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Function expansions and series
In order to clarify the discussions about Taylor series and asymptotic expansions, it is necessary to define the relevant concepts as precisely as possible.
Sympy makes a sharp distinction between expressions and functions. In the following, whenever something is defined for a function (f), there's an equivalent definition applying to the expression (f(x)).
In this section we attract attention to the structure of finite or infinite series.
A formal power series is an object of the form (S = \sum_{n=0}^{+\infty} a_n X^n) where (X) is a formal parameter. The set of formal power series over a field (K) is noted (KX) and has a ring structure.
A classical Laurent series is an object of the form (\sum_{n=+\infty}^{+\infty} a_n X^n).
A generalized formal power series, or formal Laurent series, in an object (S) such that (X^n S) is a formal power series for some integer (n). The set of formal Laurent series over a field (K) is noted (K((X))) and has a field structure.
The Taylor series at (x_0) of a smooth function (f) is the formal power series (TS_{x_0}(f) = \sum_{n = 0}^\infty \frac{f^{(n)}(x_0)}{n!} X^n). where (X = x-x_0) (f) is analytic iff there exists a neighbourhood (V) of (x_0) where (f(x) = S(x-x_0), x \in V, S = TS_{x_0}(f)).
The n-th order Taylor polynomial of f at (x_0) is the degree-n polynomial (TP_{n, x_0}(f) = \sum_{n = 0}^n \frac{f^{(k)}(x_0)}{k!} X^k). If f is analytic, (TP_{n, x_0}(f)) is the degree-n truncation of (TS_{x_0}(f)).
In this section we attract attention to the asymptotic expansion and related things.
f is dominated by g at (x_0), written (f = O_{x_0}(g)), iff (f/g) is bounded at (x_0). Similarly, (f(x) = O_{x \rightarrow x_0}(g(x))) iff (f = O_{x_0}(g)), i.e. iff (f(x)/g(x)) is bounded for (x) going to (x_0). Note that (\cdot = O_\cdot(\cdot)) is a purely notational device here, not an equality.
The set of functions dominated by a function (g) at (x_0), (O_{x_0}(g)) is defined by (f \in O_{x_0}(g)) iff (f = O_{x_0}(g)). (h + O(g)) is a set of functions defined in the usual way: (h + O(g) = {h+\alpha ; \alpha \in O(g)}). It is called an asymptotic expansion of f iff (f \in h + O(g)).
The n-th order Taylor expansion of f at (x_0) is (TP_{n, x_0}(f)(x-x_0) + O_{x \rightarrow x_0}((x-x_0)^{n+1}))
This section is not really related with expansion but related with series.
The ordinary generating function of a sequence (a_n) is ( G(a_n;x) = \sum_{n=0}^{+\infty} a_n x^n).
The exponential generating function of a sequence (a_n) is ( EG(a_n;x) = \sum_{n=0}^{+\infty} a_n \frac{x^n}{n!}).
According to its docstring, f(x).series(x, x0, n) is supposed to return the (n-1)th order generalized Taylor expansion of (f(x)) for (x \rightarrow x_0).
Actually, it works only for (x_0 = 0) and when f doesn't have a generalized Taylor expansion, it returns some arbitrarily chosen asymptotic expansion of (f(x)).
Nevertheless the big work with processing of many various cases was executed recently, also many tests have been collected and passed. This is used for limits processing.
Also there are an object Sum is defined, which represent unevaluated summation ( \sum_{k=a}^b a(n) ).