Domain of order terms (Landau symbols)
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x = x0, y = y0, …>)
O(f, x = x0) represents the Landau symbol .
Mathematically, for a function f in the variables (x, y, …), the Landau symbol
is a function in these variables with the following property: there exists a constant c and a neighborhood of the limit point (x0, y0, …) such that |g| ≤ c |f| for all values (x, y, …) in that neighborhood.
Typically, Landau symbols are used to denote the order terms
(“error terms”) of series expansions. Note, however,
that the series expansions produced by
taylor represent order terms as a part
of the data structures
they do not use the domain
With the equations
x = x0,
y = y0 etc.,
regarded as a function of the specified variables. All other identifiers
f are regarded as constant parameters.
If no variables and limit points are specified, then all identifiers
f are used as variables, each tending to the
default limit point 0.
Variables tending to 0 are not printed on the screen.
The variables of an order term may be obtained with the function
indets. The limit points
may be queried with the function
The arithmetical operations
^ are overloaded for order terms.
Automatic simplifications are currently restricted to polynomial
f. Univariate polynomial expressions
are reduced to the leading monomial of the expansion around the limit
point. In multivariate polynomial expressions, all terms are discarded
that are divisible by lower order terms. For non-polynomial expressions,
only integer factors are removed.
For polynomial expressions, certain simplifications occur:
O(x^4 + 2*x^2), O(7*x^3), O(x, x = 1)
A zero limit point is not printed on the screen:
O(1), O(1, x = 1), O(x^2/(y + 1), x = 0, y = -1, z = PI)
The arithmetical operations are overloaded for order terms:
7*O(x), O(x^2) + O(x^13), O(x^3) - O(x^3), O(x^2)^2 + O(x^4)
For multivariate polynomial expression, higher order terms are discarded if they are divisible by lower order terms:
O(15*x*y^2 + 3*x^2*y + x^2*y^2)
O(x + x^2*y) = O(x)*O(1 + x*y)
We demonstrate how to access the variables and the limit points of an order term:
a := O(x^2*y^2)
indets(a) = O::indets(a), O::points(a)
expression representing a function in
The variables: identifiers
The limit points: arithmetical expressions
Element of the domain