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Catalan

Status: Stable

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

Description

Catalan

is Catalan's constant, with numerical value ~= 0.915966.

NumericQ[Catalan]

is True, and D[Catalan, x] is 0. N[Catalan, prec]

Notes Catalan is the sum over k \>= 0 of (-1)^k (2 k + 1)^-2. It is a mathematical constant: it has attributes Constant and Protected, evaluates it to any precision.

Examples (5)

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

Applications (5)

In[1]:= N[Catalan]
Out[1]= 0.915966

In[2]:= N[Catalan, 40]
Out[2]= 0.91596559417721901505460351493238411077416

In[3]:= D[Catalan, x]
Out[3]= 0

In[4]:= NumericQ[Catalan]
Out[4]= True

In[5]:= N[8 Catalan, 30]
Out[5]= 7.32772475341775212043682811946

Implementation notes

Definition. Catalan is Catalan's constant G ≈ 0.9159655942, the alternating sum Sum_{k>=0} (-1)^k (2 k + 1)^-2 (equivalently Beta[2], the Dirichlet beta function at 2). It is a constant symbol: row { "Catalan", 0.9159655941…, fill_mpfr_catalan } of kConstants[] in src/numeric.c supplies its value and the numericalize_init loop stamps it Constant | Protected, so Attributes[Catalan] is {Constant, Protected}. Interned name SYM_Catalan (src/sym_names.c), docstring in src/info.c. It carries no DownValues, and D[Catalan, x] -> 0 follows from Constant.

Representation & numeric value. A bare EXPR_SYMBOL, kept exact under evaluation. N[Catalan] returns the tabled machine double; N[Catalan, k] runs fill_mpfr_catalan → MPFR's mpfr_const_catalan at the requested precision, returning an EXPR_MPFR. Machine value 0.915966; N[Catalan, 50] = 0.915965594177219015054603514932384110774149374281673.

Usage & limits. Catalan arises as the closed form of various definite integrals and sums; NumericQ[Catalan] is True (whitelisted in is_numeric_quantity, src/core.c). Its irrationality is open; Mathilda treats it as an exact symbol whose digits are produced on demand by N.

  • Attributes Constant, Protected. Attributes[Catalan] = {Constant, Protected}; the symbol cannot be reassigned.
  • Propagated as an exact, unevaluated symbol; NumericQ[Catalan] is True and D[Catalan, x] = 0.
  • N[Catalan] gives the machine value 0.915966; N[Catalan, prec] gives any precision (MPFR mpfr_const_catalan), e.g. N[Catalan, 50] = 0.915965594177219015054603514932384110774149374281673.

Attributes: Constant, Protected.

References

Notes & additional examples

Notes

Catalan is Catalan's constant G, the alternating sum over k >= 0 of (-1)^k (2 k + 1)^-2. It is a first-class symbolic constant: it carries the Constant attribute (so D[Catalan, x] is 0), satisfies NumericQ, and N[Catalan, prec] evaluates it to any requested precision via an internal fast-converging series. The 40-digit value matches the standard reference value, and arithmetic combinations such as 8 Catalan are held symbolically until numericalized.