Table of
x n x_{n} x n
to 50 digits for
0 ≤ n ≤ 10 0 \le n \le 10 0 ≤ n ≤ 1 0 n n n x n ( nearest 50 D ) x_{n} \; (\text{nearest } 50 \text{D}) x n ( nearest 5 0 D ) 0 1.4616321449683623412626595423257213284681962040064 1 -0.50408300826445540925826930453330249895538518236858 2 -1.5734984731623904587782860436904346126550408591168 3 -2.6107208684441446500015377157187242079510740108735 4 -3.6352933664369010978391815669460177139484238611935 5 -4.6532377617431424417145981511482073637190694161339 6 -5.6671624415568855358494741745181554247117957876948 7 -6.6784182130734267428298558886022000992046860101508 8 -7.6877883250316260374400988918437049536838217978826 9 -8.6957641638164012664887761608046458202724380849667 10 -9.7026725400018637360844267648942153185775505998219
Definitions:
Fungrim symbol Notation Short description DigammaFunctionZero x n x_{n} x n Zero of the digamma function
Source code for this entry:
Entry(ID("950e5a"),
Description("Table of", DigammaFunctionZero(n), "to 50 digits for", LessEqual(0, n, 10)),
Table(Var(n), TableValueHeadings(n, NearestDecimal(DigammaFunctionZero(n), 50)), TableSplit(1), List(Tuple(0, Decimal("1.4616321449683623412626595423257213284681962040064")), Tuple(1, Decimal("-0.50408300826445540925826930453330249895538518236858")), Tuple(2, Decimal("-1.5734984731623904587782860436904346126550408591168")), Tuple(3, Decimal("-2.6107208684441446500015377157187242079510740108735")), Tuple(4, Decimal("-3.6352933664369010978391815669460177139484238611935")), Tuple(5, Decimal("-4.6532377617431424417145981511482073637190694161339")), Tuple(6, Decimal("-5.6671624415568855358494741745181554247117957876948")), Tuple(7, Decimal("-6.6784182130734267428298558886022000992046860101508")), Tuple(8, Decimal("-7.6877883250316260374400988918437049536838217978826")), Tuple(9, Decimal("-8.6957641638164012664887761608046458202724380849667")), Tuple(10, Decimal("-9.7026725400018637360844267648942153185775505998219"))))) ψ ( x n ) = 0 \psi\!\left(x_{n}\right) = 0 ψ ( x n ) = 0 Assumptions: n ∈ Z ≥ 0 n \in \mathbb{Z}_{\ge 0} n ∈ Z ≥ 0 TeX:
\psi\!\left(x_{n}\right) = 0
n \in \mathbb{Z}_{\ge 0} Definitions:
Source code for this entry:
Entry(ID("3f15eb"),
Formula(Equal(DigammaFunction(DigammaFunctionZero(n)), 0)),
Variables(n),
Assumptions(Element(n, ZZGreaterEqual(0)))) ψ ( 1 ) = − γ \psi\!\left(1\right) = -\gamma ψ ( 1 ) = − γ TeX:
\psi\!\left(1\right) = -\gamma Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ConstGamma γ \gamma γ The constant gamma (0.577...)
Source code for this entry:
Entry(ID("ea2482"),
Formula(Equal(DigammaFunction(1), Neg(ConstGamma)))) ψ ( 2 ) = 1 − γ \psi\!\left(2\right) = 1 - \gamma ψ ( 2 ) = 1 − γ TeX:
\psi\!\left(2\right) = 1 - \gamma Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ConstGamma γ \gamma γ The constant gamma (0.577...)
Source code for this entry:
Entry(ID("ada157"),
Formula(Equal(DigammaFunction(2), Sub(1, ConstGamma)))) ψ ( 3 ) = 3 2 − γ \psi\!\left(3\right) = \frac{3}{2} - \gamma ψ ( 3 ) = 2 3 − γ TeX:
\psi\!\left(3\right) = \frac{3}{2} - \gamma Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ConstGamma γ \gamma γ The constant gamma (0.577...)
