Series

The series module implements series expansions as a function and many related functions.

class diofant.series.limits.Limit[source]

Represents a directional limit of expr at the point z0.

Parameters:
expr : Expr

algebraic expression

z : Symbol

variable of the expr

z0 : Expr

limit point, \(z_0\)

dir : {“+”, “-“, “real”}, optional

For dir="+" (default) it calculates the limit from the right (\(z\to z_0 + 0\)) and for dir="-" the limit from the left (\(z\to z_0 - 0\)). If dir="real", the limit is the bidirectional real limit. For infinite z0 (oo or -oo), the dir argument is determined from the direction of the infinity (i.e., dir="-" for oo).

Examples

>>> Limit(sin(x)/x, x, 0)
Limit(sin(x)/x, x, 0)
>>> Limit(1/x, x, 0, dir="-")
Limit(1/x, x, 0, dir='-')
doit(**hints)[source]

Evaluates limit.

Notes

First we handle some trivial cases (i.e. constant), then try Gruntz algorithm (see the gruntz module).

free_symbols

Return from the atoms of self those which are free symbols.

For most expressions, all symbols are free symbols. For some classes this is not true. e.g. Integrals use Symbols for the dummy variables which are bound variables, so Integral has a method to return all symbols except those. Derivative keeps track of symbols with respect to which it will perform a derivative; those are bound variables, too, so it has its own free_symbols method.

Any other method that uses bound variables should implement a free_symbols method.

diofant.series.limits.limit(expr, z, z0, dir='+')[source]

Compute the directional limit of expr at the point z0.

See also

Limit

Examples

>>> limit(sin(x)/x, x, 0)
1
>>> limit(1/x, x, 0, dir="+")
oo
>>> limit(1/x, x, 0, dir="-")
-oo
>>> limit(1/x, x, oo)
0
diofant.series.series.series(expr, x=None, x0=0, n=6, dir='+')[source]

Series expansion of expr in x around point x0.

diofant.series.order.O

alias of diofant.series.order.Order

class diofant.series.order.Order[source]

Represents the limiting behavior of function.

The formal definition [1] for order symbol \(O(f(x))\) (Big O) is that \(g(x) \in O(f(x))\) as \(x\to a\) iff

\[\lim\limits_{x \rightarrow a} \sup \left|\frac{g(x)}{f(x)}\right| < \infty\]
Parameters:
expr : Expr

an expression

args : sequence of Symbol’s or pairs (Symbol, Expr), optional

If only symbols are provided, i.e. no limit point are passed, then the limit point is assumed to be zero. If no symbols are passed then all symbols in the expression are used.

References

[1](1, 2) https//en.wikipedia.org/wiki/Big_O_notation

Examples

The order of a function can be intuitively thought of representing all terms of powers greater than the one specified. For example, \(O(x^3)\) corresponds to any terms proportional to \(x^3, x^4,\ldots\) and any higher power. For a polynomial, this leaves terms proportional to \(x^2\), \(x\) and constants.

>>> 1 + x + x**2 + x**3 + x**4 + O(x**3)
1 + x + x**2 + O(x**3)

O(f(x)) is automatically transformed to O(f(x).as_leading_term(x)):

>>> O(x + x**2)
O(x)
>>> O(cos(x))
O(1)

Some arithmetic operations:

>>> O(x)*x
O(x**2)
>>> O(x) - O(x)
O(x)

The Big O symbol is a set, so we support membership test:

>>> x in O(x)
True
>>> O(1) in O(1, x)
True
>>> O(1, x) in O(1)
False
>>> O(x) in O(1, x)
True
>>> O(x**2) in O(x)
True

Limit points other then zero and multivariate Big O are also supported:

>>> O(x) == O(x, (x, 0))
True
>>> O(x + x**2, (x, oo))
O(x**2, (x, oo))
>>> O(cos(x), (x, pi/2))
O(x - pi/2, (x, pi/2))
>>> O(1 + x*y)
O(1, x, y)
>>> O(1 + x*y, (x, 0), (y, 0))
O(1, x, y)
>>> O(1 + x*y, (x, oo), (y, oo))
O(x*y, (x, oo), (y, oo))
contains(expr)[source]

Membership test.

Returns:
Boolean or None

Return True if expr belongs to self. Return False if self belongs to expr. Return None if the inclusion relation cannot be determined.

free_symbols

Return from the atoms of self those which are free symbols.

For most expressions, all symbols are free symbols. For some classes this is not true. e.g. Integrals use Symbols for the dummy variables which are bound variables, so Integral has a method to return all symbols except those. Derivative keeps track of symbols with respect to which it will perform a derivative; those are bound variables, too, so it has its own free_symbols method.

Any other method that uses bound variables should implement a free_symbols method.

getO()[source]

Returns the additive O(..) symbol if there is one, else None.

removeO()[source]

Removes the additive O(..) symbol if there is one

diofant.series.residues.residue(expr, x, x0)[source]

Finds the residue of expr at the point x=x0.

The residue is defined [1] as the coefficient of \(1/(x - x_0)\) in the power series expansion around \(x=x_0\).

This notion is essential for the Residue Theorem [2]

References

[1](1, 2) https//en.wikipedia.org/wiki/Residue_%28complex_analysis%29
[2](1, 2) https//en.wikipedia.org/wiki/Residue_theorem

Examples

>>> residue(1/x, x, 0)
1
>>> residue(1/x**2, x, 0)
0
>>> residue(2/sin(x), x, 0)
2