S\right)

, represents the minimum element of the set

S

. This operator is only defined if

S

is a subset of

\mathbb{R} \cup \left\{-\infty, +\infty\right\}

and the minimum exists.

Called with 3 arguments, Minimum(f(x), x, P(x)), rendered

\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)

, represents

\min\left(\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}\right)

.

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`Minimum`	$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$	Minimum value of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension

Source code for this entry:

Entry(ID("d0cb24"),
    SymbolDefinition(Minimum, Minimum(f(x), x, P(x)), "Minimum value of a set or function"),
    Description("This operator can be called with 1 or 3 arguments."),
    Description("Called with 1 argument, ", SourceForm(Minimum(S)), ", rendered", Minimum(S), ", represents the minimum element of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), " and the minimum exists."),
    Description("Called with 3 arguments, ", SourceForm(Minimum(f(x), x, P(x))), ", rendered", Minimum(f(x), x, P(x)), ", represents", Minimum(SetBuilder(f(x), x, P(x))), "."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

65ccf2

Symbol: Maximum —

\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)

— Maximum value of a set or function

This operator can be called with 1 or 3 arguments.

Called with 1 argument, Maximum(S), rendered

\max\left(S\right)

, represents the maximum element of the set

S

. This operator is only defined if

S

is a subset of

\mathbb{R} \cup \left\{-\infty, +\infty\right\}

and the maximum exists.

Called with 3 arguments, Maximum(f(x), x, P(x)), rendered

\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)

, represents

\max\left(\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}\right)

.

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`Maximum`	$\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)$	Maximum value of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension

Source code for this entry:

Entry(ID("65ccf2"),
    SymbolDefinition(Maximum, Maximum(f(x), x, P(x)), "Maximum value of a set or function"),
    Description("This operator can be called with 1 or 3 arguments."),
    Description("Called with 1 argument, ", SourceForm(Maximum(S)), ", rendered", Maximum(S), ", represents the maximum element of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), " and the maximum exists."),
    Description("Called with 3 arguments, ", SourceForm(Maximum(f(x), x, P(x))), ", rendered", Maximum(f(x), x, P(x)), ", represents", Maximum(SetBuilder(f(x), x, P(x))), "."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

0a3e5a

Symbol: ArgMin —

\mathop{\operatorname{arg\,min}}\limits_{P\left(x\right)} f\!\left(x\right)

— Locations of minimum value

ArgMin(f(x), x, P(x)), rendered

\mathop{\operatorname{arg\,min}}\limits_{P\left(x\right)} f\!\left(x\right)

, gives the set of points

r

satisfying

P\!\left(r\right)

such that

f\!\left(r\right) = \mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)

, if the minimum value exists.

If

f\!\left(x\right)

does not attain a minimum value on the set of points defined by

P\!\left(x\right)

, the result is the empty set

\left\{\right\}

.

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`ArgMin`	$\mathop{\operatorname{arg\,min}}\limits_{P\left(x\right)} f\!\left(x\right)$	Locations of minimum value
`Minimum`	$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$	Minimum value of a set or function

Source code for this entry:

Entry(ID("0a3e5a"),
    SymbolDefinition(ArgMin, ArgMin(f(x), x, P(x)), "Locations of minimum value"),
    Description(SourceForm(ArgMin(f(x), x, P(x))), ", rendered", ArgMin(f(x), x, P(x)), ", gives the set of points", r, "satisfying", P(r), "such that", Equal(f(r), Minimum(f(x), x, P(x))), ", if the minimum value exists."),
    Description("If", f(x), "does not attain a minimum value on the set of points defined by", P(x), ", the result is the empty set", Set(), "."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

617fe3

Symbol: ArgMax —

\mathop{\operatorname{arg\,max}}\limits_{P\left(x\right)} f\!\left(x\right)

— Locations of maximum value

ArgMax(f(x), x, P(x)), rendered

\mathop{\operatorname{arg\,max}}\limits_{P\left(x\right)} f\!\left(x\right)

, gives the set of points

r

satisfying

P\!\left(r\right)

such that

f\!\left(r\right) = \mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)

, if the maximum value exists.

If

f\!\left(x\right)

does not attain a maximum value on the set of points defined by

P\!\left(x\right)

, the result is the empty set

\left\{\right\}

.

