Operators

Symbol: Sum —

\sum_{n} f(n)

— Sum

Sum(S), rendered as

\sum S

, gives the sum of the elements of the set

S

. The sum is required to be absolutely convergent.

Sum(f(n), For(n, a, b)), rendered as

\sum_{n=a}^{b} f(n)

, gives the sum of

f(n)

for integers

n

from

a

to

b

, where

a

and

b

should be integers or

-\infty

or

\infty

. If

a < b

, the sum is empty. The sum

\sum_{n=0}^{\infty} f(n)

is interpreted as

\lim_{N \to \infty} \sum_{n=0}^{N} f(n)

and can be conditionally convergent.

Sum(f(n), For(n, a, b), P(n)), rendered as

\sum_{\textstyle{n=a \atop P(n)}}^{b} f(n)

, is as above but only terms satisfying the predicate

P(n)

are included.

Sum(f(x), ForElement(x, S)), rendered as

\sum_{x \in S} f(x)

, gives the sum of

f(x)

for all

x

in the set

S

. The sum is required to be absolutely convergent.

Sum(f(x), ForElement(x, S), P(x)), rendered as

\sum_{\textstyle{x \in S \atop P(x)}} f(x)

, gives the sum of

f(x)

for all

x

in the set

S

and satisfying the predicate

P(x)

. The sum is required to be absolutely convergent.

Sum(f(x), For(x), P(x)), rendered as

\sum_{P(x)} f(x)

, gives the sum of

f(x)

for all

x

satisfying the predicate

P(x)

. The predicate

P(x)

should define the domain of

x

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

x \in S

where

S

is a known set. The sum is required to be absolutely convergent.

The empty sum is 0. The sum can range over an uncountable number of terms, as long as only countably many terms are nonzero.

The special expression For(x), For(x, a, b) or ForElement(x) declares x as a locally bound variable within the scope of the arguments to this operator. Similarly For(Tuple(x, y)), ForElement(Tuple(x, y), S) etc. defines multiple locally bound variables.

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence

Source code for this entry:

Entry(ID("044e42"),
    SymbolDefinition(Sum, Sum(f(n), For(n)), "Sum"),
    Description(SourceForm(Sum(S)), ", rendered as ", Sum(S), ", gives the sum of the elements of the set", S, ". The sum is required to be absolutely convergent."),
    Description(SourceForm(Sum(f(n), For(n, a, b))), ", rendered as ", Sum(f(n), For(n, a, b)), ", gives the sum of", f(n), "for integers", n, "from", a, "to", b, ", where", a, "and", b, "should be integers or", Neg(Infinity), "or", Infinity, ". ", "If", Less(a, b), ", the sum is empty. The sum", Sum(f(n), For(n, 0, Infinity)), "is interpreted as", SequenceLimit(Sum(f(n), For(n, 0, N)), For(N, Infinity)), "and can be conditionally convergent."),
    Description(SourceForm(Sum(f(n), For(n, a, b), P(n))), ", rendered as ", Sum(f(n), For(n, a, b), P(n)), ", is as above but", "only terms satisfying the predicate", P(n), "are included."),
    Description(SourceForm(Sum(f(x), ForElement(x, S))), ", rendered as ", Sum(f(x), ForElement(x, S)), ", gives the sum of", f(x), "for all", x, "in the set", S, ". The sum is required to be absolutely convergent."),
    Description(SourceForm(Sum(f(x), ForElement(x, S), P(x))), ", rendered as ", Sum(f(x), ForElement(x, S), P(x)), ", gives the sum of", f(x), "for all", x, "in the set", S, "and satisfying the predicate", P(x), ". The sum is required to be absolutely convergent."),
    Description(SourceForm(Sum(f(x), For(x), P(x))), ", rendered as ", Sum(f(x), For(x), P(x)), ", gives the sum of", f(x), "for all", x, "satisfying the predicate", P(x), ". ", "The predicate", P(x), "should define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "The sum is required to be absolutely convergent."),
    Description("The empty sum is", 0, ". The sum can range over an uncountable number of terms, as long as only countably many terms are nonzero."),
    Description("The special expression", SourceForm(For(x)), ", ", SourceForm(For(x, a, b)), " or", SourceForm(ForElement(x)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "Similarly", SourceForm(For(Tuple(x, y))), ", ", SourceForm(ForElement(Tuple(x, y), S)), "etc. defines multiple locally bound variables."))

1e2755

Symbol: Product —

\prod_{n} f(n)

— Product

Product(S), rendered as

\prod S

, gives the product of the elements of the set

S

. The product is required to be absolutely convergent. The empty product is 1.

Product(f(n), For(n, a, b)), rendered as

\prod_{n=a}^{b} f(n)

, gives the product of

f(n)

for integers

n

from

a

to

b

, where

a

and

b

should be integers or

-\infty

or

\infty

. If

a < b

, the product is empty. The product

\prod_{n=0}^{\infty} f(n)

is interpreted as

\lim_{N \to \infty} \prod_{n=0}^{N} f(n)

and can be conditionally convergent.

Product(f(n), For(n, a, b), P(n)), rendered as

\prod_{\textstyle{n=a \atop P(n)}}^{b} f(n)

, is as above but only terms satisfying the predicate

P(n)

are included.

Product(f(x), ForElement(x, S)), rendered as

\prod_{x \in S} f(x)

, gives the product of

f(x)

for all

x

in the set

S

. The product is required to be absolutely convergent.

Product(f(x), ForElement(x, S), P(x)), rendered as

\prod_{\textstyle{x \in S \atop P(x)}} f(x)

, gives the product of

f(x)

for all

x

in the set

S

and satisfying the predicate

P(x)

. The product is required to be absolutely convergent.

Product(f(x), For(x), P(x)), rendered as

\prod_{P(x)} f(x)

, gives the product of

f(x)

for all

x

satisfying the predicate

P(x)

. The predicate

P(x)

should define the domain of

x

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

x \in S

where

S

is a known set. The product is required to be absolutely convergent.

The empty product is 1. The product can range over an uncountable number of factors, as long as only countably many factors are nonzero.

The special expression For(x), For(x, a, b) or ForElement(x) declares x as a locally bound variable within the scope of the arguments to this operator. Similarly For(Tuple(x, y)), ForElement(Tuple(x, y), S) etc. defines multiple locally bound variables.

Definitions:

Fungrim symbol	Notation	Short description
`Product`	$\prod_{n} f(n)$	Product
`Infinity`	$\infty$	Positive infinity
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence

Source code for this entry:

Entry(ID("1e2755"),
    SymbolDefinition(Product, Product(f(n), For(n)), "Product"),
    Description(SourceForm(Product(S)), ", rendered as ", Product(S), ", gives the product of the elements of the set", S, ". The product is required to be absolutely convergent. The empty product is 1."),
    Description(SourceForm(Product(f(n), For(n, a, b))), ", rendered as ", Product(f(n), For(n, a, b)), ", gives the product of", f(n), "for integers", n, "from", a, "to", b, ", where", a, "and", b, "should be integers or", Neg(Infinity), "or", Infinity, ". ", "If", Less(a, b), ", the product is empty. The product", Product(f(n), For(n, 0, Infinity)), "is interpreted as", SequenceLimit(Product(f(n), For(n, 0, N)), Var(N), Infinity), "and can be conditionally convergent."),
    Description(SourceForm(Product(f(n), For(n, a, b), P(n))), ", rendered as ", Product(f(n), For(n, a, b), P(n)), ", is as above but", "only terms satisfying the predicate", P(n), "are included."),
    Description(SourceForm(Product(f(x), ForElement(x, S))), ", rendered as ", Product(f(x), ForElement(x, S)), ", gives the product of", f(x), "for all", x, "in the set", S, ". The product is required to be absolutely convergent."),
    Description(SourceForm(Product(f(x), ForElement(x, S), P(x))), ", rendered as ", Product(f(x), ForElement(x, S), P(x)), ", gives the product of", f(x), "for all", x, "in the set", S, "and satisfying the predicate", P(x), ". The product is required to be absolutely convergent."),
    Description(SourceForm(Product(f(x), For(x), P(x))), ", rendered as ", Product(f(x), For(x), P(x)), ", gives the product of", f(x), "for all", x, "satisfying the predicate", P(x), ". ", "The predicate", P(x), "should define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "The product is required to be absolutely convergent."),
    Description("The empty product is", 1, ". The product can range over an uncountable number of factors, as long as only countably many factors are nonzero."),
    Description("The special expression", SourceForm(For(x)), ", ", SourceForm(For(x, a, b)), " or", SourceForm(ForElement(x)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "Similarly", SourceForm(For(Tuple(x, y))), ", ", SourceForm(ForElement(Tuple(x, y), S)), "etc. defines multiple locally bound variables."))

9f703a

Symbol: PrimeSum —

\sum_{p} f(p)

— Sum over primes

PrimeSum(f(p), For(p)), rendered as

\sum_{p} f(p)

, represents the sum of

f(p)

taken over all prime numbers

p

.

