Series Expansions

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

Limits

The main purpose of this module is the computation of limits.

sympy.series.limits.limit(e, z, z0, dir='+')

Compute the limit of e(z) at the point z0.

z0 can be any expression, including oo and -oo.

For dir=”+” (default) it calculates the limit from the right (z->z0+) and for dir=”-” the limit from the left (z->z0-). For infinite z0 (oo or -oo), the dir argument doesn’t matter.

Examples:

>>> from sympy import limit, sin, Symbol, oo
>>> from sympy.abc import x
>>> 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

Strategy:

First we try some heuristics for easy and frequent cases like “x”, “1/x”, “x**2” and similar, so that it’s fast. For all other cases, we use the Gruntz algorithm (see the gruntz() function).

As is explained above, the workhorse for limit computations is the function gruntz() which implements Gruntz’ algorithm for computing limits.

The Gruntz Algorithm

This section explains the basics of the algorithm used for computing limits. Most of the time the limit() function should just work. However it is still useful to keep in mind how it is implemented in case something does not work as expected.

First we define an ordering on functions. Suppose f(x) and g(x) are two real-valued functions such that \lim_{x \to \infty} f(x) = \infty and similarly \lim_{x \to \infty} g(x) = \infty. We shall say that f(x) dominates g(x), written f(x) \succ g(x), if for all a, b \in \mathbb{R}_{>0} we have \lim_{x \to \infty} \frac{f(x)^a}{g(x)^b} = \infty. We also say that f(x) and g(x) are of the same comparability class if neither f(x) \succ g(x) nor g(x) \succ f(x) and shall denote it as f(x) \asymp g(x).

Note that whenever a, b \in \mathbb{R}_{>0} then a f(x)^b \asymp f(x), and we shall use this to extend the definition of \succ to all functions which tend to 0 or \pm \infty as x \to \infty. Thus we declare that f(x) \asymp 1/f(x) and f(x) \asymp -f(x).

It is easy to show the following examples:

  • e^x \succ x^m
  • e^{x^2} \succ e^{mx}
  • e^{e^x} \succ e^{x^m}
  • x^m \asymp x^n
  • e^{x + \frac{1}{x}} \asymp e^{x + \log{x}} \asymp e^x.

From the above definition, it is possible to prove the following property:

Suppose \omega, g_1, g_2, \dots are functions of x, \lim_{x \to \infty} \omega = 0 and \omega \succ g_i for all i. Let c_1, c_2, \dots \in \mathbb{R} with c_1 < c_2 < \dots.

Then \lim_{x \to \infty} \sum_i g_i \omega^{c_i} = \lim_{x \to \infty} g_1 \omega^{c_1}.

For g_1 = g and \omega as above we also have the following easy result:

  • \lim_{x \to \infty} g \omega^c = 0 for c > 0
  • \lim_{x \to \infty} g \omega^c = \pm \infty for c < 0, where the sign is determined by the (eventual) sign of g
  • \lim_{x \to \infty} g \omega^0 = \lim_{x \to \infty} g.

Using these results yields the following strategy for computing \lim_{x \to \infty} f(x):

  1. Find the set of most rapidly varying subexpressions (MRV set) of f(x). That is, from the set of all subexpressions of f(x), find the elements that are maximal under the relation \succ.
  2. Choose a function \omega that is in the same comparability class as the elements in the MRV set, such that \lim_{x \to \infty} \omega = 0.
  3. Expand f(x) as a series in \omega in such a way that the antecedents of the above theorem are satisfied.
  4. Apply the theorem and conclude the computation of \lim_{x \to \infty} f(x), possibly by recursively working on g_1(x).

Notes

This exposition glossed over several details. Many are described in the file gruntz.py, and all can be found in Gruntz’ very readable thesis. The most important points that have not been explained are:

  1. Given f(x) and g(x), how do we determine if f(x) \succ g(x), g(x) \succ f(x) or g(x) \asymp f(x)?
  2. How do we find the MRV set of an expression?
  3. How do we compute series expansions?
  4. Why does the algorithm terminate?

If you are interested, be sure to take a look at Gruntz Thesis.

More Intuitive Series Expansion

This is achieved by creating a wrapper around Basic.series(). This allows for the use of series(x*cos(x),x), which is possibly more intuative than (x*cos(x)).series(x).

Examples

>>> from sympy import Symbol, cos, series
>>> x = Symbol('x')
>>> series(cos(x),x)
1 - x**2/2 + x**4/24 + O(x**6)

Order Terms

This module also implements automatic keeping track of the order of your expansion.

Examples

>>> from sympy import Symbol, Order
>>> x = Symbol('x')
>>> Order(x) + x**2
O(x)
>>> Order(x) + 1
1 + O(x)

Series Acceleration

TODO

Residues

TODO

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