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`ArgMax`	$\mathop{\operatorname{arg\,max}}\limits_{P\left(x\right)} f\!\left(x\right)$	Locations of maximum value
`Maximum`	$\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)$	Maximum value of a set or function

Source code for this entry:

Entry(ID("617fe3"),
    SymbolDefinition(ArgMax, ArgMax(f(x), x, P(x)), "Locations of maximum value"),
    Description(SourceForm(ArgMax(f(x), x, P(x))), ", rendered", ArgMax(f(x), x, P(x)), ", gives the set of points", r, "satisfying", P(r), "such that", Equal(f(r), Maximum(f(x), x, P(x))), ", if the maximum value exists."),
    Description("If", f(x), "does not attain a maximum value on the set of points defined by", P(x), ", the result is the empty set", Set(), "."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

f4fbb8

Symbol: ArgMinUnique —

\mathop{\operatorname{arg\,min*}}\limits_{P\left(x\right)} f\!\left(x\right)

— Unique location of minimum value

ArgMinUnique(f(x), x, P(x)) represents the unique point

r

satisfying

P\!\left(r\right)

such that

f\!\left(r\right) = \mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)

. This operation is only defined if such a unique point exists.

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`ArgMinUnique`	$\mathop{\operatorname{arg\,min*}}\limits_{P\left(x\right)} f\!\left(x\right)$	Unique location of minimum value
`Minimum`	$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$	Minimum value of a set or function

Source code for this entry:

Entry(ID("f4fbb8"),
    SymbolDefinition(ArgMinUnique, ArgMinUnique(f(x), x, P(x)), "Unique location of minimum value"),
    Description(SourceForm(ArgMinUnique(f(x), x, P(x))), "represents the unique point", r, "satisfying", P(r), "such that", Equal(f(r), Minimum(f(x), x, P(x))), ". This operation is only defined if such a unique point exists."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

be4926

Symbol: ArgMaxUnique —

\mathop{\operatorname{arg\,max*}}\limits_{P\left(x\right)} f\!\left(x\right)

— Unique location of maximum value

ArgMaxUnique(f(x), x, P(x)) represents the unique point

r

satisfying

P\!\left(r\right)

such that

f\!\left(r\right) = \mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)

. This operation is only defined if such a unique point exists.

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`ArgMaxUnique`	$\mathop{\operatorname{arg\,max*}}\limits_{P\left(x\right)} f\!\left(x\right)$	Unique location of maximum value
`Maximum`	$\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)$	Maximum value of a set or function

Source code for this entry:

Entry(ID("be4926"),
    SymbolDefinition(ArgMaxUnique, ArgMaxUnique(f(x), x, P(x)), "Unique location of maximum value"),
    Description(SourceForm(ArgMaxUnique(f(x), x, P(x))), "represents the unique point", r, "satisfying", P(r), "such that", Equal(f(r), Maximum(f(x), x, P(x))), ". This operation is only defined if such a unique point exists."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

26ea9f

Symbol: Limit —

\lim_{x \to a} f\!\left(x\right)

— Limiting value

Limit(f(x), x, a, P(x)) rendered as

\lim_{x \to a, P\left(x\right)} f\!\left(x\right)

represents the limiting value of

f\!\left(x\right)

for every sequence of

x

satisfying

P\!\left(x\right)

and approaching the limit point

a

.

If the predicate

P\!\left(x\right)

is omitted, the expression renders correctly to LaTeX, but this form should be avoided since it is ambiguous whether it denotes a sequence limit, real limit or complex limit (or some other kind of limit). It is better to use SequenceLimit, RealLimit, LeftLimit, RightLimit or ComplexLimit.

The limit is always a deleted limit. That is, the value of

f\!\left(a\right)

does not need to be equal to the limit and does not even need to be defined.

