Symbolic expressions

Table of contents: Introduction - Language documentation - Markup for Fungrim entries - Variable-generating expressions - Sequence-generating expressions - Cosmetic markup

Introduction

Formulas in Fungrim are represented using symbolic expressions that encode mathematical objects and operations in a semantic way. The symbolic expression language is called Grim. The LaTeX output rendered on the Fungrim website is produced automatically from the symbolic Grim expressions.

Language documentation

See: The Grim formula language

Markup for Fungrim entries

6db9b2

Symbol: Entry —

\ldots

— Entry

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("6db9b2"),
    SymbolDefinition(Entry, Ellipsis, "Entry"))

0d9952

Symbol: ID —

\ldots

— Entry ID

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("0d9952"),
    SymbolDefinition(ID, Ellipsis, "Entry ID"))

c2fb41

Symbol: Formula —

\ldots

— Formula

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("c2fb41"),
    SymbolDefinition(Formula, Ellipsis, "Formula"))

f37520

Symbol: Variables —

\ldots

— Declaration of variables

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("f37520"),
    SymbolDefinition(Variables, Ellipsis, "Declaration of variables"))

3c8d31

Symbol: Assumptions —

\ldots

— Assumptions (domain declaration) for the variables

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("3c8d31"),
    SymbolDefinition(Assumptions, Ellipsis, "Assumptions (domain declaration) for the variables"))

8af90e

Symbol: References —

\ldots

— References

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("8af90e"),
    SymbolDefinition(References, Ellipsis, "References"))

b976df

Symbol: Description —

\ldots

— Text description

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("b976df"),
    SymbolDefinition(Description, Ellipsis, "Text description"))

Variable-generating expressions

43cc72

Symbol: For —

\ldots

— General-purpose generator

For(x) declares the given symbol as locally bound variable in the scope of the parent call. For example, f(a, For(x), b) declares

x

as a locally bound variable that may be used within the expressions

a

and

b

. The interpretation of the variable is left to the parent operator

f

Called with a tuple of symbols, For(Tuple(x, y, z)), each symbol becomes a locally bound variable.

Called with several arguments, for example For(x, a, b, c), the additional parameters

a

b

c

specify information about the range of

x

. The interpretation of the parameters is up to the parent operator

f

. Most operators recognize For() with two additional parameters as specifying an iteration range: for example, Sum(Factorial(n), For(n, 2, 10)) gives

\sum_{n=2}^{10} n !

. (When For(n, a, b) is used in this sense, the endpoints

a

and

b

must be integers or possibly

a = -\infty

and/or

b = \infty

where an infinite sequence makes sense. The iteration sequence is empty if

b < a

There are various exceptions. For example, Integral understands two parameters as representing the endpoints (not necessarily integers) of a directed line segment to integrate over: Integral(Cos(x), For(x, Neg(Pi), Pi)) becomes

\int_{-\pi}^{\pi} \cos(x) \, dx

. Derivative takes one or two parameters denoting the evaluation point and optionally the order of differentiation: Derivative(Sin(x), For(x, y)) becomes

\sin'(y)

and Derivative(Sin(x), For(x, y, 2)) becomes

\sin''(y)

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Sin`	$\sin(z)$	Sine

Source code for this entry:

Entry(ID("43cc72"),
    SymbolDefinition(For, Ellipsis, "General-purpose generator"),
    Description(SourceForm(For(x)), "declares the given symbol as locally bound variable in the
    scope of the parent call. For example,", SourceForm(f(a, For(x), b)), "declares", x, "as a locally bound variable that may be used within the expressions ", a, "and", b, ". The interpretation of the variable is left to the
    parent operator", f, "."),
    Description("Called with a tuple of symbols,", SourceForm(For(Tuple(x, y, z))), ", each symbol
    becomes a locally bound variable."),
    Description("
    Called with several arguments, for example ", SourceForm(For(x, a, b, c)), ", the additional parameters", a, b, c, "specify information about the range of", x, ".", "The interpretation of the parameters is up to the parent operator", f, ". Most operators recognize ", SourceForm(For()), "with two additional parameters as
    specifying an iteration range: for example, ", SourceForm(Sum(Factorial(n), For(n, 2, 10))), "gives", Sum(Factorial(n), For(n, 2, 10)), ".", "(When", SourceForm(For(n, a, b)), " is used in this sense, the endpoints", a, "and", b, "must be integers or possibly", Equal(a, Neg(Infinity)), "and/or", Equal(b, Infinity), "where an infinite sequence makes sense. ", "The iteration sequence is empty if", Less(b, a), ".)"),
    Description("
    There are various exceptions. For example,", SourceForm(Integral), "understands two
    parameters as representing the endpoints (not necessarily integers)
    of a directed line segment to integrate over: ", SourceForm(Integral(Cos(x), For(x, Neg(Pi), Pi))), "becomes", Integral(Cos(x), For(x, Neg(Pi), Pi)), ". ", SourceForm(Derivative), "takes one or two parameters", "denoting the evaluation point and optionally the order of differentiation:", SourceForm(Derivative(Sin(x), For(x, y))), "becomes", Derivative(Sin(x), For(x, y)), "and", SourceForm(Derivative(Sin(x), For(x, y, 2))), "becomes", Derivative(Sin(x), For(x, y, 2)), "."))

978576

Symbol: ForElement —

\ldots

— Generator for all the elements of a set

ForElement(x, S) declares the variable x just like For(x) and additionally tells the parent operator that

x

is to range over the elements of the set

S

. Examples: Set(Mul(2, n), ForElement(n, ZZ)) becomes

\left\{ 2 n : n \in \mathbb{Z} \right\}

. Sum(Div(1, Pow(n, 2)), ForElement(n, SetMinus(ZZ, Set(0)))) becomes

\sum_{n \in \mathbb{Z} \setminus \left\{0\right\}} \frac{1}{{n}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("978576"),
    SymbolDefinition(ForElement, Ellipsis, "Generator for all the elements of a set"),
    Description(SourceForm(ForElement(x, S)), "declares the variable", SourceForm(x), "just like", SourceForm(For(x)), "and additionally tells the parent operator", "that", x, "is to range over the elements of the set", S, ". ", "Examples: ", SourceForm(Set(Mul(2, n), ForElement(n, ZZ))), "becomes", Set(Mul(2, n), ForElement(n, ZZ)), ". ", SourceForm(Sum(Div(1, Pow(n, 2)), ForElement(n, SetMinus(ZZ, Set(0))))), "becomes", Sum(Div(1, Pow(n, 2)), ForElement(n, SetMinus(ZZ, Set(0)))), ". "))

Sequence-generating expressions

82c978

Symbol: Repeat —

\underbrace{x, \ldots, x}_{n \text{ times}}

— Repeating sequence

Represents the first arguments repeated the number of times specified by the last argument. This expression does not represent a mathematical object: it only exists at the expression level, and injects the sequence between surrounding arguments. To construct a mathematical object, we must pass the generator expression to a function such as Tuple. Example: Formula(Tuple(Repeat(1, N), 0, Repeat(1, 2, 3, M), 1, 2)) renders as

\left(\underbrace{1, \ldots, 1}_{N \text{ times}}, 0, \underbrace{1, 2, 3, \ldots, 1, 2, 3}_{\left(1, 2, 3\right) \; M \text{ times}}, 1, 2\right)

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("82c978"),
    SymbolDefinition(Repeat, Repeat(x, n), "Repeating sequence"),
    Description("Represents the first arguments repeated the number of times ", "specified by the last argument. This expression does not represent ", "a mathematical object: it only exists at the expression level, and ", "injects the sequence between surrounding arguments. ", "To construct a mathematical object, we must pass the generator expression ", "to a function such as", SourceForm(Tuple), ". Example: ", SourceForm(Formula(Tuple(Repeat(1, N), 0, Repeat(1, 2, 3, M), 1, 2))), "renders as ", Tuple(Repeat(1, N), 0, Repeat(1, 2, 3, M), 1, 2), "."))