PrimeSum(f(p), For(p), P(p)), rendered as

\sum_{P(p)} f(p)

, represents the sum of

f(p)

taken over all prime numbers

p

satisfying the predicate

P(p)

.

The special expression For(p) defines p as a locally bound variable.

The empty sum is equal to zero. Sums taken over an infinite number of terms are required to be absolutely convergent.

Definitions:

Fungrim symbol	Notation	Short description
`PrimeSum`	$\sum_{p} f(p)$	Sum over primes

Source code for this entry:

Entry(ID("9f703a"),
    SymbolDefinition(PrimeSum, PrimeSum(f(p), For(p)), "Sum over primes"),
    Description(SourceForm(PrimeSum(f(p), For(p))), ", rendered as ", PrimeSum(f(p), For(p)), ", represents the sum of", f(p), "taken over all prime numbers", p, "."),
    Description(SourceForm(PrimeSum(f(p), For(p), P(p))), ", rendered as ", PrimeSum(f(p), For(p), P(p)), ", represents the sum of", f(p), "taken over all prime numbers", p, "satisfying the predicate", P(p), "."),
    Description("The special expression", SourceForm(For(p)), "defines", SourceForm(p), "as a locally bound variable."),
    Description("The empty sum is equal to zero. Sums taken over an infinite number of terms are required to be absolutely convergent."))

2a896d

Symbol: PrimeProduct —

\prod_{p} f(p)

— Product over primes

PrimeProduct(f(p), For(p)), rendered as

\prod_{p} f(p)

, represents the product of

f(p)

taken over all prime numbers

p

.

PrimeProduct(f(p), For(p), P(p)), rendered as

\prod_{P(p)} f(p)

, represents the product of

f(p)

taken over all prime numbers

p

satisfying the predicate

P(p)

.

The special expression For(p) defines p as a locally bound variable.

The empty product is equal to one. Products taken over an infinite number of factors are required to be absolutely convergent.

Definitions:

Fungrim symbol	Notation	Short description
`PrimeProduct`	$\prod_{p} f(p)$	Product over primes

Source code for this entry:

Entry(ID("2a896d"),
    SymbolDefinition(PrimeProduct, PrimeProduct(f(p), For(p)), "Product over primes"),
    Description(SourceForm(PrimeProduct(f(p), For(p))), ", rendered as ", PrimeProduct(f(p), For(p)), ", represents the product of", f(p), "taken over all prime numbers", p, "."),
    Description(SourceForm(PrimeProduct(f(p), For(p), P(p))), ", rendered as ", PrimeProduct(f(p), For(p), P(p)), ", represents the product of", f(p), "taken over all prime numbers", p, "satisfying the predicate", P(p), "."),
    Description("The special expression", SourceForm(For(p)), "defines", SourceForm(p), "as a locally bound variable."),
    Description("The empty product is equal to one. Products taken over an infinite number of factors are required to be absolutely convergent."))

8baf79

Symbol: DivisorSum —

\sum_{k \mid n} f(k)

— Sum over divisors

DivisorSum(f(k), For(k, n)), rendered as

\sum_{k \mid n} f(k)

, represents the sum of

f(k)

taken over all positive integers

k

dividing the integer

n

.

DivisorSum(f(k), For(k, n), P(k)), rendered as

\sum_{k \mid n,\, P(k)} f(k)

, represents the sum of

f(k)

taken over all positive integers

k

dividing the integer

n

and satisfying the predicate

P(k)

.

The special expression For(k, n) defines k as a locally bound variable.

The empty sum is equal to zero.

Definitions:

Fungrim symbol	Notation	Short description
`DivisorSum`	$\sum_{k \mid n} f(k)$	Sum over divisors

Source code for this entry:

Entry(ID("8baf79"),
    SymbolDefinition(DivisorSum, DivisorSum(f(k), For(k, n)), "Sum over divisors"),
    Description(SourceForm(DivisorSum(f(k), For(k, n))), ", rendered as ", DivisorSum(f(k), For(k, n)), ", represents the sum of", f(k), "taken over all positive integers", k, "dividing the integer", n, "."),
    Description(SourceForm(DivisorSum(f(k), For(k, n), P(k))), ", rendered as ", DivisorSum(f(k), For(k, n), P(k)), ", represents the sum of", f(k), "taken over all positive integers", k, "dividing the integer", n, "and satisfying the predicate", P(k), "."),
    Description("The special expression", SourceForm(For(k, n)), "defines", SourceForm(k), "as a locally bound variable."),
    Description("The empty sum is equal to zero."))

5830eb

Symbol: DivisorProduct —

\prod_{k \mid n} f(k)

— Product over divisors

DivisorProduct(f(k), For(k, n)), rendered as

\prod_{k \mid n} f(k)

, represents the product of

f(k)

taken over all positive integers

k

dividing the integer

n

.

DivisorProduct(f(k), For(k, n), P(k)), rendered as

\prod_{k \mid n,\, P(k)} f(k)

, represents the product of

f(k)

taken over all positive integers

k

dividing the integer

n

and satisfying the predicate

P(k)

.

The special expression For(k, n) defines k as a locally bound variable.

The empty product is equal to one.

Definitions:

Fungrim symbol	Notation	Short description
`DivisorProduct`	$\prod_{k \mid n} f(k)$	Product over divisors

Source code for this entry:

Entry(ID("5830eb"),
    SymbolDefinition(DivisorProduct, DivisorProduct(f(k), For(k, n)), "Product over divisors"),
    Description(SourceForm(DivisorProduct(f(k), For(k, n))), ", rendered as ", DivisorProduct(f(k), For(k, n)), ", represents the product of", f(k), "taken over all positive integers", k, "dividing the integer", n, "."),
    Description(SourceForm(DivisorProduct(f(k), For(k, n), P(k))), ", rendered as ", DivisorProduct(f(k), For(k, n), P(k)), ", represents the product of", f(k), "taken over all positive integers", k, "dividing the integer", n, "and satisfying the predicate", P(k), "."),
    Description("The special expression", SourceForm(For(k, n)), "defines", SourceForm(k), "as a locally bound variable."),
    Description("The empty product is equal to one."))

f7ce46

Symbol: Zeros —

\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)

— Zeros (roots) of function

Zeros(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)

, represents the set of values

x \in S

satisfying

f(x) = 0

.

Zeros(f(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{zeros}\,}\limits_{x \in S,\,P(x)} f(x)

, represents the set of values

x \in S

satisfying

P(x)

and

f(x) = 0

.

Zeros(f(x), For(x), P(x)), rendered

\mathop{\operatorname{zeros}\,}\limits_{P(x)} f(x)

, represents the set of values

x

satisfying

P(x)

and

f(x) = 0

.

Zeros(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

\mathop{\operatorname{zeros}\,}\limits_{P\left(x, y\right)} f\!\left(x, y\right)

, represents the set of tuples

\left(x, y\right)

satisfying

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

and

f\!\left(x, y\right) = 0

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function

Source code for this entry:

Entry(ID("f7ce46"),
    SymbolDefinition(Zeros, Zeros(f(x), ForElement(x, S)), "Zeros (roots) of function"),
    Description(SourceForm(Zeros(f(x), ForElement(x, S))), ", rendered", Zeros(f(x), ForElement(x, S)), ", represents the set of values", Element(x, S), "satisfying", Equal(f(x), 0), "."),
    Description(SourceForm(Zeros(f(x), ForElement(x, S), P(x))), ", rendered", Zeros(f(x), ForElement(x, S), P(x)), ", represents the set of values", Element(x, S), "satisfying", P(x), "and", Equal(f(x), 0), "."),
    Description(SourceForm(Zeros(f(x), For(x), P(x))), ", rendered", Zeros(f(x), For(x), P(x)), ", represents the set of values", x, "satisfying", P(x), "and", Equal(f(x), 0), "."),
    Description(SourceForm(Zeros(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Zeros(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents the set of tuples", Tuple(x, y), "satisfying", P(x, y), "and", Equal(f(x, y), 0), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

d2714b

Symbol: UniqueZero —

\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)

— Unique zero (root) of function

UniqueZero(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)

, represents the unique value

x \in S

satisfying

f(x) = 0

.

This operation is undefined if such a value does not exist or is not unique.

UniqueZero(f(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{zero*}\,}\limits_{x \in S,\,P(x)} f(x)

, represents the unique value

x \in S

satisfying

P(x)

and

f(x) = 0

.

UniqueZero(f(x), For(x), P(x)), rendered

\mathop{\operatorname{zero*}\,}\limits_{P(x)} f(x)

, represents the unique value

x

satisfying

P(x)

and

f(x) = 0

.