The expression f(x) is not required to be defined for all

x

satisfying

P\!\left(x\right)

. It only needs to be defined for all

x

in some neighborhood of the limit point and also satisfying

P\!\left(x\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`Limit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value
`SequenceLimit`	$\lim_{n \to a} f\!\left(n\right)$	Limiting value of sequence
`RealLimit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value, real variable
`LeftLimit`	$\lim_{x \to {a}^{-}} f\!\left(x\right)$	Limiting value, from the left
`RightLimit`	$\lim_{x \to {a}^{+}} f\!\left(x\right)$	Limiting value, from the right
`ComplexLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable

Source code for this entry:

Entry(ID("26ea9f"),
    SymbolDefinition(Limit, Limit(f(x), x, a), "Limiting value"),
    Description(SourceForm(Limit(f(x), x, a, P(x))), "rendered as", Limit(f(x), x, a, P(x)), "represents the limiting value of", f(x), "for every sequence of", x, "satisfying", P(x), "and approaching the limit point", a, "."),
    Description("If the predicate", P(x), "is omitted, the expression renders correctly to LaTeX, ", "but this form should be avoided since it is ambiguous whether it denotes a sequence limit, ", "real limit or complex limit (or some other kind of limit). It is better to use", SourceForm(SequenceLimit), ",", SourceForm(RealLimit), ",", SourceForm(LeftLimit), ",", SourceForm(RightLimit), "or", SourceForm(ComplexLimit), "."),
    Description("The limit is always a deleted limit. That is, the value of", f(a), "does not need to be equal to the limit and does not even need to be defined."),
    Description("The expression", SourceForm(f(x)), "is not required to be defined for all", x, "satisfying", P(x), ".", "It only needs to be defined for all", x, "in some neighborhood of the limit point and also satisfying", P(x), "."))

1d2ee5

Symbol: SequenceLimit —

\lim_{n \to a} f\!\left(n\right)

— Limiting value of sequence

This operator can be called with three or four arguments.

SequenceLimit(f(n), n, a), rendered as

\lim_{n \to a} f\!\left(n\right)

, is equivalent to Limit(f(n), n, a, Element(n, ZZ)) but renders to LaTeX without displaying the predicate

n \in \mathbb{Z}

which readers will typically understand from context.

SequenceLimit(f(n), n, a, P(n)), rendered as

\lim_{n \to a, P\left(n\right)} f\!\left(n\right)

, is equivalent to Limit(f(n), n, a, And(Element(n, ZZ), P(n))).

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimit`	$\lim_{n \to a} f\!\left(n\right)$	Limiting value of sequence
`Limit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("1d2ee5"),
    SymbolDefinition(SequenceLimit, SequenceLimit(f(n), n, a), "Limiting value of sequence"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(SequenceLimit(f(n), n, a)), ", rendered as", SequenceLimit(f(n), n, a), ", is equivalent to", SourceForm(Limit(f(n), n, a, Element(n, ZZ))), "but renders to LaTeX without displaying the predicate", Element(n, ZZ), " which readers will typically understand from context."),
    Description(SourceForm(SequenceLimit(f(n), n, a, P(n))), ", rendered as", SequenceLimit(f(n), n, a, P(n)), ", is equivalent to", SourceForm(Limit(f(n), n, a, And(Element(n, ZZ), P(n)))), "."))

6fe5c1

Symbol: RealLimit —

\lim_{x \to a} f\!\left(x\right)

— Limiting value, real variable

This operator can be called with three or four arguments.

RealLimit(f(x), x, a), rendered as

\lim_{x \to a} f\!\left(x\right)

, is equivalent to Limit(f(x), x, a, Element(x, RR)) but renders to LaTeX without displaying the predicate

x \in \mathbb{R}

which readers will typically understand from context.

RealLimit(f(x), x, a, P(x)), rendered as

\lim_{x \to a, P\left(x\right)} f\!\left(x\right)

, is equivalent to Limit(f(x), x, a, And(Element(x, RR), P(x))).

Definitions:

Fungrim symbol	Notation	Short description
`RealLimit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value, real variable
`Limit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("6fe5c1"),
    SymbolDefinition(RealLimit, RealLimit(f(x), x, a), "Limiting value, real variable"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(RealLimit(f(x), x, a)), ", rendered as", RealLimit(f(x), x, a), ", is equivalent to", SourceForm(Limit(f(x), x, a, Element(x, RR))), "but renders to LaTeX without displaying the predicate", Element(x, RR), " which readers will typically understand from context."),
    Description(SourceForm(RealLimit(f(x), x, a, P(x))), ", rendered as", RealLimit(f(x), x, a, P(x)), ", is equivalent to", SourceForm(Limit(f(x), x, a, And(Element(x, RR), P(x)))), "."))

c8a5f0

Symbol: LeftLimit —

\lim_{x \to {a}^{-}} f\!\left(x\right)

— Limiting value, from the left

This operator can be called with three or four arguments.