73f5e7

Symbol: Step —

f(a), f\!\left(a + 1\right), \ldots, f(b)

— Enumerated sequence

Step(f(n), For(n, a, b)) represents the sequence of values

f(n)

for

n

between the integers

a

and

b

. The sequence is empty if

b < a

. This expression does not represent a mathematical object: it only exists at the expression level, and injects the sequence between a surrounding arguments. To construct a mathematical object, we must pass the generator expression to a function such as Tuple. Examples:

f(Step(Pow(k, 2), For(k, 1, 10)))

f\!\left({1}^{2}, {2}^{2}, \ldots, {10}^{2}\right)

Tuple(Step(Repeat(n, n), For(n, 0, N)))

\left(\underbrace{0, \ldots, 0}_{0 \text{ times}}, \underbrace{1, \ldots, 1}_{1 \text{ times}}, \ldots, \underbrace{N, \ldots, N}_{N \text{ times}}\right)

Tuple(1, 2, 2, Step(Repeat(n, n), For(n, 3, N)))

\left(1, 2, 2, \underbrace{3, \ldots, 3}_{3 \text{ times}}, \underbrace{4, \ldots, 4}_{4 \text{ times}}, \ldots, \underbrace{N, \ldots, N}_{N \text{ times}}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("73f5e7"),
    SymbolDefinition(Step, Step(f(n), For(n, a, b)), "Enumerated sequence"),
    Description(SourceForm(Step(f(n), For(n, a, b))), " represents the sequence of values ", f(n), "for ", n, "between ", "the integers", a, "and", b, ". ", "The sequence is empty if ", Less(b, a), ". ", "This expression does not represent ", "a mathematical object: it only exists at the expression level, and ", "injects the sequence between a surrounding arguments. ", "To construct a mathematical object, we must pass the generator expression ", "to a function such as", SourceForm(Tuple), ". Examples: "),
    Description(SourceForm(f(Step(Pow(k, 2), For(k, 1, 10))))),
    Description(f(Step(Pow(k, 2), For(k, 1, 10)))),
    Description(SourceForm(Tuple(Step(Repeat(n, n), For(n, 0, N))))),
    Description(Tuple(Step(Repeat(n, n), For(n, 0, N)))),
    Description(SourceForm(Tuple(1, 2, 2, Step(Repeat(n, n), For(n, 3, N))))),
    Description(Tuple(1, 2, 2, Step(Repeat(n, n), For(n, 3, N)))))

Cosmetic markup

ecfe08

Symbol: Parentheses —

\left(\ldots\right)

— Parentheses

Hints that the enclosed expression should be rendered surrounded by parentheses. Semantically represents the identity function: Parentheses(x) is equivalent to x.

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("ecfe08"),
    SymbolDefinition(Parentheses, Parentheses(Ellipsis), "Parentheses"),
    Description("Hints that the enclosed expression should be rendered surrounded by parentheses. ", "Semantically represents the identity function: ", SourceForm(Parentheses(x)), " is equivalent to ", SourceForm(x), "."))

d14265

Symbol: Brackets —

\left[\ldots\right]

— Square brackets

Hints that the enclosed expression should be rendered surrounded by square brackets. Semantically represents the identity function: Brackets(x) is equivalent to x.

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("d14265"),
    SymbolDefinition(Brackets, Brackets(Ellipsis), "Square brackets"),
    Description("Hints that the enclosed expression should be rendered surrounded by square brackets. ", "Semantically represents the identity function: ", SourceForm(Brackets(x)), " is equivalent to ", SourceForm(x), "."))

ca1edc

Symbol: Braces —

\left\{\ldots\right\}

— Curly braces

Hints that the enclosed expression should be rendered surrounded by curly braces. Semantically represents the identity function: Braces(x) is equivalent to x.

Definitions:

Fungrim symbol	Notation	Short description

Source code for this entry:

Entry(ID("ca1edc"),
    SymbolDefinition(Braces, Braces(Ellipsis), "Curly braces"),
    Description("Hints that the enclosed expression should be rendered surrounded by curly braces. ", "Semantically represents the identity function: ", SourceForm(Braces(x)), " is equivalent to ", SourceForm(x), "."))