UniqueZero(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

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

, represents the unique tuple

\left(x, y\right)

such that

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

and

f\!\left(x, y\right) = 0

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function

Source code for this entry:

Entry(ID("d2714b"),
    SymbolDefinition(UniqueZero, UniqueZero(f(x), ForElement(x, S)), "Unique zero (root) of function"),
    Description(SourceForm(UniqueZero(f(x), ForElement(x, S))), ", rendered", UniqueZero(f(x), ForElement(x, S)), ", represents the unique value", Element(x, S), "satisfying", Equal(f(x), 0), "."),
    Description("This operation is undefined if such a value does not exist or is not unique."),
    Description(SourceForm(UniqueZero(f(x), ForElement(x, S), P(x))), ", rendered", UniqueZero(f(x), ForElement(x, S), P(x)), ", represents the unique value", Element(x, S), "satisfying", P(x), "and", Equal(f(x), 0), "."),
    Description(SourceForm(UniqueZero(f(x), For(x), P(x))), ", rendered", UniqueZero(f(x), For(x), P(x)), ", represents the unique value", x, "satisfying", P(x), "and", Equal(f(x), 0), "."),
    Description(SourceForm(UniqueZero(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", UniqueZero(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents the unique tuple", Tuple(x, y), "such that", P(x, y), "and", Equal(f(x, y), 0), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

5862bb

Symbol: Solutions —

\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)

— Solution set

Solutions(Q(x), ForElement(x, S)), rendered

\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)

, represents the set of values

x \in S

satisfying

Q(x)

.

Solutions(Q(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{solutions}\,}\limits_{x \in S,\,P(x)} Q(x)

, represents the set of values

x \in S

satisfying

P(x)

and

Q(x)

.

Solutions(Q(x), For(x), P(x)), rendered

\mathop{\operatorname{solutions}\,}\limits_{P(x)} Q(x)

, represents the set of values

x

satisfying

P(x)

and

Q(x)

.

Solutions(Q(x, y), For(Tuple(x, y)), P(x, y)), rendered

\mathop{\operatorname{solutions}\,}\limits_{P\left(x, y\right)} Q\!\left(x, y\right)

, represents the set of tuples

\left(x, y\right)

satisfying

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

and

Q\!\left(x, y\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`Solutions`	$\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$	Solution set

Source code for this entry:

Entry(ID("5862bb"),
    SymbolDefinition(Solutions, Solutions(Q(x), ForElement(x, S)), "Solution set"),
    Description(SourceForm(Solutions(Q(x), ForElement(x, S))), ", rendered", Solutions(Q(x), ForElement(x, S)), ", represents the set of values", Element(x, S), "satisfying", Q(x), "."),
    Description(SourceForm(Solutions(Q(x), ForElement(x, S), P(x))), ", rendered", Solutions(Q(x), ForElement(x, S), P(x)), ", represents the set of values", Element(x, S), "satisfying", P(x), "and", Q(x), "."),
    Description(SourceForm(Solutions(Q(x), For(x), P(x))), ", rendered", Solutions(Q(x), For(x), P(x)), ", represents the set of values", x, "satisfying", P(x), "and", Q(x), "."),
    Description(SourceForm(Solutions(Q(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Solutions(Q(x, y), For(Tuple(x, y)), P(x, y)), ", represents the set of tuples", Tuple(x, y), "satisfying", P(x, y), "and", Q(x, y), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

f5ae93

Symbol: UniqueSolution —

\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)

— Unique solution

UniqueSolution(Q(x), ForElement(x, S)), rendered

\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)

, represents the unique value

x \in S

satisfying

Q(x)

.

This operation is undefined if such a value does not exist or is not unique.

UniqueSolution(Q(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{solution*}\,}\limits_{x \in S,\,P(x)} Q(x)

, represents the unique value

x \in S

satisfying

P(x)

and

Q(x)

.

UniqueSolution(Q(x), For(x), P(x)), rendered

\mathop{\operatorname{solution*}\,}\limits_{P(x)} Q(x)

, represents the unique value

x

satisfying

P(x)

and

Q(x)

.

UniqueSolution(Q(x, y), For(Tuple(x, y)), P(x, y)), rendered

\mathop{\operatorname{solution*}\,}\limits_{P\left(x, y\right)} Q\!\left(x, y\right)

, represents the unique tuple

\left(x, y\right)

satisfying

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

and

Q\!\left(x, y\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`UniqueSolution`	$\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)$	Unique solution

Source code for this entry:

Entry(ID("f5ae93"),
    SymbolDefinition(UniqueSolution, UniqueSolution(Q(x), ForElement(x, S)), "Unique solution"),
    Description(SourceForm(UniqueSolution(Q(x), ForElement(x, S))), ", rendered", UniqueSolution(Q(x), ForElement(x, S)), ", represents the unique value", Element(x, S), "satisfying", Q(x), "."),
    Description("This operation is undefined if such a value does not exist or is not unique."),
    Description(SourceForm(UniqueSolution(Q(x), ForElement(x, S), P(x))), ", rendered", UniqueSolution(Q(x), ForElement(x, S), P(x)), ", represents the unique value", Element(x, S), "satisfying", P(x), "and", Q(x), "."),
    Description(SourceForm(UniqueSolution(Q(x), For(x), P(x))), ", rendered", UniqueSolution(Q(x), For(x), P(x)), ", represents the unique value", x, "satisfying", P(x), "and", Q(x), "."),
    Description(SourceForm(UniqueSolution(Q(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", UniqueSolution(Q(x, y), For(Tuple(x, y)), P(x, y)), ", represents the unique tuple", Tuple(x, y), "satisfying", P(x, y), "and", Q(x, y), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

6ec976

Symbol: Supremum —

\mathop{\operatorname{sup}}\limits_{x \in S} f(x)

— Supremum of a set or function

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

.

Supremum(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{sup}}\limits_{x \in S} f(x)

, represents

\operatorname{sup} \left\{ f(x) : x \in S \right\}

.

Supremum(f(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{sup}}\limits_{x \in S,\,P(x)} f(x)

, represents

\operatorname{sup} \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}

.

Supremum(f(x), For(x), P(x)), rendered

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

, represents

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

.

Supremum(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

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

, represents

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

where

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

is a predicate defining the range of

x

and

y

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

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

Source code for this entry:

Entry(ID("6ec976"),
    SymbolDefinition(Supremum, Supremum(f(x), ForElement(x, S)), "Supremum of a set or function"),
    Description(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(SourceForm(Supremum(f(x), ForElement(x, S))), ", rendered", Supremum(f(x), ForElement(x, S)), ", represents", Supremum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Supremum(f(x), ForElement(x, S), P(x))), ", rendered", Supremum(f(x), ForElement(x, S), P(x)), ", represents", Supremum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Supremum(f(x), For(x), P(x))), ", rendered", Supremum(f(x), For(x), P(x)), ", represents", Supremum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Supremum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Supremum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Supremum(Set(f(x, y), For(Tuple(x, y)), P(x, y))), "where", P(x, y), "is a predicate defining the range of", x, "and", y, ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

bbeb35

Symbol: Infimum —

\mathop{\operatorname{inf}}\limits_{x \in S} f(x)

— Infimum of a set or function

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) = b

.

Infimum(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{inf}}\limits_{x \in S} f(x)

, represents

\operatorname{inf} \left\{ f(x) : x \in S \right\}

.

Infimum(f(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{inf}}\limits_{x \in S,\,P(x)} f(x)

, represents

\operatorname{inf} \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}

.

Infimum(f(x), For(x), P(x)), rendered

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

, represents

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

.

Infimum(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

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

, represents

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

where

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

is a predicate defining the range of

x

and

y

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

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

Source code for this entry:

Entry(ID("bbeb35"),
    SymbolDefinition(Infimum, Infimum(f(x), ForElement(x, S)), "Infimum of a set or function"),
    Description(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), b), "."),
    Description(SourceForm(Infimum(f(x), ForElement(x, S))), ", rendered", Infimum(f(x), ForElement(x, S)), ", represents", Infimum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Infimum(f(x), ForElement(x, S), P(x))), ", rendered", Infimum(f(x), ForElement(x, S), P(x)), ", represents", Infimum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Infimum(f(x), For(x), P(x))), ", rendered", Infimum(f(x), For(x), P(x)), ", represents", Infimum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Infimum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Infimum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Infimum(Set(f(x, y), For(Tuple(x, y)), P(x, y))), "where", P(x, y), "is a predicate defining the range of", x, "and", y, ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

d0cb24

Symbol: Minimum —

\mathop{\min}\limits_{x \in S} f(x)

— Minimum value of a set or function

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.

Minimum(f(x), ForElement(x, S)), rendered

\mathop{\min}\limits_{x \in S} f(x)

, represents

\min \left\{ f(x) : x \in S \right\}

.

Minimum(f(x), ForElement(x, S), P(x)), rendered

\mathop{\min}\limits_{x \in S,\,P(x)} f(x)

, represents

\min \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}

.

Minimum(f(x), For(x), P(x)), rendered

\mathop{\min}\limits_{P(x)} f(x)

, represents

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

.

Minimum(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

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

, represents

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

where

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

is a predicate defining the range of

x

and

y

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`Minimum`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("d0cb24"),
    SymbolDefinition(Minimum, Minimum(f(x), ForElement(x, S)), "Minimum value of a set or function"),
    Description(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(SourceForm(Minimum(f(x), ForElement(x, S))), ", rendered", Minimum(f(x), ForElement(x, S)), ", represents", Minimum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Minimum(f(x), ForElement(x, S), P(x))), ", rendered", Minimum(f(x), ForElement(x, S), P(x)), ", represents", Minimum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Minimum(f(x), For(x), P(x))), ", rendered", Minimum(f(x), For(x), P(x)), ", represents", Minimum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Minimum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Minimum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Minimum(Set(f(x, y), For(Tuple(x, y)), P(x, y))), "where", P(x, y), "is a predicate defining the range of", x, "and", y, ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

65ccf2

Symbol: Maximum —

\mathop{\max}\limits_{x \in S} f(x)

— Maximum value of a set or function

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.