LeftLimit(f(x), x, a), rendered as

\lim_{x \to {a}^{-}} f\!\left(x\right)

, is equivalent to Limit(f(x), x, a, Element(x, OpenInterval(Neg(Infinity), a))).

LeftLimit(f(x), x, a, P(x)), rendered as

\lim_{x \to {a}^{-}, P\left(x\right)} f\!\left(x\right)

, is equivalent to Limit(f(x), x, a, And(Element(x, OpenInterval(Neg(Infinity), a)), P(x))).

Definitions:

Fungrim symbol	Notation	Short description
`LeftLimit`	$\lim_{x \to {a}^{-}} f\!\left(x\right)$	Limiting value, from the left
`Limit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("c8a5f0"),
    SymbolDefinition(LeftLimit, LeftLimit(f(x), x, a), "Limiting value, from the left"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(LeftLimit(f(x), x, a)), ", rendered as", LeftLimit(f(x), x, a), ", is equivalent to", SourceForm(Limit(f(x), x, a, Element(x, OpenInterval(Neg(Infinity), a)))), "."),
    Description(SourceForm(LeftLimit(f(x), x, a, P(x))), ", rendered as", LeftLimit(f(x), x, a, P(x)), ", is equivalent to", SourceForm(Limit(f(x), x, a, And(Element(x, OpenInterval(Neg(Infinity), a)), P(x)))), "."))

afd5ca

Symbol: RightLimit —

\lim_{x \to {a}^{+}} f\!\left(x\right)

— Limiting value, from the right

This operator can be called with three or four arguments.

RightLimit(f(x), x, a), rendered as

\lim_{x \to {a}^{+}} f\!\left(x\right)

, is equivalent to Limit(f(x), x, a, Element(x, OpenInterval(a, Infinity))).

RightLimit(f(x), x, a, P(x)), rendered as

\lim_{x \to {a}^{+}, P\left(x\right)} f\!\left(x\right)

, is equivalent to Limit(f(x), x, a, And(Element(x, OpenInterval(a, Infinity)), P(x))).

Definitions:

Fungrim symbol	Notation	Short description
`RightLimit`	$\lim_{x \to {a}^{+}} f\!\left(x\right)$	Limiting value, from the right
`Limit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("afd5ca"),
    SymbolDefinition(RightLimit, RightLimit(f(x), x, a), "Limiting value, from the right"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(RightLimit(f(x), x, a)), ", rendered as", RightLimit(f(x), x, a), ", is equivalent to", SourceForm(Limit(f(x), x, a, Element(x, OpenInterval(a, Infinity)))), "."),
    Description(SourceForm(RightLimit(f(x), x, a, P(x))), ", rendered as", RightLimit(f(x), x, a, P(x)), ", is equivalent to", SourceForm(Limit(f(x), x, a, And(Element(x, OpenInterval(a, Infinity)), P(x)))), "."))

05a3ee

Symbol: ComplexLimit —

\lim_{z \to a} f\!\left(z\right)

— Limiting value, complex variable

This operator can be called with three or four arguments.

ComplexLimit(f(z), z, a), rendered as

\lim_{z \to a} f\!\left(z\right)

, is equivalent to Limit(f(z), z, a, Element(z, CC)) but renders to LaTeX without displaying the predicate

z \in \mathbb{C}

which readers will typically understand from context.

ComplexLimit(f(z), z, a, P(z)), rendered as

\lim_{z \to a, P\left(z\right)} f\!\left(z\right)

, is equivalent to Limit(f(z), z, a, And(Element(z, CC), P(z))).

