RootSum¶

Status: Stable

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

Description¶

RootSum[Function[t, p[t]], Function[t, body[t]]]
    The formal sum of body[\[Alpha]] over the roots \[Alpha] of
    p[\[Alpha]] == 0.  Held symbolic form, used by the rational
    integrator's NaiveLogPart fallback when the logarithmic part
    cannot be expressed in closed-form real elementary functions.
    Differentiation threads through the body Function:
      D[RootSum[f1, Function[t, body]], x]
        == RootSum[f1, Function[t, D[body, x]]].

Examples¶

No verified examples yet for this function.

Implementation notes¶

Algorithm. RootSum[Function[d], Function[body]] denotes the sum of body over the roots of the squarefree polynomial d. It is a held symbolic head (ATTR_HOLDALL); builtin_rootsum does not expand the sum but tries one closed-form collapse via rootsum_try_lagrange. The recognised identity is the Hermite/Lagrange interpolation collapse

  Σ_i  a(α_i) / (d'(α_i) (x − α_i))  ==  a(x) / d(x)

valid for squarefree d with roots α_i and deg(a) < deg(d). This is exactly the form produced by differentiating the log part of a rational integral (D[RootSum[Function[d], Log[x−#]/d'(#)&], x]).

The collapse works in Slot[1] form: it scans the body for a literal Plus[x, Times[-1, Slot[1]]] factor to identify the external variable x (find_x_minus_slot1), computes d' as D[d, Slot[1]], reconstructs a(#) = body · (x−#) · d'(#) and simplifies it with Cancel[Together[...]] (simplify_rational). If the simplified a still contains x the body did not fit the shape and it bails (NULL). Otherwise it substitutes Slot[1] -> x in both a and d (subst_slot1) and returns Together[a(x)/d(x)].

Construction. root_make_rootsum (called from the rational-integration log part, src/intrat.c) canonicalises the bound variable into the 1-arg Function[...Slot[1]...] form via substitute_bvar_with_slot, avoiding leaked context-qualified symbols. D over RootSum threads through the body in src/deriv.c.

Limits. Only the single Lagrange shape is recognised; any other RootSum (including numeric expansion over actual roots) is left unevaluated.

Attributes: HoldAll, Protected.

Implementation status¶

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

References¶

Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd ed. (Springer, 2005).
Source: src/root.c
Specification: docs/spec/builtins/solutions-of-equations.md

Notes & additional examples¶

Worked examples¶

In[1]:= RootSum[Function[t, t^2 + t + 1], Function[t, t^3]]
Out[1]= RootSum[Function[t, t^2 + t + 1], Function[t, t^3]]

In[2]:= RootSum[#^2 + 1 &, # &]
Out[2]= RootSum[#1^2 + 1 &, #1 &]

In[1]:= Integrate[1/(x^3 - x + 1), x]
Out[1]= RootSum[1 + #1^3 - #1 &, Log[-#1 + x]/(-1 + 3 #1^2) &]

In[1]:= D[RootSum[Function[t, t^3 + t + 1], Function[t, Log[x - t]/t]], x]
Out[1]= RootSum[Function[t, t^3 + t + 1], Function[t, 1/(t (-t + x))]]

Notes¶

RootSum[f, form] denotes the formal sum of form[a] over the roots a of f[a] == 0 and is kept as a held symbolic object. It is produced by the rational integrator's NaiveLogPart fallback when the logarithmic part of an integral cannot be written with real elementary functions; differentiation threads through the form function.