Maximum(f(x), ForElement(x, S)), rendered

\mathop{\max}\limits_{x \in S} f(x)

, represents

\max \left\{ f(x) : x \in S \right\}

.

Maximum(f(x), ForElement(x, S), P(x)), rendered

\mathop{\max}\limits_{x \in S,\,P(x)} f(x)

, represents

\max \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}

.

Maximum(f(x), For(x), P(x)), rendered

\mathop{\max}\limits_{P(x)} f(x)

, represents

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

.

Maximum(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

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

, represents

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

where

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

is a predicate defining the range of

x

and

y

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`Maximum`	$\mathop{\max}\limits_{x \in S} f(x)$	Maximum value of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("65ccf2"),
    SymbolDefinition(Maximum, Maximum(f(x), ForElement(x, S)), "Maximum value of a set or function"),
    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(SourceForm(Maximum(f(x), ForElement(x, S))), ", rendered", Maximum(f(x), ForElement(x, S)), ", represents", Maximum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Maximum(f(x), ForElement(x, S), P(x))), ", rendered", Maximum(f(x), ForElement(x, S), P(x)), ", represents", Maximum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Maximum(f(x), For(x), P(x))), ", rendered", Maximum(f(x), For(x), P(x)), ", represents", Maximum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Maximum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Maximum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Maximum(Set(f(x, y), For(Tuple(x, y)), P(x, y))), "where", P(x, y), "is a predicate defining the range of", x, "and", y, ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

0a3e5a

Symbol: ArgMin —

\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x)

— Locations of minimum value

ArgMin(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x)

, gives the set of values

x \in S

such that

f(x) = \mathop{\min}\limits_{s \in S} f(s)

.

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

\mathop{\operatorname{arg\,min}}\limits_{x \in S,\,P(x)} f(x)

, gives the set of values

x \in S

satisfying

P(x)

and such that

f(x) = \mathop{\min}\limits_{s \in S,\,P(s)} f(s)

.

If

f(x)

does not attain a minimum value satisfying the conditions, the result is the empty set {}.

ArgMin(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

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

, gives the set of tuples

\left(x, y\right)

satisfying

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

such that

f\!\left(x, y\right) = \mathop{\min}\limits_{P\left(s, t\right)} f\!\left(s, t\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

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

Source code for this entry:

Entry(ID("0a3e5a"),
    SymbolDefinition(ArgMin, ArgMin(f(x), ForElement(x, S)), "Locations of minimum value"),
    Description(SourceForm(ArgMin(f(x), ForElement(x, S))), ", rendered", ArgMin(f(x), ForElement(x, S)), ", gives the set of values", Element(x, S), "such that", Equal(f(x), Minimum(f(s), ForElement(s, S))), "."),
    Description(SourceForm(ArgMin(f(x), ForElement(x, S), P(x))), ", rendered", ArgMin(f(x), ForElement(x, S), P(x)), ", gives the set of values", Element(x, S), "satisfying", P(x), "and such that", Equal(f(x), Minimum(f(s), ForElement(s, S), P(s))), "."),
    Description("If", f(x), "does not attain a minimum value satisfying the conditions, the result is the empty set", Set(), "."),
    Description(SourceForm(ArgMin(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", ArgMin(f(x, y), For(Tuple(x, y)), P(x, y)), ", gives the set of tuples", Tuple(x, y), "satisfying", P(x, y), "such that", Equal(f(x, y), Minimum(f(s, t), For(Tuple(s, t)), P(s, t))), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

617fe3

Symbol: ArgMax —

\mathop{\operatorname{arg\,max}}\limits_{x \in S} f(x)

— Locations of maximum value

ArgMax(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{arg\,max}}\limits_{x \in S} f(x)

, gives the set of values

x \in S

such that

f(x) = \mathop{\max}\limits_{s \in S} f(s)

.

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

\mathop{\operatorname{arg\,max}}\limits_{x \in S,\,P(x)} f(x)

, gives the set of values

x \in S

satisfying

P(x)

and such that

f(x) = \mathop{\max}\limits_{s \in S,\,P(s)} f(s)

.

If

f(x)

does not attain a maximum value satisfying the conditions, the result is the empty set {}.

ArgMax(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

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

, gives the set of tuples

\left(x, y\right)

satisfying

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

such that

f\!\left(x, y\right) = \mathop{\max}\limits_{P\left(s, t\right)} f\!\left(s, t\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

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

Source code for this entry:

Entry(ID("617fe3"),
    SymbolDefinition(ArgMax, ArgMax(f(x), ForElement(x, S)), "Locations of maximum value"),
    Description(SourceForm(ArgMax(f(x), ForElement(x, S))), ", rendered", ArgMax(f(x), ForElement(x, S)), ", gives the set of values", Element(x, S), "such that", Equal(f(x), Maximum(f(s), ForElement(s, S))), "."),
    Description(SourceForm(ArgMax(f(x), ForElement(x, S), P(x))), ", rendered", ArgMax(f(x), ForElement(x, S), P(x)), ", gives the set of values", Element(x, S), "satisfying", P(x), "and such that", Equal(f(x), Maximum(f(s), ForElement(s, S), P(s))), "."),
    Description("If", f(x), "does not attain a maximum value satisfying the conditions, the result is the empty set", Set(), "."),
    Description(SourceForm(ArgMax(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", ArgMax(f(x, y), For(Tuple(x, y)), P(x, y)), ", gives the set of tuples", Tuple(x, y), "satisfying", P(x, y), "such that", Equal(f(x, y), Maximum(f(s, t), For(Tuple(s, t)), P(s, t))), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

f4fbb8

Symbol: ArgMinUnique —

\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x)

— Unique location of minimum value

ArgMinUnique(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x)

, represents the unique value

x \in S

such that

f(x) = \mathop{\min}\limits_{s \in S} f(s)

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

ArgMinUnique(f(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{arg\,min*}}\limits_{x \in S,\,P(x)} f(x)

, represents the unique value

x \in S

satisfying

P(x)

and such that

f(x) = \mathop{\min}\limits_{s \in S} f(s)

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

ArgMinUnique(f(x, y), For(Tuple(x, y)), P(x, y)) represents the unique tuple

\left(x, y\right)

satisfying

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

such that

f\!\left(x, y\right) = \mathop{\min}\limits_{P\left(s, t\right)} f\!\left(s, t\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

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

Source code for this entry:

Entry(ID("f4fbb8"),
    SymbolDefinition(ArgMinUnique, ArgMinUnique(f(x), ForElement(x, S)), "Unique location of minimum value"),
    Description(SourceForm(ArgMinUnique(f(x), ForElement(x, S))), ", rendered", ArgMinUnique(f(x), ForElement(x, S)), ", ", "represents the unique value", Element(x, S), "such that", Equal(f(x), Minimum(f(s), ForElement(s, S))), ". This operation is only defined if such a unique value exists."),
    Description(SourceForm(ArgMinUnique(f(x), ForElement(x, S), P(x))), ", rendered", ArgMinUnique(f(x), ForElement(x, S), P(x)), ", ", "represents the unique value", Element(x, S), "satisfying", P(x), "and", "such that", Equal(f(x), Minimum(f(s), ForElement(s, S))), ". This operation is only defined if such a unique value exists."),
    Description(SourceForm(ArgMinUnique(f(x, y), For(Tuple(x, y)), P(x, y))), "represents the unique tuple", Tuple(x, y), "satisfying", P(x, y), "such that", Equal(f(x, y), Minimum(f(s, t), For(Tuple(s, t)), P(s, t))), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

be4926

Symbol: ArgMaxUnique —

\mathop{\operatorname{arg\,max*}}\limits_{x \in S} f(x)

— Unique location of maximum value

ArgMaxUnique(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{arg\,max*}}\limits_{x \in S} f(x)

, represents the unique value

x \in S

such that

f(x) = \mathop{\max}\limits_{s \in S} f(s)

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

ArgMaxUnique(f(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{arg\,max*}}\limits_{x \in S,\,P(x)} f(x)

, represents the unique value

x \in S

satisfying

P(x)

and such that

f(x) = \mathop{\max}\limits_{s \in S} f(s)

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

ArgMaxUnique(f(x, y), For(Tuple(x, y)), P(x, y)) represents the unique tuple

\left(x, y\right)

satisfying

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

such that

f\!\left(x, y\right) = \mathop{\max}\limits_{P\left(s, t\right)} f\!\left(s, t\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

,

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

, etc.