Definitions:

Fungrim symbol	Notation	Short description
`ComplexLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable
`Limit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("05a3ee"),
    SymbolDefinition(ComplexLimit, ComplexLimit(f(z), z, a), "Limiting value, complex variable"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(ComplexLimit(f(z), z, a)), ", rendered as", ComplexLimit(f(z), z, a), ", is equivalent to", SourceForm(Limit(f(z), z, a, Element(z, CC))), "but renders to LaTeX without displaying the predicate", Element(z, CC), " which readers will typically understand from context."),
    Description(SourceForm(ComplexLimit(f(z), z, a, P(z))), ", rendered as", ComplexLimit(f(z), z, a, P(z)), ", is equivalent to", SourceForm(Limit(f(z), z, a, And(Element(z, CC), P(z)))), "."))

2be0b5

Symbol: MeromorphicLimit —

\lim_{z \to a} f\!\left(z\right)

— Limiting value, allowing poles

This operator is equivalent to ComplexLimit except that whereas ComplexLimit in general is undefined when

a

is a pole (because the direction of the resulting infinity depends on the direction of approach), MeromorphicLimit is taken to give UnsignedInfinity (

{\tilde \infty}

) when

a

is a pole.

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, allowing poles
`ComplexLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("2be0b5"),
    SymbolDefinition(MeromorphicLimit, MeromorphicLimit(f(z), z, a), "Limiting value, allowing poles"),
    Description("This operator is equivalent to", SourceForm(ComplexLimit), "except that whereas", SourceForm(ComplexLimit), "in general is undefined when", a, "is a pole (because the direction of the resulting infinity depends on the direction of approach),", SourceForm(MeromorphicLimit), "is taken to give", SourceForm(UnsignedInfinity), "(", UnsignedInfinity, ")", "when", a, "is a pole."))

1b6a57

Symbol: Derivative —

\frac{d}{d z}\, f\!\left(z\right)

— Derivative

This operator can be called with three or four arguments.

Derivative(f(z), z, a), rendered as

\left[ \frac{d}{d z}\, f\!\left(z\right) \right]_{z = a}

or

f'(a)

, represents the derivative of

f\!\left(z\right)

evaluated at

z = a

.

Derivative(f(z), z, a, n), rendered as

\left[ \frac{d^{n}}{{d z}^{n}} f\!\left(z\right) \right]_{z = a}

or

{f}^{(n)}(a)

, represents the order

n

derivative of

f\!\left(z\right)

evaluated at

z = a

.

The second argument

z

defines a locally bound variable for the expression in the first argument. With the evaluation point set to

a = z

, Derivative(f(z), z, z) may render more simply as

\frac{d}{d z}\, f\!\left(z\right)

.

This operator is ambiguous since the intended meaning could be a real derivative, a complex derivative, or some other form of derivative. It is better to use RealDerivative, ComplexDerivative, ComplexBranchDerivative, or MeromorphicDerivative.

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`RealDerivative`	$\frac{d}{d x}\, f\!\left(x\right)$	Real derivative
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles

Source code for this entry:

Entry(ID("1b6a57"),
    SymbolDefinition(Derivative, Derivative(Call(f, z), z, z), "Derivative"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(Derivative(f(z), z, a)), ", rendered as ", Derivative(Call(f, z), z, a), "or", Derivative(f(z), z, a), ", represents the derivative of", f(z), "evaluated at", Equal(z, a), "."),
    Description(SourceForm(Derivative(f(z), z, a, n)), ", rendered as ", Derivative(Call(f, z), z, a, n), "or", Derivative(f(z), z, a, n), ", represents the order", n, "derivative of", f(z), "evaluated at", Equal(z, a), "."),
    Description("The second argument", z, "defines a locally bound variable for the expression in the first argument. With the evaluation point set to", Equal(a, z), ",", SourceForm(Derivative(f(z), z, z)), "may render more simply as", Derivative(Call(f, z), z, z), "."),
    Description("This operator is ambiguous since the intended meaning could be a real derivative, a complex derivative, or some other form of derivative.", "It is better to use", SourceForm(RealDerivative), ",", SourceForm(ComplexDerivative), ",", SourceForm(ComplexBranchDerivative), ", or", SourceForm(MeromorphicDerivative), "."))

452407

Symbol: RealDerivative —

\frac{d}{d x}\, f\!\left(x\right)

— Real derivative

The call syntax for this operator is the same as for Derivative.

The result is defined as

f'(x) = \lim_{h \to 0} \frac{f\!\left(x + h\right) - f\!\left(x\right)}{h}

where the limit is taken with respect to a real variable

h

( RealLimit ).