Source code for this entry:
Entry(ID("75f9bf"),
Formula(Equal(DigammaFunction(3), Sub(Div(3, 2), ConstGamma)))) ψ ( n ) = H n − 1 − γ \psi\!\left(n\right) = H_{n - 1} - \gamma ψ ( n ) = H n − 1 − γ Assumptions: n ∈ Z ≥ 1 n \in \mathbb{Z}_{\ge 1} n ∈ Z ≥ 1 TeX:
\psi\!\left(n\right) = H_{n - 1} - \gamma
n \in \mathbb{Z}_{\ge 1} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ConstGamma γ \gamma γ The constant gamma (0.577...) ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("00c02a"),
Formula(Equal(DigammaFunction(n), Sub(HarmonicNumber(Sub(n, 1)), ConstGamma))),
Variables(n),
Assumptions(Element(n, ZZGreaterEqual(1)))) ψ ( 1 2 ) = − 2 log ( 2 ) − γ \psi\!\left(\frac{1}{2}\right) = -2 \log(2) - \gamma ψ ( 2 1 ) = − 2 log ( 2 ) − γ TeX:
\psi\!\left(\frac{1}{2}\right) = -2 \log(2) - \gamma Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Log log ( z ) \log(z) log ( z ) Natural logarithm ConstGamma γ \gamma γ The constant gamma (0.577...)
Source code for this entry:
Entry(ID("89bed3"),
Formula(Equal(DigammaFunction(Div(1, 2)), Sub(Neg(Mul(2, Log(2))), ConstGamma)))) ψ ( 1 3 ) = − 3 π 6 − γ − 3 log ( 3 ) 2 \psi\!\left(\frac{1}{3}\right) = -\frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2} ψ ( 3 1 ) = − 6 3 π − γ − 2 3 log ( 3 ) TeX:
\psi\!\left(\frac{1}{3}\right) = -\frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Sqrt z \sqrt{z} z Principal square root Pi π \pi π The constant pi (3.14...) ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm
Source code for this entry:
Entry(ID("98f642"),
Formula(Equal(DigammaFunction(Div(1, 3)), Sub(Sub(Neg(Div(Mul(Sqrt(3), Pi), 6)), ConstGamma), Div(Mul(3, Log(3)), 2))))) ψ ( 2 3 ) = 3 π 6 − γ − 3 log ( 3 ) 2 \psi\!\left(\frac{2}{3}\right) = \frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2} ψ ( 3 2 ) = 6 3 π − γ − 2 3 log ( 3 ) TeX:
\psi\!\left(\frac{2}{3}\right) = \frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Sqrt z \sqrt{z} z Principal square root Pi π \pi π The constant pi (3.14...) ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm
Source code for this entry:
Entry(ID("45a969"),
Formula(Equal(DigammaFunction(Div(2, 3)), Sub(Sub(Div(Mul(Sqrt(3), Pi), 6), ConstGamma), Div(Mul(3, Log(3)), 2))))) ψ ( 1 4 ) = − π 2 − γ − 3 log ( 2 ) \psi\!\left(\frac{1}{4}\right) = -\frac{\pi}{2} - \gamma - 3 \log(2) ψ ( 4 1 ) = − 2 π − γ − 3 log ( 2 ) TeX:
\psi\!\left(\frac{1}{4}\right) = -\frac{\pi}{2} - \gamma - 3 \log(2) Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pi π \pi π The constant pi (3.14...) ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm
Source code for this entry:
Entry(ID("7ec4f0"),
Formula(Equal(DigammaFunction(Div(1, 4)), Sub(Sub(Neg(Div(Pi, 2)), ConstGamma), Mul(3, Log(2)))))) ψ ( 3 4 ) = π 2 − γ − 3 log ( 2 ) \psi\!\left(\frac{3}{4}\right) = \frac{\pi}{2} - \gamma - 3 \log(2) ψ ( 4 3 ) = 2 π − γ − 3 log ( 2 ) TeX:
\psi\!\left(\frac{3}{4}\right) = \frac{\pi}{2} - \gamma - 3 \log(2) Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pi π \pi π The constant pi (3.14...) ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm
Source code for this entry:
Entry(ID("f93bae"),
Formula(Equal(DigammaFunction(Div(3, 4)), Sub(Sub(Div(Pi, 2), ConstGamma), Mul(3, Log(2)))))) ψ ( 1 6 ) = − 3 π 2 − γ − 2 log ( 2 ) − 3 log ( 3 ) 2 \psi\!\left(\frac{1}{6}\right) = -\frac{\sqrt{3} \pi}{2} - \gamma - 2 \log(2) - \frac{3 \log(3)}{2} ψ ( 6 1 ) = − 2 3 π − γ − 2 log ( 2 ) − 2 3 log ( 3 ) TeX:
\psi\!\left(\frac{1}{6}\right) = -\frac{\sqrt{3} \pi}{2} - \gamma - 2 \log(2) - \frac{3 \log(3)}{2} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Sqrt z \sqrt{z} z Principal square root Pi π \pi π The constant pi (3.14...) ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm
Source code for this entry:
Entry(ID("177de7"),
Formula(Equal(DigammaFunction(Div(1, 6)), Sub(Sub(Sub(Neg(Div(Mul(Sqrt(3), Pi), 2)), ConstGamma), Mul(2, Log(2))), Div(Mul(3, Log(3)), 2))))) ψ ( 5 6 ) = 3 π 2 − γ − 2 log ( 2 ) − 3 log ( 3 ) 2 \psi\!\left(\frac{5}{6}\right) = \frac{\sqrt{3} \pi}{2} - \gamma - 2 \log(2) - \frac{3 \log(3)}{2} ψ ( 6 5 ) = 2 3 π − γ − 2 log ( 2 ) − 2 3 log ( 3 ) TeX:
\psi\!\left(\frac{5}{6}\right) = \frac{\sqrt{3} \pi}{2} - \gamma - 2 \log(2) - \frac{3 \log(3)}{2} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Sqrt z \sqrt{z} z Principal square root Pi π \pi π The constant pi (3.14...) ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm
Source code for this entry:
Entry(ID("967bbb"),
Formula(Equal(DigammaFunction(Div(5, 6)), Sub(Sub(Sub(Div(Mul(Sqrt(3), Pi), 2), ConstGamma), Mul(2, Log(2))), Div(Mul(3, Log(3)), 2))))) ψ ( 1 8 ) = − π 2 ( 2 + 1 ) − γ − 4 log ( 2 ) − log ( 2 + 2 ) − log ( 2 − 2 ) 2 \psi\!\left(\frac{1}{8}\right) = -\frac{\pi}{2} \left(\sqrt{2} + 1\right) - \gamma - 4 \log(2) - \frac{\log\!\left(2 + \sqrt{2}\right) - \log\!\left(2 - \sqrt{2}\right)}{\sqrt{2}} ψ ( 8 1 ) = − 2 π ( 2 + 1 ) − γ − 4 log ( 2 ) − 2 log ( 2 + 2 ) − log ( 2 − 2 ) TeX:
\psi\!\left(\frac{1}{8}\right) = -\frac{\pi}{2} \left(\sqrt{2} + 1\right) - \gamma - 4 \log(2) - \frac{\log\!\left(2 + \sqrt{2}\right) - \log\!\left(2 - \sqrt{2}\right)}{\sqrt{2}} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pi π \pi π The constant pi (3.14...) Sqrt z \sqrt{z} z Principal square root ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm
Source code for this entry:
Entry(ID("8c368f"),
