StieltjesGamma

The Stieltjes constants $\gamma_n$, the coefficients of the Laurent expansion of $\zeta(s)$ at $s = 1$: $\zeta(s) = \dfrac{1}{s-1} + \sum_{n\ge0} \dfrac{(-1)^n}{n!}\gamma_n (s-1)^n$; with a second argument, the generalized $\gamma_n(a)$ for HurwitzZeta.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaStieltjes constants↗MathWorldStieltjesConstants↗WikidataQ1816316↗DLMF25.2.E5↗FungrimStieltjesGamma✓↗Fungrim entryWolfram LanguageStieltjesGamma!↗

StieltjesGamma(n)the $n$-th Stieltjes constant $\gamma_n$.

SageMathstieltjes(n)SymPystieltjes(n)mpmathstieltjes(n)

StieltjesGamma(n, a)the generalized Stieltjes constant $\gamma_n(a)$, from the expansion of $\zeta(s, a)$ at $s = 1$.

SymPystieltjes(n, a)mpmathstieltjes(n, a)

Typed signatures

Type	Package	Applies
(integer, number?) -> number	analyticreplaces compute-engine	always
(integer, number?) -> number	compute-engine	always

Domain: Special functions

Details

• $\gamma_0 = \gamma = 0.5772\ldots$ is Euler's constant; $\gamma_1 = -0.0728158\ldots$, $\gamma_2 = -0.0096903\ldots$. They have no closed forms beyond $\gamma_0$.
• $\gamma_n = \lim_{m\to\infty}\left(\sum_{k=1}^m \dfrac{\ln^n k}{k} - \dfrac{\ln^{n+1} m}{n+1}\right)$; for $n = 0$ this is the definition of Euler's constant.
• Generalized: $\zeta(s, a) = \dfrac{1}{s-1} + \sum_n \dfrac{(-1)^n}{n!}\gamma_n(a)(s-1)^n$, so $\gamma_n(1) = \gamma_n$ and $\gamma_0(a) = -\psi(a)$, the negated digamma. Shift: $\gamma_n(a+1) = \gamma_n(a) - \ln^n(a)/a$. Poles at $a = 0, -1, -2, \dots$.
• Same normalisation as Wolfram's $\mathrm{StieltjesGamma}[n, a]$ and mpmath's stieltjes(n, a). Complex $a$ supported.
• Numerically, Euler–Maclaurin on $\ln^n(x)/x$. In double precision the partial sum and the subtracted $\ln^{n+1}$ term cancel more and more with $n$: about $10^{-12}$ relative through $n = 15$, $10^{-10}$ at $n = 20$, $10^{-8}$ at $n = 30$ — and orders past 30 are left unevaluated rather than returned wrong.
• Certified digits: for an integer order $n \le 30$ and an exact real $a > 0$, $\mathrm{N}(\gamma_n(a), d)$ is proven. The Euler–Maclaurin remainder is bounded by integrating each coefficient of $|f^{(2M)}|$ on its own: the usual "next term" bound needs $f^{(2M)}$ to keep one sign past the tail point, and for $n \ge 1$ it does not. See N.

Plot

Examples

3 more kept as data, checked against the oracles

Basic #

$\gamma_0 = \gamma$, Euler's constant

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In$\mathrm{StieltjesGamma}(0)$Out$γ$

No closed form past $\gamma_0$ — stays symbolic until N(), which gives $-0.0728158\ldots$

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In$\mathrm{StieltjesGamma}(1)$Out$\mathrm{StieltjesGamma}(1)$

$\gamma_1(\tfrac32)$ numerically — a floating-point argument asks for a number

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In$\mathrm{StieltjesGamma}(1, 1.5)$Out$0.032\,834\,680\,314\,949\,4$

$\gamma_3(\tfrac12) = -0.66742\ldots$

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In$\mathrm{StieltjesGamma}(3, 0.5)$Out$-0.667\,424\,273\,711\,379\,8$

$\gamma_2(\tfrac32)$

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In$\mathrm{StieltjesGamma}(2, 1.5)$Out$0.007\,958\,447\,383\,888\,047$

$\gamma_1 = -0.0728158\ldots$ under N()

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In$N(\mathrm{StieltjesGamma}(1))$Out$-0.072\,815\,845\,483\,676\,52$

The first Stieltjes constant $\gamma_1$ to 50 digits

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In$N(\mathrm{StieltjesGamma}(1), 50)$Out$-0.072\,815\,845\,483\,676\,724\,860\,586\,375\,874\,901\,319\,137\,736\,338\,334\,338$