Note that

x

can be complex and that the "real derivative" can be complex-valued; the "real" qualifier just refers to the direction in which the limit is computed.

Definitions:

Fungrim symbol	Notation	Short description
`RealDerivative`	$\frac{d}{d x}\, f\!\left(x\right)$	Real derivative
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`RealLimit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value, real variable

Source code for this entry:

Entry(ID("452407"),
    SymbolDefinition(RealDerivative, RealDerivative(Call(f, x), x, x), "Real derivative"),
    Description("The call syntax for this operator is the same as for", SourceForm(Derivative), "."),
    Description("The result is defined as", Equal(RealDerivative(f(x), x, x), RealLimit(Div(Sub(f(Add(x, h)), f(x)), h), h, 0)), "where the limit is taken with respect to a real variable", h, "(", SourceForm(RealLimit), ")."),
    Description("Note that", x, "can be complex and that the \"real derivative\" can be complex-valued; the \"real\" qualifier just refers to the direction in which the limit is computed."))

b4b319

Symbol: ComplexDerivative —

\frac{d}{d z}\, f\!\left(z\right)

— Complex derivative

The call syntax for this operator is the same as for Derivative.

The result is defined as

f'(z) = \lim_{h \to 0} \frac{f\!\left(z + h\right) - f\!\left(z\right)}{h}

where the limit is taken with respect to a complex variable

h

( ComplexLimit ).

If this limit exists (and is finite), then

f

is holomorphic at

z

.

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`ComplexLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable

Source code for this entry:

Entry(ID("b4b319"),
    SymbolDefinition(ComplexDerivative, ComplexDerivative(Call(f, z), z, z), "Complex derivative"),
    Description("The call syntax for this operator is the same as for", SourceForm(Derivative), "."),
    Description("The result is defined as", Equal(ComplexDerivative(f(z), z, z), ComplexLimit(Div(Sub(f(Add(z, h)), f(z)), h), h, 0)), "where the limit is taken with respect to a complex variable", h, "(", SourceForm(ComplexLimit), ")."),
    Description("If this limit exists (and is finite), then", f, "is holomorphic at", z, "."))

96f695

Symbol: ComplexBranchDerivative —

\frac{d}{d z}\, f\!\left(z\right)

— Complex derivative, allowing branch cuts

The call syntax for this operator is the same as for Derivative.

This operator is equivalent to ComplexDerivative except that whereas ComplexDerivative is undefined on a branch cut (where the function is not complex differentiable), ComplexBranchDerivative gives the complex derivative of the analytically continued function across the branch cut.

Definitions:

Fungrim symbol	Notation	Short description
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative

Source code for this entry:

Entry(ID("96f695"),
    SymbolDefinition(ComplexBranchDerivative, ComplexBranchDerivative(Call(f, z), z, z), "Complex derivative, allowing branch cuts"),
    Description("The call syntax for this operator is the same as for", SourceForm(Derivative), "."),
    Description("This operator is equivalent to", SourceForm(ComplexDerivative), "except that whereas", SourceForm(ComplexDerivative), "is undefined on a branch cut (where the function is not complex differentiable),", SourceForm(ComplexBranchDerivative), "gives the complex derivative of the analytically continued function across the branch cut."))

4c6780

Symbol: MeromorphicDerivative —

\frac{d}{d z}\, f\!\left(z\right)

— Complex derivative, allowing poles

The call syntax for this operator is the same as for Derivative.

This operator is equivalent to ComplexDerivative except that whereas ComplexDerivative is undefined at a pole, MeromorphicDerivative gives UnsignedInfinity (

{\tilde \infty}

) at a pole.

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("4c6780"),
    SymbolDefinition(MeromorphicDerivative, MeromorphicDerivative(Call(f, z), z, z), "Complex derivative, allowing poles"),
    Description("The call syntax for this operator is the same as for", SourceForm(Derivative), "."),
    Description("This operator is equivalent to", SourceForm(ComplexDerivative), "except that whereas", SourceForm(ComplexDerivative), "is undefined at a pole,", SourceForm(MeromorphicDerivative), "gives", SourceForm(UnsignedInfinity), "(", UnsignedInfinity, ")", "at a pole."))

Extreme values

Limits

Derivatives