Formula(Equal(DigammaFunction(Div(1, 8)), Sub(Sub(Sub(Neg(Mul(Div(Pi, 2), Add(Sqrt(2), 1))), ConstGamma), Mul(4, Log(2))), Div(Sub(Log(Add(2, Sqrt(2))), Log(Sub(2, Sqrt(2)))), Sqrt(2)))))) ψ ( p q ) = − γ − log ( 2 q ) − π 2 cot ( π p q ) + 2 ∑ k = 1 ⌊ ( q − 1 ) / 2 ⌋ cos ( 2 π k p q ) log ( sin ( π k q ) ) \psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right) ψ ( q p ) = − γ − log ( 2 q ) − 2 π cot ( q π p ) + 2 k = 1 ∑ ⌊ ( q − 1 ) / 2 ⌋ cos ( q 2 π k p ) log ( sin ( q π k ) ) Assumptions: q ∈ Z ≥ 2 and p ∈ { 1 , 2 , … , q − 1 } q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; p \in \{1, 2, \ldots, q - 1\} q ∈ Z ≥ 2 a n d p ∈ { 1 , 2 , … , q − 1 } TeX:
\psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right)
q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; p \in \{1, 2, \ldots, q - 1\} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm Pi π \pi π The constant pi (3.14...) Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum Cos cos ( z ) \cos(z) cos ( z ) Cosine Sin sin ( z ) \sin(z) sin ( z ) Sine ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n Range { a , a + 1 , … , b } \{a, a + 1, \ldots, b\} { a , a + 1 , … , b } Integers between given endpoints
Source code for this entry:
Entry(ID("3fe553"),
Formula(Equal(DigammaFunction(Div(p, q)), Add(Sub(Sub(Neg(ConstGamma), Log(Mul(2, q))), Mul(Div(Pi, 2), Cot(Div(Mul(Pi, p), q)))), Mul(2, Sum(Mul(Cos(Div(Mul(Mul(Mul(2, Pi), k), p), q)), Log(Sin(Div(Mul(Pi, k), q)))), For(k, 1, Floor(Div(Sub(q, 1), 2)))))))),
Variables(p, q),
Assumptions(And(Element(q, ZZGreaterEqual(2)), Element(p, Range(1, Sub(q, 1)))))) ψ ′ ( 1 ) = π 2 6 \psi'\!\left(1\right) = \frac{{\pi}^{2}}{6} ψ ′ ( 1 ) = 6 π 2 TeX:
\psi'\!\left(1\right) = \frac{{\pi}^{2}}{6} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("babd3c"),
Formula(Equal(DigammaFunction(1, 1), Div(Pow(Pi, 2), 6)))) ψ ′ ′ ( 1 ) = − 2 ζ ( 3 ) \psi''\!\left(1\right) = -2 \zeta\!\left(3\right) ψ ′ ′ ( 1 ) = − 2 ζ ( 3 ) TeX:
\psi''\!\left(1\right) = -2 \zeta\!\left(3\right) Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function
Source code for this entry:
Entry(ID("4a30f1"),
Formula(Equal(DigammaFunction(1, 2), Neg(Mul(2, RiemannZeta(3)))))) ψ ( m ) ( 1 ) = ( − 1 ) m + 1 m ! ζ ( m + 1 ) \psi^{(m)}\!\left(1\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1\right) ψ ( m ) ( 1 ) = ( − 1 ) m + 1 m ! ζ ( m + 1 ) Assumptions: m ∈ Z ≥ 1 m \in \mathbb{Z}_{\ge 1} m ∈ Z ≥ 1 TeX:
\psi^{(m)}\!\left(1\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1\right)
m \in \mathbb{Z}_{\ge 1} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Factorial n ! n ! n ! Factorial RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("a62320"),
Formula(Equal(DigammaFunction(1, m), Mul(Mul(Pow(-1, Add(m, 1)), Factorial(m)), RiemannZeta(Add(m, 1))))),
Variables(m),
Assumptions(Element(m, ZZGreaterEqual(1)))) ψ ′ ( 2 ) = π 2 6 − 1 \psi'\!\left(2\right) = \frac{{\pi}^{2}}{6} - 1 ψ ′ ( 2 ) = 6 π 2 − 1 TeX:
\psi'\!\left(2\right) = \frac{{\pi}^{2}}{6} - 1 Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("fa0292"),
Formula(Equal(DigammaFunction(2, 1), Sub(Div(Pow(Pi, 2), 6), 1)))) ψ ( m ) ( n ) = ( − 1 ) m + 1 m ! ζ ( m + 1 , n ) \psi^{(m)}\!\left(n\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, n\right) ψ ( m ) ( n ) = ( − 1 ) m + 1 m ! ζ ( m + 1 , n ) Assumptions: n ∈ Z ≥ 1 and m ∈ Z ≥ 1 n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1} n ∈ Z ≥ 1 a n d m ∈ Z ≥ 1 TeX:
\psi^{(m)}\!\left(n\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, n\right)
n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Factorial n ! n ! n ! Factorial HurwitzZeta ζ ( s , a ) \zeta\!\left(s, a\right) ζ ( s , a ) Hurwitz zeta function ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("90b26f"),
Formula(Equal(DigammaFunction(n, m), Mul(Mul(Pow(-1, Add(m, 1)), Factorial(m)), HurwitzZeta(Add(m, 1), n)))),
Variables(n, m),
Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(m, ZZGreaterEqual(1))))) ψ ′ ( 1 2 ) = π 2 2 \psi'\!\left(\frac{1}{2}\right) = \frac{{\pi}^{2}}{2} ψ ′ ( 2 1 ) = 2 π 2 TeX:
\psi'\!\left(\frac{1}{2}\right) = \frac{{\pi}^{2}}{2} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("595f46"),
Formula(Equal(DigammaFunction(Div(1, 2), 1), Div(Pow(Pi, 2), 2)))) ψ ′ ′ ( 1 2 ) = − 14 ζ ( 3 ) \psi''\!\left(\frac{1}{2}\right) = -14 \zeta\!\left(3\right) ψ ′ ′ ( 2 1 ) = − 1 4 ζ ( 3 ) TeX:
\psi''\!\left(\frac{1}{2}\right) = -14 \zeta\!\left(3\right) Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function
Source code for this entry:
Entry(ID("b31fd2"),
Formula(Equal(DigammaFunction(Div(1, 2), 2), Neg(Mul(14, RiemannZeta(3)))))) ψ ′ ′ ′ ( 1 2 ) = π 4 \psi'''\!\left(\frac{1}{2}\right) = {\pi}^{4} ψ ′ ′ ′ ( 2 1 ) = π 4 TeX:
\psi'''\!\left(\frac{1}{2}\right) = {\pi}^{4} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("2251c6"),
Formula(Equal(DigammaFunction(Div(1, 2), 3), Pow(Pi, 4)))) ψ ( m ) ( 1 2 ) = ( − 1 ) m + 1 ( 2 m + 1 − 1 ) m ! ζ ( m + 1 ) \psi^{(m)}\!\left(\frac{1}{2}\right) = {\left(-1\right)}^{m + 1} \left({2}^{m + 1} - 1\right) m ! \zeta\!\left(m + 1\right) ψ ( m ) ( 2 1 ) = ( − 1 ) m + 1 ( 2 m + 1 − 1 ) m ! ζ ( m + 1 ) Assumptions: m ∈ Z ≥ 1 m \in \mathbb{Z}_{\ge 1} m ∈ Z ≥ 1 TeX:
\psi^{(m)}\!\left(\frac{1}{2}\right) = {\left(-1\right)}^{m + 1} \left({2}^{m + 1} - 1\right) m ! \zeta\!\left(m + 1\right)
m \in \mathbb{Z}_{\ge 1} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Factorial n ! n ! n ! Factorial RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("5ce30b"),
Formula(Equal(DigammaFunction(Div(1, 2), m), Mul(Mul(Mul(Pow(-1, Add(m, 1)), Sub(Pow(2, Add(m, 1)), 1)), Factorial(m)), RiemannZeta(Add(m, 1))))),
Variables(m),