Definitions:

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

Source code for this entry:

Entry(ID("be4926"),
    SymbolDefinition(ArgMaxUnique, ArgMaxUnique(f(x), ForElement(x, S)), "Unique location of maximum value"),
    Description(SourceForm(ArgMaxUnique(f(x), ForElement(x, S))), ", rendered", ArgMaxUnique(f(x), ForElement(x, S)), ", ", "represents the unique value", Element(x, S), "such that", Equal(f(x), Maximum(f(s), ForElement(s, S))), ". This operation is only defined if such a unique value exists."),
    Description(SourceForm(ArgMaxUnique(f(x), ForElement(x, S), P(x))), ", rendered", ArgMaxUnique(f(x), ForElement(x, S), P(x)), ", ", "represents the unique value", Element(x, S), "satisfying", P(x), "and", "such that", Equal(f(x), Maximum(f(s), ForElement(s, S))), ". This operation is only defined if such a unique value exists."),
    Description(SourceForm(ArgMaxUnique(f(x, y), For(Tuple(x, y)), P(x, y))), "represents the unique tuple", Tuple(x, y), "satisfying", P(x, y), "such that", Equal(f(x, y), Maximum(f(s, t), For(Tuple(s, t)), P(s, t))), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

26ea9f

Symbol: Limit —

\lim_{x \to a} f(x)

— Limiting value

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

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

represents the limiting value of

f(x)

for every sequence of

x

satisfying

P(x)

and approaching the limit point

a

.

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 SequenceLimit, RealLimit, LeftLimit, RightLimit or ComplexLimit.

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.

The expression 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)

.

The expression For(x, a) declares x as a locally bound variable within the scope of the arguments to this operator.

Definitions:

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

Source code for this entry:

Entry(ID("26ea9f"),
    SymbolDefinition(Limit, Limit(f(x), For(x, a)), "Limiting value"),
    Description(SourceForm(Limit(f(x), For(x, a), P(x))), "rendered as", Limit(f(x), For(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), "."),
    Description("The expression", SourceForm(For(x, a)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator."))

1d2ee5

Symbol: SequenceLimit —

\lim_{n \to a} f(n)

— Limiting value of sequence

This operator can be called with three or four arguments.

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

\lim_{n \to a} f(n)

, is equivalent to Limit(f(n), For(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), For(n, a), P(n)), rendered as

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

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

The expression For(x, a) declares x as a locally bound variable within the scope of the arguments to this operator.

Definitions:

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

Source code for this entry:

Entry(ID("1d2ee5"),
    SymbolDefinition(SequenceLimit, SequenceLimit(f(n), For(n, a)), "Limiting value of sequence"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(SequenceLimit(f(n), For(n, a))), ", rendered as", SequenceLimit(f(n), For(n, a)), ", is equivalent to", SourceForm(Limit(f(n), For(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), For(n, a), P(n))), ", rendered as", SequenceLimit(f(n), For(n, a), P(n)), ", is equivalent to", SourceForm(Limit(f(n), For(n, a), And(Element(n, ZZ), P(n)))), "."),
    Description("The expression", SourceForm(For(x, a)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator."))

6fe5c1

Symbol: RealLimit —

\lim_{x \to a} f(x)

— Limiting value, real variable

This operator can be called with three or four arguments.

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

\lim_{x \to a} f(x)

, is equivalent to Limit(f(x), For(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), For(x, a), P(x)), rendered as

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

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

The expression For(x, a) declares x as a locally bound variable within the scope of the arguments to this operator.

Definitions:

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

Source code for this entry:

Entry(ID("6fe5c1"),
    SymbolDefinition(RealLimit, RealLimit(f(x), For(x, a)), "Limiting value, real variable"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(RealLimit(f(x), For(x, a))), ", rendered as", RealLimit(f(x), For(x, a)), ", is equivalent to", SourceForm(Limit(f(x), For(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), For(x, a), P(x))), ", rendered as", RealLimit(f(x), For(x, a), P(x)), ", is equivalent to", SourceForm(Limit(f(x), For(x, a), And(Element(x, RR), P(x)))), "."),
    Description("The expression", SourceForm(For(x, a)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator."))

c8a5f0

Symbol: LeftLimit —

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

— Limiting value, from the left

This operator can be called with three or four arguments.

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

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

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

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

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

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

The expression For(x, a) declares x as a locally bound variable within the scope of the arguments to this operator.

Definitions:

Fungrim symbol	Notation	Short description
`LeftLimit`	$\lim_{x \to {a}^{-}} f(x)$	Limiting value, from the left
`Limit`	$\lim_{x \to a} f(x)$	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), For(x, a)), "Limiting value, from the left"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(LeftLimit(f(x), For(x, a))), ", rendered as", LeftLimit(f(x), For(x, a)), ", is equivalent to", SourceForm(Limit(f(x), For(x, a), Element(x, OpenInterval(Neg(Infinity), a)))), "."),
    Description(SourceForm(LeftLimit(f(x), For(x, a), P(x))), ", rendered as", LeftLimit(f(x), For(x, a), P(x)), ", is equivalent to", SourceForm(Limit(f(x), For(x, a), And(Element(x, OpenInterval(Neg(Infinity), a)), P(x)))), "."),
    Description("The expression", SourceForm(For(x, a)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator."))

afd5ca

Symbol: RightLimit —

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

— Limiting value, from the right

This operator can be called with three or four arguments.

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

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

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

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

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

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

The expression For(x, a) declares x as a locally bound variable within the scope of the arguments to this operator.

Definitions:

Fungrim symbol	Notation	Short description
`RightLimit`	$\lim_{x \to {a}^{+}} f(x)$	Limiting value, from the right
`Limit`	$\lim_{x \to a} f(x)$	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), For(x, a)), "Limiting value, from the right"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(RightLimit(f(x), For(x, a))), ", rendered as", RightLimit(f(x), For(x, a)), ", is equivalent to", SourceForm(Limit(f(x), For(x, a), Element(x, OpenInterval(a, Infinity)))), "."),
    Description(SourceForm(RightLimit(f(x), For(x, a), P(x))), ", rendered as", RightLimit(f(x), For(x, a), P(x)), ", is equivalent to", SourceForm(Limit(f(x), For(x, a), And(Element(x, OpenInterval(a, Infinity)), P(x)))), "."),
    Description("The expression", SourceForm(For(x, a)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator."))

05a3ee

Symbol: ComplexLimit —

\lim_{z \to a} f(z)

— Limiting value, complex variable

This operator can be called with three or four arguments.

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

\lim_{z \to a} f(z)

, is equivalent to Limit(f(z), For(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), For(z, a), P(z)), rendered as

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

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

The expression For(z, a) declares z as a locally bound variable within the scope of the arguments to this operator.

Definitions:

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

Source code for this entry:

Entry(ID("05a3ee"),
    SymbolDefinition(ComplexLimit, ComplexLimit(f(z), For(z, a)), "Limiting value, complex variable"),
    Description("This operator can be called with three or four arguments."),
    Description(SourceForm(ComplexLimit(f(z), For(z, a))), ", rendered as", ComplexLimit(f(z), For(z, a)), ", is equivalent to", SourceForm(Limit(f(z), For(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), For(z, a), P(z))), ", rendered as", ComplexLimit(f(z), For(z, a), P(z)), ", is equivalent to", SourceForm(Limit(f(z), For(z, a), And(Element(z, CC), P(z)))), "."),
    Description("The expression", SourceForm(For(z, a)), "declares", SourceForm(z), "as a locally bound variable within the scope of the arguments to this operator."))

2be0b5

Symbol: MeromorphicLimit —

\lim_{z \to a} f(z)

— 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`	$\lim_{z \to a} f(z)$	Limiting value, allowing poles
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("2be0b5"),
    SymbolDefinition(MeromorphicLimit, MeromorphicLimit(f(z), For(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."))

6cd302

Symbol: SequenceLimitInferior —

\liminf_{n \to a} f(n)

— Limit inferior of sequence

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimitInferior`	$\liminf_{n \to a} f(n)$	Limit inferior of sequence

Source code for this entry:

Entry(ID("6cd302"),
    SymbolDefinition(SequenceLimitInferior, SequenceLimitInferior(f(n), For(n, a)), "Limit inferior of sequence"))

fdf152

Symbol: SequenceLimitSuperior —

\limsup_{n \to a} f(n)

— Limit superior of sequence

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimitSuperior`	$\limsup_{n \to a} f(n)$	Limit superior of sequence

Source code for this entry:

Entry(ID("fdf152"),
    SymbolDefinition(SequenceLimitSuperior, SequenceLimitSuperior(f(n), For(n, a)), "Limit superior of sequence"))

1b6a57

Symbol: Derivative —

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

— Derivative

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

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

or

f'(a)

, represents the derivative of

f(z)

evaluated at

z = a

.

