Stieltjes constants

Symbol: StieltjesGamma —

\gamma_{n}\!\left(a\right)

— Stieltjes constant

StieltjesGamma(n), rendered as

\gamma_{n}

, represents the Stieltjes constant of index

n

.

StieltjesGamma(n, a), rendered as

\gamma_{n}\!\left(a\right)

, represents the generalized Stieltjes constant of index

n

with parameter

a

.

Domain	Codomain
$n \in \mathbb{Z}_{\ge 0}$	$\gamma_{n} \in \mathbb{R}$
$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}$	$\gamma_{n}\!\left(a\right) \in \mathbb{C}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \;\implies\; \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("d10029"),
    SymbolDefinition(StieltjesGamma, StieltjesGamma(n, a), "Stieltjes constant"),
    Description(SourceForm(StieltjesGamma(n)), ", rendered as", StieltjesGamma(n), ", represents the Stieltjes constant of index", n, "."),
    Description(SourceForm(StieltjesGamma(n, a)), ", rendered as", StieltjesGamma(n, a), ", represents the generalized Stieltjes constant of index", n, " with parameter", a, "."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(Element(n, ZZGreaterEqual(0)), Element(StieltjesGamma(n), RR)), Tuple(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), NotElement(a, ZZLessEqual(0))), Element(StieltjesGamma(n, a), CC)))))

b1a2e1

\zeta\!\left(s\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n} {\left(s - 1\right)}^{n}

Assumptions:

s \in \mathbb{C}

TeX:

\zeta\!\left(s\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n} {\left(s - 1\right)}^{n}

s \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b1a2e1"),
    Formula(Equal(RiemannZeta(s), Add(Div(1, Sub(s, 1)), Sum(Mul(Mul(Div(Pow(-1, n), Factorial(n)), StieltjesGamma(n)), Pow(Sub(s, 1), n)), For(n, 0, Infinity))))),
    Variables(s),
    Assumptions(Element(s, CC)))

60c6da

\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

TeX:

\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("60c6da"),
    Formula(Equal(HurwitzZeta(s, a), Add(Div(1, Sub(s, 1)), Sum(Mul(Mul(Div(Pow(-1, n), Factorial(n)), StieltjesGamma(n, a)), Pow(Sub(s, 1), n)), For(n, 0, Infinity))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

411f3b

\gamma_{n}\!\left(a\right) = \lim_{N \to \infty} \left[\left(\sum_{k=0}^{N} \frac{\log^{n}\!\left(k + a\right)}{k + a}\right) - \frac{\log^{n + 1}\!\left(N + a\right)}{n + 1}\right]

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

TeX:

\gamma_{n}\!\left(a\right) = \lim_{N \to \infty} \left[\left(\sum_{k=0}^{N} \frac{\log^{n}\!\left(k + a\right)}{k + a}\right) - \frac{\log^{n + 1}\!\left(N + a\right)}{n + 1}\right]

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("411f3b"),
    Formula(Equal(StieltjesGamma(n, a), SequenceLimit(Brackets(Sub(Parentheses(Sum(Div(Pow(Log(Add(k, a)), n), Add(k, a)), For(k, 0, N))), Div(Pow(Log(Add(N, a)), Add(n, 1)), Add(n, 1)))), For(N, Infinity)))),
    Variables(n, a),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

51206a

\gamma_{n}\!\left(1\right) = \gamma_{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\gamma_{n}\!\left(1\right) = \gamma_{n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("51206a"),
    Formula(Equal(StieltjesGamma(n, 1), StieltjesGamma(n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

8ae153

\gamma_{0}\!\left(1\right) = \gamma_{0} = \gamma

TeX:

\gamma_{0}\!\left(1\right) = \gamma_{0} = \gamma

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`ConstGamma`	$\gamma$	The constant gamma (0.577...)