A generalised Stieltjes constant at $a=\frac12$ to 50 digits

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In$N(\mathrm{StieltjesGamma}(2, \frac{1}{2}), 50)$Out$0.968\,864\,475\,220\,290\,711\,421\,711\,062\,323\,780\,654\,182\,598\,044\,760\,42$

Scope #

Complex $a$

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In$N(\mathrm{StieltjesGamma}(2, 1+i))$Out$0.170\,304\,201\,468\,587\,9+0.373\,177\,195\,757\,495\,25i$

Listable in the order $n$: threads over a list

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In$\mathrm{StieltjesGamma}(\bigl\lbrack1, 2, 3\bigr\rbrack, 0.5)$Out$\bigl\lbrack-1.353\,459\,680\,804\,941\,5, 0.968\,864\,475\,220\,290\,7, -0.667\,424\,273\,711\,380\,7\bigr\rbrack$

Interval arithmetic in $a$

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In$\mathrm{StieltjesGamma}(2, \mathrm{Interval}(2.34, 2.35))$Out$\mathrm{Interval}(-0.067\,241\,280\,355\,087\,15, -0.065\,140\,403\,663\,072\,42)$

To 30 significant digits, past a double's reach

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In$N(\mathrm{StieltjesGamma}(1), 30)$Out$-0.072\,815\,845\,483\,676\,724\,860\,586\,375\,874\,9$

Applications #

$\gamma_1$ to 40 digits, proven: the Euler–Maclaurin remainder is bounded, and both ends of the enclosure round to these digits

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In$N(\mathrm{StieltjesGamma}(1), 40)$Out$-0.072\,815\,845\,483\,676\,724\,860\,586\,375\,874\,901\,319\,137\,74$

$\gamma_2(1/3)$ to 40 digits, proven the same way

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In$N(\mathrm{StieltjesGamma}(2, \frac{1}{3}), 40)$Out$3.619\,163\,909\,618\,963\,629\,887\,182\,216\,185\,941\,703\,294$

Properties #

$\gamma_0(a) = -\psi(a)$, symbolically in $a$

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In$\mathrm{StieltjesGamma}(0, a)$Out$-\mathrm{PolyGamma}(0, a)$

$\gamma_n(1) = \gamma_n$: the two-argument form reduces to it

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In$\mathrm{StieltjesGamma}(1, 1)$Out$\mathrm{StieltjesGamma}(1)$

$\gamma_0(1) = -\psi(1) = \gamma$ -- PolyGamma(0, z) routes through Digamma(z) (#113), which has $\psi(1) = -\gamma$ exactly

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In$\mathrm{StieltjesGamma}(0, 1)$Out$γ$

$\gamma_0(\tfrac12) = -\psi(\tfrac12) = \gamma + 2\ln 2$, now that $\psi(\tfrac12)$ itself has a closed form

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In$\mathrm{StieltjesGamma}(0, \frac{1}{2})$Out$γ+2\ln(2)$

$\gamma_3(1) = \gamma_3$ — order 1 at $a=1$ is just the ordinary Stieltjes constant.

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In$\mathrm{StieltjesGamma}(3, 1)$Out$\mathrm{StieltjesGamma}(3)$

The constants are tiny: $\gamma_{10}$ is positive and below $10^{-3}$

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In$0\lt\mathrm{StieltjesGamma}(10)\lt10^{-3}$Out$\mathrm{True}$

Possible issues #

$\gamma_n(a)$ has poles at the nonpositive integers, like $\zeta(s, a)$

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In$\mathrm{StieltjesGamma}(2, -1)$Out$\tilde∞$

Orders past 30 stay unevaluated even under N(): the double-precision kernel cannot be trusted there

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In$\mathrm{StieltjesGamma}(40)$Out$\mathrm{StieltjesGamma}(40)$

With $a=1$ it is the ordinary constant $\gamma_2$

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In$\mathrm{StieltjesGamma}(2, 1)$Out$\mathrm{StieltjesGamma}(2)$

A symbolic index cannot be evaluated

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In$N(\mathrm{StieltjesGamma}(2n))$Out$\mathrm{StieltjesGamma}(2n)$

The index stays symbolic; only the shift becomes a float

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In$N(\mathrm{StieltjesGamma}(2n, \frac{3}{2}))$Out$\mathrm{StieltjesGamma}(2n, 1.5)$

About

Examples	28
Checked against	Wolfram Language 26 agree1 differSageMath 7 agree1 differ1 unevaluatedSymPy 21 agree4 differ2 unevaluated1 errormpmath 18 agree2 differ2 unevaluated1 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×8Wolfram LanguageSageMathSymPy ×2mpmath ×2

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/stieltjes.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

StieltjesGamma[n, a]; mpmath.stieltjes(n, a).

mappedsympypackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

See also: Zeta, HurwitzZeta, EulerGamma, Digamma