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Normal

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Normal[expr]

converts expr to a normal expression. If expr is a SeriesData object, the O-term is dropped and the truncated polynomial (or Laurent/Puiseux sum) is returned. Normal[SparseArray[...]] gives the dense nested List the sparse specification denotes, Normal[assoc] its list of rules and Normal[ndarray] its nested List. Other expressions pass through unchanged.

Examples (6)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (2)

In[1]:= Normal[{<|"a" -> 1|>, f[<|"b" -> 2|>]}]
Out[1]= {{"a" -> 1}, f[{"b" -> 2}]}

In[2]:= Normal[<|"a" -> <|"b" -> 1|>|>]
Out[2]= {"a" -> <|"b" -> 1|>}

Applications (4)

In[3]:= Normal[Series[Exp[x], {x, 0, 5}]]
Out[3]= 1 + x + 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + 1/120 x^5

In[4]:= Normal[Series[Sin[x]/x, {x, 0, 6}]]
Out[4]= 1 - 1/6 x^2 + 1/120 x^4 - 1/5040 x^6

In[5]:= Normal[Series[Tan[x], {x, 0, 7}]]
Out[5]= x + 1/3 x^3 + 2/15 x^5 + 17/315 x^7

In[6]:= Normal[Series[Log[1 + x], {x, 0, 5}]]
Out[6]= x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 + 1/5 x^5

Algorithm

============================================================================ series.c - Series and SeriesData ============================================================================

This module implements the power-series machinery for Mathilda.

SeriesData[x, x0, {a0, ..., a_{k-1}}, nmin, nmax, den] is the data head that represents a truncated power series. The i-th coefficient multiplies (x - x0)^((nmin + i)/den) and an O[x - x0]^(nmax/den) term captures the dropped higher-order terms.

Series[f, {x, x0, n}]  expands f as a power series in (x - x0) up to

order n. Series also accepts the leading-term form Series[f, x -> x0] and the iterated multivariate form Series[f, {x, x0, nx}, {y, y0, ny}, ...]. The algorithm is a recursive "series algebra": primitive subexpressions become SeriesObj's, algebraic heads (Plus, Times, Power) combine them, and elementary heads (Exp, Log, Sin, Cos, Sinh, Cosh, Tan, Tanh) apply their known series kernels. Unknown heads fall back to naive Taylor via D[...]. Expansion about Infinity is handled by substituting x -> 1/u internally and presenting the result with Power[x, -1] as the series variable.

Normal[s] drops the O-term from a SeriesData and returns an ordinary sum.

Implementation notes

Algorithm. builtin_normal converts a SeriesData[x, x0, {a0,...,a_{k-1}}, nmin, nmax, den] into an ordinary polynomial by dropping the O-term. It builds the base (x - x0) (series_build_xmx0), then for each non-zero coefficient a_i forms the term a_i (x - x0)^((nmin+i)/den) ā€” using an integer exponent when den == 1, otherwise Rational[num, den], and emitting the coefficient bare when the exponent is 0 ā€” and sums the terms (Plus, or the single term / literal 0 for degenerate cases), evaluating the result. Any argument that is not a 6-element SeriesData is passed through unchanged (expr_copy).

Data structures. A direct read of the SeriesData arg slots (coefficient List, nmin, den); no SeriesObj is reconstructed.

Attributes: Protected.

References

See also: Hold, HoldComplete

Notes & additional examples

Notes

Normal[expr] converts expr to a normal expression. Applied to a SeriesData object it drops the O-term and returns the truncated polynomial (or Laurent/Puiseux sum) as an ordinary Plus expression, ready to be added, differentiated, or substituted into. Expressions that are already normal pass through unchanged.