Derivative(f(z), For(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(z)

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), For(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), For(z, z)), "Derivative"),
    Description(SourceForm(Derivative(f(z), For(z, a))), ", rendered as ", Derivative(Call(f, z), For(z, a)), "or", Derivative(f(z), For(z, a)), ", represents the derivative of", f(z), "evaluated at", Equal(z, a), "."),
    Description(SourceForm(Derivative(f(z), For(z, a, n))), ", rendered as ", Derivative(Call(f, z), For(z, a, n)), "or", Derivative(f(z), For(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), For(z, z))), "may render more simply as", Derivative(Call(f, z), For(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(x)}{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`	$\lim_{x \to a} f(x)$	Limiting value, real variable

Source code for this entry:

Entry(ID("452407"),
    SymbolDefinition(RealDerivative, RealDerivative(Call(f, x), For(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), For(x, x)), RealLimit(Div(Sub(f(Add(x, h)), f(x)), h), For(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(z)}{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`	$\lim_{z \to a} f(z)$	Limiting value, complex variable

Source code for this entry:

Entry(ID("b4b319"),
    SymbolDefinition(ComplexDerivative, ComplexDerivative(Call(f, z), For(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), For(z, z)), ComplexLimit(Div(Sub(f(Add(z, h)), f(z)), h), For(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), For(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), For(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."))

c285c7

Symbol: Integral —

\int_{a}^{b} f(x) \, dx

— Integral

Integral(f(x), For(x, a, b)), rendered as

\int_{a}^{b} f(x) \, dx

, represents the integral of

f(x)

from

a

to

b

. The order is significant:

\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx

.

Integral(f(x), ForElement(x, S)), rendered as

\int_{x \in S} f(x) \, dx

, represents the integral of

f(x)

over the set

S

.

The special expression For(x, a, b) or ForElement(x, S) defines a locally bound variable.

The precise class of integrals allowed by this operator is yet to be defined, but should normally encompass Lebesgue integrals.

The integrand is allowed to be undefined on a subset of measure of zero.

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral

Source code for this entry:

Entry(ID("c285c7"),
    SymbolDefinition(Integral, Integral(f(x), For(x, a, b)), "Integral"),
    Description(SourceForm(Integral(f(x), For(x, a, b))), ", rendered as ", Integral(f(x), For(x, a, b)), ", represents the integral of", f(x), "from", a, "to", b, ". ", "The order is significant: ", Equal(Integral(f(x), For(x, a, b)), Neg(Integral(f(x), For(x, b, a)))), "."),
    Description(SourceForm(Integral(f(x), ForElement(x, S))), ", rendered as ", Integral(f(x), ForElement(x, S)), ", represents the integral of", f(x), "over the set", S, "."),
    Description("The special expression", SourceForm(For(x, a, b)), "or", SourceForm(ForElement(x, S)), "defines a locally bound variable."),
    Description("The precise class of integrals allowed by this operator is yet to be defined, but should normally encompass Lebesgue integrals."),
    Description("The integrand is allowed to be undefined on a subset of measure of zero."))

2e4fbc

Symbol: IndefiniteIntegralEqual —

\int f(x) \, dx = g(x) + \mathcal{C}

— Indefinite integral

IndefiniteIntegralEqual(f(x), g(x), x, c), rendered as

\int f(x) \, dx = g(x) + \mathcal{C}, x = c

, expresses that

g(x)

is an antiderivative of

f(x)

at the point

c

, or formally that

g'(c) = f(c)

. In other words,

g(x)

belongs to the equivalence class of antiderivatives of

f(x)

at the point

c

. This is rendered as a statement of equality (with an arbitrary constant of integration) to follow the conventional notation for indefinite integrals.

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 RealIndefiniteIntegralEqual or ComplexIndefiniteIntegralEqual.

The argument x defines a locally bound variable used in the expressions

f(x)

and

g(x)

. If this operator is called more simply as IndefiniteIntegralEqual(f(x), g(x), x), the meaning is the same as IndefiniteIntegralEqual(f(x), g(x), x, x), where the x appearing in

f(x)

and

g(x)

is understood as a new dummy variable. This dummy variable is evaluated at the value x defined in the surrounding context only after the functions have been constructed.

Definitions:

Fungrim symbol	Notation	Short description
`IndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`RealIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, real derivative
`ComplexIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, complex derivative

Source code for this entry:

Entry(ID("2e4fbc"),
    SymbolDefinition(IndefiniteIntegralEqual, IndefiniteIntegralEqual(f(x), g(x), x), "Indefinite integral"),
    Description(SourceForm(IndefiniteIntegralEqual(f(x), g(x), x, c)), ", rendered as ", IndefiniteIntegralEqual(f(x), g(x), x, c), ", expresses that", g(x), "is an antiderivative of", f(x), "at the point", c, ", or formally that", Equal(Derivative(g(x), For(x, c)), f(c)), ".", "In other words,", g(x), "belongs to the equivalence class of antiderivatives of", f(x), "at the point", c, ". This is rendered as a statement of equality (with an arbitrary constant of integration) to follow the conventional notation for indefinite integrals."),
    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(RealIndefiniteIntegralEqual), "or", SourceForm(ComplexIndefiniteIntegralEqual), "."),
    Description("The argument", SourceForm(x), "defines a locally bound variable used in the expressions", f(x), "and", g(x), ". ", "If this operator is called more simply as", SourceForm(IndefiniteIntegralEqual(f(x), g(x), x)), ", ", "the meaning is the same as", SourceForm(IndefiniteIntegralEqual(f(x), g(x), x, x)), ", where the", SourceForm(x), "appearing in", f(x), "and", g(x), "is understood as a new dummy variable. This dummy variable is evaluated at the value", SourceForm(x), "defined in the surrounding context only after the functions have been constructed."))

0be77d

Symbol: RealIndefiniteIntegralEqual —

\int f(x) \, dx = g(x) + \mathcal{C}

— Indefinite integral, real derivative

RealIndefiniteIntegralEqual(f(x), g(x), x, c), rendered as

\int f(x) \, dx = g(x) + \mathcal{C}, x = c

, expresses that

g(x)

is an antiderivative of

f(x)

at the point

c

, or formally that

g'(c) = f(c)

. In other words,

g(x)

belongs to the equivalence class of antiderivatives of

f(x)

at the point

c

. This is rendered as a statement of equality (with an arbitrary constant of integration) to follow the conventional notation for indefinite integrals.

The argument x defines a locally bound variable used in the expressions

f(x)

and

g(x)

. If this operator is called more simply as RealIndefiniteIntegralEqual(f(x), g(x), x), the meaning is the same as RealIndefiniteIntegralEqual(f(x), g(x), x, x), where the x appearing in

f(x)

and

g(x)

is understood as a new dummy variable. This dummy variable is evaluated at the value x defined in the surrounding context only after the functions have been constructed.

Definitions:

Fungrim symbol	Notation	Short description
`RealIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, real derivative
`RealDerivative`	$\frac{d}{d x}\, f\!\left(x\right)$	Real derivative

Source code for this entry:

Entry(ID("0be77d"),
    SymbolDefinition(RealIndefiniteIntegralEqual, RealIndefiniteIntegralEqual(f(x), g(x), x), "Indefinite integral, real derivative"),
    Description(SourceForm(RealIndefiniteIntegralEqual(f(x), g(x), x, c)), ", rendered as ", RealIndefiniteIntegralEqual(f(x), g(x), x, c), ", expresses that", g(x), "is an antiderivative of", f(x), "at the point", c, ", or formally that", Equal(RealDerivative(g(x), For(x, c)), f(c)), ".", "In other words,", g(x), "belongs to the equivalence class of antiderivatives of", f(x), "at the point", c, ". This is rendered as a statement of equality (with an arbitrary constant of integration) to follow the conventional notation for indefinite integrals."),
    Description("The argument", SourceForm(x), "defines a locally bound variable used in the expressions", f(x), "and", g(x), ". ", "If this operator is called more simply as", SourceForm(RealIndefiniteIntegralEqual(f(x), g(x), x)), ", ", "the meaning is the same as", SourceForm(RealIndefiniteIntegralEqual(f(x), g(x), x, x)), ", where the", SourceForm(x), "appearing in", f(x), "and", g(x), "is understood as a new dummy variable. This dummy variable is evaluated at the value", SourceForm(x), "defined in the surrounding context only after the functions have been constructed."))

fb2790

Symbol: ComplexIndefiniteIntegralEqual —

\int f(x) \, dx = g(x) + \mathcal{C}

— Indefinite integral, complex derivative

ComplexIndefiniteIntegralEqual(f(x), g(x), x, c), rendered as

\int f(x) \, dx = g(x) + \mathcal{C}, x = c

, expresses that

g(x)

is an antiderivative of

f(x)

at the point

c

, or formally that

g'(c) = f(c)

. In other words,

g(x)

belongs to the equivalence class of antiderivatives of

f(x)

at the point

c

. This is rendered as a statement of equality (with an arbitrary constant of integration) to follow the conventional notation for indefinite integrals.

The argument x defines a locally bound variable used in the expressions

f(x)

and

g(x)

. If this operator is called more simply as ComplexIndefiniteIntegralEqual(f(x), g(x), x), the meaning is the same as ComplexIndefiniteIntegralEqual(f(x), g(x), x, x), where the x appearing in

f(x)

and

g(x)

is understood as a new dummy variable. This dummy variable is evaluated at the value x defined in the surrounding context only after the functions have been constructed.

