ó ¡¼™\c@ sÖddlmZmZddlmZmZmZmZmZddlm Z m Z m Z m Z ddl mZddlmZmZddlmZddlmZddlmZd efd „ƒYZeZd S( iÿÿÿÿ(tprint_functiontdivision(tStsympifytExprtRationaltDummy(tAddtMultexpand_power_baset expand_log(tcacheit(tdefault_sort_keyt is_sequence(tTuple(t Complement(tuniqtOrdercB s×eZdZeZgZed„ƒZd„Ze d„ƒZ e d„ƒZ e d„ƒZ e d„ƒZ d„Zd„Zd „Zd „Zed „ƒZd „Zd „Zd„Zd„Zd„Zd„ZRS(sä Represents the limiting behavior of some function The order of a function characterizes the function based on the limiting behavior of the function as it goes to some limit. Only taking the limit point to be a number is currently supported. This is expressed in big O notation [1]_. The formal definition for the order of a function `g(x)` about a point `a` is such that `g(x) = O(f(x))` as `x \rightarrow a` if and only if for any `\delta > 0` there exists a `M > 0` such that `|g(x)| \leq M|f(x)|` for `|x-a| < \delta`. This is equivalent to `\lim_{x \rightarrow a} \sup |g(x)/f(x)| < \infty`. Let's illustrate it on the following example by taking the expansion of `\sin(x)` about 0: .. math :: \sin(x) = x - x^3/3! + O(x^5) where in this case `O(x^5) = x^5/5! - x^7/7! + \cdots`. By the definition of `O`, for any `\delta > 0` there is an `M` such that: .. math :: |x^5/5! - x^7/7! + ....| <= M|x^5| \text{ for } |x| < \delta or by the alternate definition: .. math :: \lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| < \infty which surely is true, because .. math :: \lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| = 1/5! As it is usually used, 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. Examples ======== >>> from sympy import O, oo, cos, pi >>> from sympy.abc import x, y >>> O(x + x**2) O(x) >>> O(x + x**2, (x, 0)) O(x) >>> O(x + x**2, (x, oo)) O(x**2, (x, oo)) >>> 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)) >>> 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 >>> O(x)*x O(x**2) >>> O(x) - O(x) O(x) >>> O(cos(x)) O(1) >>> O(cos(x), (x, pi/2)) O(x - pi/2, (x, pi/2)) References ========== .. [1] `Big O notation `_ Notes ===== In ``O(f(x), x)`` the expression ``f(x)`` is assumed to have a leading term. ``O(f(x), x)`` is automatically transformed to ``O(f(x).as_leading_term(x),x)``. ``O(expr*f(x), x)`` is ``O(f(x), x)`` ``O(expr, x)`` is ``O(1)`` ``O(0, x)`` is 0. Multivariate O is also supported: ``O(f(x, y), x, y)`` is transformed to ``O(f(x, y).as_leading_term(x,y).as_leading_term(y), x, y)`` In the multivariate case, it is assumed the limits w.r.t. the various symbols commute. 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