Source code for this entry:

Entry(ID("8ae153"),
    Formula(Equal(StieltjesGamma(0, 1), StieltjesGamma(0), ConstGamma)))

b6808d

\gamma_{0}\!\left(a\right) = -\psi\!\left(a\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

TeX:

\gamma_{0}\!\left(a\right) = -\psi\!\left(a\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("b6808d"),
    Formula(Equal(StieltjesGamma(0, a), Neg(DigammaFunction(a)))),
    Variables(a),
    Assumptions(And(Element(a, CC), NotElement(a, ZZLessEqual(0)))))

70a705

\gamma_{1}\!\left(\frac{1}{2}\right) = \gamma_{1} - 2 \gamma \log(2) - \log^{2}\!\left(2\right)

TeX:

\gamma_{1}\!\left(\frac{1}{2}\right) = \gamma_{1} - 2 \gamma \log(2) - \log^{2}\!\left(2\right)

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Log`	$\log(z)$	Natural logarithm
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("70a705"),
    Formula(Equal(StieltjesGamma(1, Div(1, 2)), Sub(Sub(StieltjesGamma(1), Mul(Mul(2, ConstGamma), Log(2))), Pow(Log(2), 2)))))

e5bd3c

Table of

\gamma_{n}

to 50 digits for

0 \le n \le 30

$n$	$\gamma_{n} \; (\text{nearest } 50 \text{D})$
0	0.57721566490153286060651209008240243104215933593992
1	-0.072815845483676724860586375874901319137736338334338
2	-0.0096903631928723184845303860352125293590658061013407
3	0.0020538344203033458661600465427533842857158044454106
4	0.0023253700654673000574681701775260680009044694137849
5	0.00079332381730106270175333487744444483073153940458489
6	-0.00023876934543019960987242184190800427778371515635808
7	-0.00052728956705775104607409750547885828199625347296990
8	-0.00035212335380303950960205216500120874172918053379235
9	-3.4394774418088048177914623798227390620789538594442 · 10^-5
10	0.00020533281490906479468372228923706530295985377416676
11	0.00027018443954390352667290208206795567382784205868840
12	0.00016727291210514019335350154334118344660780663280557
13	-2.7463806603760158860007603693355181526785337670396 · 10^-5
14	-0.00020920926205929994583713969734458495783154421150607
15	-0.00028346865532024144664293447499712697706870298071768
16	-0.00019969685830896977470778456320324039191576497403406
17	2.6277037109918336699466597630510122816078692929114 · 10^-5
18	0.00030736840814925282659275475194862564552381129073146
19	0.00050360545304735562905559643771716003532126980764950
20	0.00046634356151155944940059482443355052511314347392569
21	0.00010443776975600011581079567436772049104442825070555
22	-0.00054159958220399770165519617317410558454386092870075
23	-0.0012439620904082457792997415995371658091470281139646
24	-0.0015885112789035615619061966115211158573187228221441
25	-0.0010745919527384888247242919873531730892739793314532
26	0.00065680351863715443150477300335621524888606506047754
27	0.0034778369136185382090073595742588115476629156638859
28	0.0064000685317006294581072282219458636666371981445885
29	0.0073711517704722391344124024235594021578413274885128
30	0.0035577288555731609479135377489084026108096506495221

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant

Source code for this entry:

Entry(ID("e5bd3c"),
    Description("Table of", StieltjesGamma(n), "to 50 digits for", LessEqual(0, n, 30)),
    Table(Var(n), TableValueHeadings(n, NearestDecimal(StieltjesGamma(n), 50)), TableSplit(1), List(Tuple(0, Decimal("0.57721566490153286060651209008240243104215933593992")), Tuple(1, Decimal("-0.072815845483676724860586375874901319137736338334338")), Tuple(2, Decimal("-0.0096903631928723184845303860352125293590658061013407")), Tuple(3, Decimal("0.0020538344203033458661600465427533842857158044454106")), Tuple(4, Decimal("0.0023253700654673000574681701775260680009044694137849")), Tuple(5, Decimal("0.00079332381730106270175333487744444483073153940458489")), Tuple(6, Decimal("-0.00023876934543019960987242184190800427778371515635808")), Tuple(7, Decimal("-0.00052728956705775104607409750547885828199625347296990")), Tuple(8, Decimal("-0.00035212335380303950960205216500120874172918053379235")), Tuple(9, Decimal("-3.4394774418088048177914623798227390620789538594442e-5")), Tuple(10, Decimal("0.00020533281490906479468372228923706530295985377416676")), Tuple(11, Decimal("0.00027018443954390352667290208206795567382784205868840")), Tuple(12, Decimal("0.00016727291210514019335350154334118344660780663280557")), Tuple(13, Decimal("-2.7463806603760158860007603693355181526785337670396e-5")), Tuple(14, Decimal("-0.00020920926205929994583713969734458495783154421150607")), Tuple(15, Decimal("-0.00028346865532024144664293447499712697706870298071768")), Tuple(16, Decimal("-0.00019969685830896977470778456320324039191576497403406")), Tuple(17, Decimal("2.6277037109918336699466597630510122816078692929114e-5")), Tuple(18, Decimal("0.00030736840814925282659275475194862564552381129073146")), Tuple(19, Decimal("0.00050360545304735562905559643771716003532126980764950")), Tuple(20, Decimal("0.00046634356151155944940059482443355052511314347392569")), Tuple(21, Decimal("0.00010443776975600011581079567436772049104442825070555")), Tuple(22, Decimal("-0.00054159958220399770165519617317410558454386092870075")), Tuple(23, Decimal("-0.0012439620904082457792997415995371658091470281139646")), Tuple(24, Decimal("-0.0015885112789035615619061966115211158573187228221441")), Tuple(25, Decimal("-0.0010745919527384888247242919873531730892739793314532")), Tuple(26, Decimal("0.00065680351863715443150477300335621524888606506047754")), Tuple(27, Decimal("0.0034778369136185382090073595742588115476629156638859")), Tuple(28, Decimal("0.0064000685317006294581072282219458636666371981445885")), Tuple(29, Decimal("0.0073711517704722391344124024235594021578413274885128")), Tuple(30, Decimal("0.0035577288555731609479135377489084026108096506495221")))))

569d5c

Table of

\gamma_{{10}^{n}}

to 50 digits for

0 \le n \le 20

$n$	$\gamma_{{10}^{n}} \; (\text{nearest } 50 \text{D})$
0	-0.072815845483676724860586375874901319137736338334338
1	0.00020533281490906479468372228923706530295985377416676
2	-425340157170802696.23144385197278358247028931053473
3	-1.5709538442047449345494023425120825242380299554570 · 10⁴⁸⁶
4	-2.2104970567221060862971082857536501900234397174729 · 10⁶⁸⁸³
5	1.9919273063125410956582272431568589205211659777533 · 10⁸³⁴³²
6	-4.4209504730980210273285480902514758066667150603243 · 10⁹⁴⁷³⁵²
7	-2.7882974834697458134414289662704891120456603986498 · 10^10390401
8	2.7324629454457814909592178706122081982218137653871 · 10^111591574
9	2.1048416655418517821363600001419516191053973500980 · 10^1181965380
10	7.5883621237131051948224033799125486921750410324510 · 10^12397849705
11	3.4076163168007069203916546697422088077748515862016 · 10^129115149508
12	-1.1713923594956898094830946178584108308869819425684 · 10^{1337330792656}
13	5.1442844004429501778205029347495102582243127602569 · 10^{13792544216233}
14	-5.8565687699062182176274937548885177768345135170181 · 10^{141762672271719}
15	1.8441017255847322907032695598351364885675746553316 · 10^{1452992510427658}
16	1.0887949866822670316936532894122644696901837098117 · 10^{14857814744168222}
17	-9.0932573236841531922129808939176217547651121139948 · 10^{151633823511792145}
18	2.6314370018873515830151010192294307578971415626833 · 10^{1544943249673388947}
19	4.8807917914447513336887536981308809263719031557975 · 10^{15718277029330950920}
20	2.3945266166432844875810628102042011083015231233149 · 10^{159718433793014252763}