Definitions:

Fungrim symbol	Notation	Short description
`ComplexIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, complex derivative
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative

Source code for this entry:

Entry(ID("fb2790"),
    SymbolDefinition(ComplexIndefiniteIntegralEqual, ComplexIndefiniteIntegralEqual(f(x), g(x), x), "Indefinite integral, complex derivative"),
    Description(SourceForm(ComplexIndefiniteIntegralEqual(f(x), g(x), x, c)), ", rendered as ", ComplexIndefiniteIntegralEqual(f(x), g(x), x, c), ", expresses that", g(x), "is an antiderivative of", f(x), "at the point", c, ", or formally that", Equal(ComplexDerivative(g(x), For(x, c)), f(c)), ".", "In other words,", g(x), "belongs to the equivalence class of antiderivatives of", f(x), "at the point", c, ". This is rendered as a statement of equality (with an arbitrary constant of integration) to follow the conventional notation for indefinite integrals."),
    Description("The argument", SourceForm(x), "defines a locally bound variable used in the expressions", f(x), "and", g(x), ". ", "If this operator is called more simply as", SourceForm(ComplexIndefiniteIntegralEqual(f(x), g(x), x)), ", ", "the meaning is the same as", SourceForm(ComplexIndefiniteIntegralEqual(f(x), g(x), x, x)), ", where the", SourceForm(x), "appearing in", f(x), "and", g(x), "is understood as a new dummy variable. This dummy variable is evaluated at the value", SourceForm(x), "defined in the surrounding context only after the functions have been constructed."))

499bdf

Symbol: IsHolomorphic —

f(z) \text{ is holomorphic at } z = c

— Holomorphic predicate

IsHolomorphic(f(z), For(z, c)), rendered

f(z) \text{ is holomorphic at } z = c

, represents the predicate that

f(z)

is complex differentiable in some open neighborhood of the point

c

.

IsHolomorphic(f(z), ForElement(z, S)), rendered

f(z) \text{ is holomorphic on } z \in S

, represents the predicate that

f(z)

is complex differentiable in some open neighborhood of every point in the set

S

.

As a special case

f(z) \text{ is holomorphic at } z = {\tilde \infty}

is equivalent to

f\!\left(\frac{1}{z}\right) \text{ is holomorphic at } z = 0

.

As a special case

f(z) \text{ is holomorphic at } z = i \infty

represents the predicate that

f(z)

is a periodic function on the upper half plane that is holomorphic at infinity (in the sense of modular function theory)

Definitions:

Fungrim symbol	Notation	Short description
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("499bdf"),
    SymbolDefinition(IsHolomorphic, IsHolomorphic(f(z), For(z, c)), "Holomorphic predicate"),
    Description(SourceForm(IsHolomorphic(f(z), For(z, c))), ", rendered", IsHolomorphic(f(z), For(z, c)), ", represents the predicate", "that", f(z), "is complex differentiable in some open neighborhood of the point", c, "."),
    Description(SourceForm(IsHolomorphic(f(z), ForElement(z, S))), ", rendered", IsHolomorphic(f(z), ForElement(z, S)), ", represents the predicate", "that", f(z), "is complex differentiable in some open neighborhood of every point in the set", S, "."),
    Description("As a special case", IsHolomorphic(f(z), For(z, UnsignedInfinity)), " is equivalent to", IsHolomorphic(f(Div(1, z)), For(z, 0)), "."),
    Description("As a special case", IsHolomorphic(f(z), For(z, Mul(ConstI, Infinity))), " represents the predicate that", f(z), "is a periodic function on the upper half plane that is holomorphic at infinity (in the sense of modular function theory)"))

0895b1

Symbol: IsMeromorphic —

f(z) \text{ is meromorphic at } z = c

— Meromorphic predicate

IsMeromorphic(f(z), For(z, c)), rendered

f(z) \text{ is meromorphic at } z = c

, represents the predicate that

f(z)

is meromorphic in some open neighborhood of the point

c

.

IsMeromorphic(f(z), ForElement(z, S)), rendered

f(z) \text{ is meromorphic on } z \in S

, represents the predicate that

f(z)

is meromorphic in some open neighborhood of every point in the set

S

.

As a special case

f(z) \text{ is meromorphic at } z = {\tilde \infty}

is equivalent to

f\!\left(\frac{1}{z}\right) \text{ is meromorphic at } z = 0

.

As a special case

f(z) \text{ is meromorphic at } z = i \infty

represents the predicate that

f(z)

is a periodic function on the upper half plane that is meromorphic at infinity (in the sense of modular function theory)

Definitions:

Fungrim symbol	Notation	Short description
`IsMeromorphic`	$f(z) \text{ is meromorphic at } z = c$	Meromorphic predicate
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("0895b1"),
    SymbolDefinition(IsMeromorphic, IsMeromorphic(f(z), For(z, c)), "Meromorphic predicate"),
    Description(SourceForm(IsMeromorphic(f(z), For(z, c))), ", rendered", IsMeromorphic(f(z), For(z, c)), ", represents the predicate", "that", f(z), "is meromorphic in some open neighborhood of the point", c, "."),
    Description(SourceForm(IsMeromorphic(f(z), ForElement(z, S))), ", rendered", IsMeromorphic(f(z), ForElement(z, S)), ", represents the predicate", "that", f(z), "is meromorphic in some open neighborhood of every point in the set", S, "."),
    Description("As a special case", IsMeromorphic(f(z), For(z, UnsignedInfinity)), " is equivalent to", IsMeromorphic(f(Div(1, z)), For(z, 0)), "."),
    Description("As a special case", IsMeromorphic(f(z), For(z, Mul(ConstI, Infinity))), " represents the predicate that", f(z), "is a periodic function on the upper half plane that is meromorphic at infinity (in the sense of modular function theory)"))

231a99

Symbol: ComplexZeroMultiplicity —

\mathop{\operatorname{ord}}\limits_{z=c} f(z)

— Multiplicity (order) of complex zero

ComplexZeroMultiplicity(f(z), For(z, c)), rendered

\mathop{\operatorname{ord}}\limits_{z=c} f(z)

, gives the root multiplicity (order of vanishing) of

f(z)

at the point

z = c

.

If

f

is holomorphic at

c

and

f(c) \ne 0

, the multiplicity is zero.

If

z = c

is a pole of

f(z)

, returns

-n

where

n

is the order of the pole.

In other words, this operator returns the order of the first nonzero term in the Laurent series of

f(z)

at

z = c

.

In the special case where

f(z) = 0

in a neighborhood of

c

, the order is

\infty

.

The result is undefined if

f(z)

is not meromorphic at

c

.

The special expression For(z, c) declares z as a locally bound variable within the scope of the arguments to this operator.

Definitions:

Fungrim symbol	Notation	Short description
`ComplexZeroMultiplicity`	$\mathop{\operatorname{ord}}\limits_{z=c} f(z)$	Multiplicity (order) of complex zero
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("231a99"),
    SymbolDefinition(ComplexZeroMultiplicity, ComplexZeroMultiplicity(f(z), For(z, c)), "Multiplicity (order) of complex zero"),
    Description(SourceForm(ComplexZeroMultiplicity(f(z), For(z, c))), ", rendered", ComplexZeroMultiplicity(f(z), For(z, c)), ", gives the root multiplicity (order of vanishing) of", f(z), "at the point", Equal(z, c), "."),
    Description("If", f, "is holomorphic at", c, "and", NotEqual(f(c), 0), ", the multiplicity is zero."),
    Description("If", Equal(z, c), "is a pole of", f(z), ", returns", Neg(n), "where", n, "is the order of the pole."),
    Description("In other words, this operator returns the order of the first nonzero term in the Laurent series of", f(z), "at", Equal(z, c), "."),
    Description("In the special case where", Equal(f(z), 0), "in a neighborhood of", c, ", the order is", Infinity, "."),
    Description("The result is undefined if", f(z), "is not meromorphic at", c, "."),
    Description("The special expression", SourceForm(For(z, c)), "declares", SourceForm(z), "as a locally bound variable within the scope of the arguments to this operator."))

c54261

Symbol: Residue —

\mathop{\operatorname{res}}\limits_{z=c} f(z)

— Complex residue

Residue(f(z), For(z, c)), rendered

\mathop{\operatorname{res}}\limits_{z=c} f(z)

, gives the complex residue of

f(z)

at the point

z = c

.

Definitions:

Fungrim symbol	Notation	Short description
`Residue`	$\mathop{\operatorname{res}}\limits_{z=c} f(z)$	Complex residue

Source code for this entry:

Entry(ID("c54261"),
    SymbolDefinition(Residue, Residue(f(z), For(z, c)), "Complex residue"),
    Description(SourceForm(Residue(f(z), For(z, c))), ", rendered", Residue(f(z), For(z, c)), ", gives the complex residue of", f(z), "at the point", Equal(z, c), "."))

4d0e14

Symbol: Path —

a \rightsquigarrow b \rightsquigarrow c

— Line path

This object represents the path formed by connecting the given points or paths by line segments. A path is a formal object, semantically different from a set of points: for a path object, the direction is significant, and it is undefined whether a path segment corresponds to an open interval or a closed interval between the points. The typical application is to represent a path of integration.

►Path(1, -1) —

1 \rightsquigarrow -1

— Represents the path going left from 1 to -1.

►Path(a, Path(b, c)) —

a \rightsquigarrow b \rightsquigarrow c

— Equivalent to Path(a, b, c).

►Path(1, ConstI, -1, Neg(ConstI), 1) —

1 \rightsquigarrow i \rightsquigarrow -1 \rightsquigarrow -i \rightsquigarrow 1

— Represents a diamond-shaped loop around the origin in the counterclockwise direction.

►Path(Neg(Mul(ConstI, Infinity)), Mul(ConstI, Infinity)) —

-i \infty \rightsquigarrow i \infty

— Represents the imaginary axis traversed upwards.

►Path(1, Mul(Exp(Div(Mul(Pi, ConstI), 4)), Infinity)) —

1 \rightsquigarrow {e}^{\pi i / 4} \infty

— Represents the ray from 1 to infinity along a 45 degree angle.

►Path(Tuple(2, 1), Tuple(0, 0)) —

\left(2, 1\right) \rightsquigarrow \left(0, 0\right)

— Represents the line segment from (2, 1) to the origin in

{\mathbb{R}}^{2}

.

Definitions:

Fungrim symbol	Notation	Short description
`Path`	$a \rightsquigarrow b \rightsquigarrow c$	Line path
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("4d0e14"),
    SymbolDefinition(Path, Path(a, b, c), "Line path"),
    Description("This object represents the path formed by connecting the given points or paths by line segments. ", "A path is a formal object, semantically different from a set of points: for a path object, the direction is significant, and it is undefined whether a path segment corresponds to an open interval or a closed interval between the points.", "The typical application is to represent a path of integration."),
    CodeExample(Path(1, -1), "Represents the path going left from", 1, "to", -1, "."),
    CodeExample(Path(a, Path(b, c)), "Equivalent to", SourceForm(Path(a, b, c)), "."),
    CodeExample(Path(1, ConstI, -1, Neg(ConstI), 1), "Represents a diamond-shaped loop around the origin in the counterclockwise direction."),
    CodeExample(Path(Neg(Mul(ConstI, Infinity)), Mul(ConstI, Infinity)), "Represents the imaginary axis traversed upwards."),
    CodeExample(Path(1, Mul(Exp(Div(Mul(Pi, ConstI), 4)), Infinity)), "Represents the ray from", 1, "to infinity along a 45 degree angle."),
    CodeExample(Path(Tuple(2, 1), Tuple(0, 0)), "Represents the line segment from", Tuple(2, 1), "to the origin in", Pow(RR, 2), "."))

bf8f37

Symbol: CurvePath —

\left(f(t),\, t : a \rightsquigarrow b\right)

— Path along a curve

►CurvePath(f(t), For(t, a, b)) —

\left(f(t),\, t : a \rightsquigarrow b\right)

— Represents the path traced by

f(t)

as

t

follows the path

a \rightsquigarrow b

.

►CurvePath(Mul(R, Exp(Mul(ConstI, t))), For(t, 0, Mul(2, Pi))) —

\left(R {e}^{i t},\, t : 0 \rightsquigarrow 2 \pi\right)

— Represents the circular path counterclockwise around the origin, starting at

R

.

►CurvePath(Mul(R, Exp(Mul(ConstI, t))), For(t, 0, Neg(Mul(2, Pi)))) —

\left(R {e}^{i t},\, t : 0 \rightsquigarrow -2 \pi\right)

— Represents the circular path clockwise around the origin, starting at

R

.

►Path(Pos(Infinity), CurvePath(Exp(Mul(ConstI, t)), For(t, Div(Pi, 2), Div(Mul(3, Pi), 2))), Pos(Infinity)) —

+\infty \rightsquigarrow \left({e}^{i t},\, t : \pi / 2 \rightsquigarrow 3 \pi / 2\right) \rightsquigarrow +\infty

— Represents the Hankel contour starting at

+\infty

, moving along a straight line above the real axis to

i

, moving in a half-circle around the origin to

-i

, and returning to

+\infty

along a straight line below the real axis.

Definitions:

Fungrim symbol	Notation	Short description
`CurvePath`	$\left(f(t),\, t : a \rightsquigarrow b\right)$	Path along a curve
`Path`	$a \rightsquigarrow b \rightsquigarrow c$	Line path
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("bf8f37"),
    SymbolDefinition(CurvePath, CurvePath(f(t), For(t, a, b)), "Path along a curve"),
    CodeExample(CurvePath(f(t), For(t, a, b)), "Represents the path traced by", f(t), "as", t, "follows the path", Path(a, b), "."),
    CodeExample(CurvePath(Mul(R, Exp(Mul(ConstI, t))), For(t, 0, Mul(2, Pi))), "Represents the circular path counterclockwise around the origin, starting at", R, "."),
    CodeExample(CurvePath(Mul(R, Exp(Mul(ConstI, t))), For(t, 0, Neg(Mul(2, Pi)))), "Represents the circular path clockwise around the origin, starting at", R, "."),
    CodeExample(Path(Pos(Infinity), CurvePath(Exp(Mul(ConstI, t)), For(t, Div(Pi, 2), Div(Mul(3, Pi), 2))), Pos(Infinity)), "Represents the Hankel contour starting at", Pos(Infinity), ", moving along a straight line above the real axis to", i, ", moving in a half-circle around the origin to", Neg(i), ", and returning to", Pos(Infinity), "along a straight line below the real axis."))

457aaa

Symbol: AnalyticContinuation —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)

— Analytic continuation

Represents the value (or limiting value)

g(b)

where

g(z)

is the unique analytic continuation along the path from

a

to

b

for the function initially represented by

f(z)

. It is assumed that the expression

f(z)

represents a holomorphic function of

z

in a neighborhood of the initial point

a

. More generally,

a

is allowed to be a pole, branch point or even an essential singularity as long as

f(z)

is holomorphic locally in a cone around the path radiating from

a

. Infinite endpoints are allowed, with the obvious interpretation. Analytic continuation paths are allowed to pass through (isolated) poles of the analytically continued function. The path is not allowed to pass through intermediate branch points, but may end at a branch point.

►AnalyticContinuation(f(z), For(z, a, b)) —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)

— Represents the analytic continuation of

f(z)

along the straight-line path from

a

to

b

.

►AnalyticContinuation(f(z), For(z, P)) —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: P}} \, f(z)

— Represents the analytic continuation of

f(z)

along the path object

P

.

►AnalyticContinuation(f(z), For(z, Path(a, b, c))) —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b \rightsquigarrow c}} \, f(z)

— Represents the analytic continuation of

f(z)

along the straight-line path

a \rightsquigarrow b \rightsquigarrow c

.

►AnalyticContinuation(f(z), For(z, CurvePath(Exp(Mul(ConstI, t)), For(t, 0, theta)))) —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left({e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, f(z)

— Represents the analytic continuation of

f(z)

along the circular path starting at

z = 1

and rotating counterclockwise by the phase

\theta

.

Definitions:

Fungrim symbol	Notation	Short description
`AnalyticContinuation`	$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$	Analytic continuation
`Path`	$a \rightsquigarrow b \rightsquigarrow c$	Line path
`CurvePath`	$\left(f(t),\, t : a \rightsquigarrow b\right)$	Path along a curve
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("457aaa"),
    SymbolDefinition(AnalyticContinuation, AnalyticContinuation(f(z), For(z, a, b)), "Analytic continuation"),
    Description("Represents the value (or limiting value)", g(b), "where", g(z), "is the unique analytic continuation along the path from", a, "to", b, "for the function initially represented by", f(z), ". ", "It is assumed that the expression", f(z), "represents a holomorphic function of", z, "in a neighborhood of the initial point", a, ". ", "More generally, ", a, "is allowed to be a pole, branch point or even an essential singularity as long as", f(z), "is holomorphic locally in a cone around", "the path radiating from", a, ". ", "Infinite endpoints are allowed, with the obvious interpretation. Analytic continuation paths are allowed to pass through (isolated) poles of the analytically continued function. ", "The path is not allowed to pass through intermediate branch points, but may end at a branch point."),
    CodeExample(AnalyticContinuation(f(z), For(z, a, b)), "Represents the analytic continuation of", f(z), "along the straight-line path from", a, "to", b, "."),
    CodeExample(AnalyticContinuation(f(z), For(z, P)), "Represents the analytic continuation of", f(z), "along the path object", P, "."),
    CodeExample(AnalyticContinuation(f(z), For(z, Path(a, b, c))), "Represents the analytic continuation of", f(z), "along the straight-line path", Path(a, b, c), "."),
    CodeExample(AnalyticContinuation(f(z), For(z, CurvePath(Exp(Mul(ConstI, t)), For(t, 0, theta)))), "Represents the analytic continuation of", f(z), "along the circular path starting at", Equal(z, 1), "and rotating ", "counterclockwise by the phase", theta, "."))

Sums and products

Solutions and zeros

Extreme values

Limits

Derivatives

Integrals

Indefinite integrals

Holomorphic functions

Paths and analytic continuation