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("569d5c"),
    Description("Table of", StieltjesGamma(Pow(10, n)), "to 50 digits for", LessEqual(0, n, 20)),
    Table(Var(n), TableValueHeadings(n, NearestDecimal(StieltjesGamma(Pow(10, n)), 50)), TableSplit(1), List(Tuple(0, Decimal("-0.072815845483676724860586375874901319137736338334338")), Tuple(1, Decimal("0.00020533281490906479468372228923706530295985377416676")), Tuple(2, Decimal("-425340157170802696.23144385197278358247028931053473")), Tuple(3, Decimal("-1.5709538442047449345494023425120825242380299554570e+486")), Tuple(4, Decimal("-2.2104970567221060862971082857536501900234397174729e+6883")), Tuple(5, Decimal("1.9919273063125410956582272431568589205211659777533e+83432")), Tuple(6, Decimal("-4.4209504730980210273285480902514758066667150603243e+947352")), Tuple(7, Decimal("-2.7882974834697458134414289662704891120456603986498e+10390401")), Tuple(8, Decimal("2.7324629454457814909592178706122081982218137653871e+111591574")), Tuple(9, Decimal("2.1048416655418517821363600001419516191053973500980e+1181965380")), Tuple(10, Decimal("7.5883621237131051948224033799125486921750410324510e+12397849705")), Tuple(11, Decimal("3.4076163168007069203916546697422088077748515862016e+129115149508")), Tuple(12, Decimal("-1.1713923594956898094830946178584108308869819425684e+1337330792656")), Tuple(13, Decimal("5.1442844004429501778205029347495102582243127602569e+13792544216233")), Tuple(14, Decimal("-5.8565687699062182176274937548885177768345135170181e+141762672271719")), Tuple(15, Decimal("1.8441017255847322907032695598351364885675746553316e+1452992510427658")), Tuple(16, Decimal("1.0887949866822670316936532894122644696901837098117e+14857814744168222")), Tuple(17, Decimal("-9.0932573236841531922129808939176217547651121139948e+151633823511792145")), Tuple(18, Decimal("2.6314370018873515830151010192294307578971415626833e+1544943249673388947")), Tuple(19, Decimal("4.8807917914447513336887536981308809263719031557975e+15718277029330950920")), Tuple(20, Decimal("2.3945266166432844875810628102042011083015231233149e+159718433793014252763")))))

687b4d

\gamma_{n}\!\left(a + 1\right) = \gamma_{n}\!\left(a\right) - \frac{\log^{n}\!\left(a\right)}{a}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

TeX:

\gamma_{n}\!\left(a + 1\right) = \gamma_{n}\!\left(a\right) - \frac{\log^{n}\!\left(a\right)}{a}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("687b4d"),
    Formula(Equal(StieltjesGamma(n, Add(a, 1)), Sub(StieltjesGamma(n, a), Div(Pow(Log(a), n), a)))),
    Variables(n, a),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

a41c92

\gamma_{n}\!\left(a\right) = -\frac{\pi}{2 \left(n + 1\right)} \int_{0}^{\infty} \frac{\log^{n + 1}\!\left(a - \frac{1}{2} + i x\right) + \log^{n + 1}\!\left(a - \frac{1}{2} - i x\right)}{\cosh^{2}\!\left(\pi x\right)} \, dx

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}

TeX:

\gamma_{n}\!\left(a\right) = -\frac{\pi}{2 \left(n + 1\right)} \int_{0}^{\infty} \frac{\log^{n + 1}\!\left(a - \frac{1}{2} + i x\right) + \log^{n + 1}\!\left(a - \frac{1}{2} - i x\right)}{\cosh^{2}\!\left(\pi x\right)} \, dx

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("a41c92"),
    Formula(Equal(StieltjesGamma(n, a), Neg(Mul(Div(Pi, Mul(2, Add(n, 1))), Integral(Div(Add(Pow(Log(Add(Sub(a, Div(1, 2)), Mul(ConstI, x))), Add(n, 1)), Pow(Log(Sub(Sub(a, Div(1, 2)), Mul(ConstI, x))), Add(n, 1))), Pow(Cosh(Mul(Pi, x)), 2)), For(x, 0, Infinity)))))),
    Variables(n, a),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Greater(Re(a), Div(1, 2)))))

1dec0d

\left|\gamma_{n}\right| < {10}^{-4} {e}^{n \log\left(\log(n)\right)}

Assumptions:

n \in \mathbb{Z}_{\ge 5}

TeX:

\left|\gamma_{n}\right| < {10}^{-4} {e}^{n \log\left(\log(n)\right)}

n \in \mathbb{Z}_{\ge 5}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Log`	$\log(z)$	Natural logarithm
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("1dec0d"),
    Formula(Less(Abs(StieltjesGamma(n)), Mul(Pow(10, -4), Exp(Mul(n, Log(Log(n))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(5))))

Definitions

Generating functions

Limit representations

Specific values

Recurrence relations

Integral representations

Bounds and inequalities