Assumptions(Element(m, ZZGreaterEqual(1)))) ψ ′ ( 1 4 ) = π 2 + 8 G \psi'\!\left(\frac{1}{4}\right) = {\pi}^{2} + 8 G ψ ′ ( 4 1 ) = π 2 + 8 G TeX:
\psi'\!\left(\frac{1}{4}\right) = {\pi}^{2} + 8 G Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) ConstCatalan G G G Catalan's constant
Source code for this entry:
Entry(ID("807c7d"),
Formula(Equal(DigammaFunction(Div(1, 4), 1), Add(Pow(Pi, 2), Mul(8, ConstCatalan))))) ψ ′ ( 3 4 ) = π 2 − 8 G \psi'\!\left(\frac{3}{4}\right) = {\pi}^{2} - 8 G ψ ′ ( 4 3 ) = π 2 − 8 G TeX:
\psi'\!\left(\frac{3}{4}\right) = {\pi}^{2} - 8 G Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) ConstCatalan G G G Catalan's constant
Source code for this entry:
Entry(ID("d2f9fb"),
Formula(Equal(DigammaFunction(Div(3, 4), 1), Sub(Pow(Pi, 2), Mul(8, ConstCatalan))))) ψ ′ ′ ( 1 4 ) = − 2 π 3 − 56 ζ ( 3 ) \psi''\!\left(\frac{1}{4}\right) = -2 {\pi}^{3} - 56 \zeta\!\left(3\right) ψ ′ ′ ( 4 1 ) = − 2 π 3 − 5 6 ζ ( 3 ) TeX:
\psi''\!\left(\frac{1}{4}\right) = -2 {\pi}^{3} - 56 \zeta\!\left(3\right) Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function
Source code for this entry:
Entry(ID("03aca0"),
Formula(Equal(DigammaFunction(Div(1, 4), 2), Sub(Neg(Mul(2, Pow(Pi, 3))), Mul(56, RiemannZeta(3)))))) ψ ′ ′ ( 3 4 ) = 2 π 3 − 56 ζ ( 3 ) \psi''\!\left(\frac{3}{4}\right) = 2 {\pi}^{3} - 56 \zeta\!\left(3\right) ψ ′ ′ ( 4 3 ) = 2 π 3 − 5 6 ζ ( 3 ) TeX:
\psi''\!\left(\frac{3}{4}\right) = 2 {\pi}^{3} - 56 \zeta\!\left(3\right) Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function
Source code for this entry:
Entry(ID("e83059"),
Formula(Equal(DigammaFunction(Div(3, 4), 2), Sub(Mul(2, Pow(Pi, 3)), Mul(56, RiemannZeta(3)))))) ψ ′ ′ ( 1 6 ) = − 182 ζ ( 3 ) − 4 3 π 3 \psi''\!\left(\frac{1}{6}\right) = -182 \zeta\!\left(3\right) - 4 \sqrt{3} {\pi}^{3} ψ ′ ′ ( 6 1 ) = − 1 8 2 ζ ( 3 ) − 4 3 π 3 TeX:
\psi''\!\left(\frac{1}{6}\right) = -182 \zeta\!\left(3\right) - 4 \sqrt{3} {\pi}^{3} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function Sqrt z \sqrt{z} z Principal square root Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("bb88c8"),
Formula(Equal(DigammaFunction(Div(1, 6), 2), Sub(Neg(Mul(182, RiemannZeta(3))), Mul(Mul(4, Sqrt(3)), Pow(Pi, 3)))))) ψ ′ ′ ( 5 6 ) = − 182 ζ ( 3 ) + 4 3 π 3 \psi''\!\left(\frac{5}{6}\right) = -182 \zeta\!\left(3\right) + 4 \sqrt{3} {\pi}^{3} ψ ′ ′ ( 6 5 ) = − 1 8 2 ζ ( 3 ) + 4 3 π 3 TeX:
\psi''\!\left(\frac{5}{6}\right) = -182 \zeta\!\left(3\right) + 4 \sqrt{3} {\pi}^{3} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function Sqrt z \sqrt{z} z Principal square root Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("921d61"),
Formula(Equal(DigammaFunction(Div(5, 6), 2), Add(Neg(Mul(182, RiemannZeta(3))), Mul(Mul(4, Sqrt(3)), Pow(Pi, 3)))))) Im ( ψ ( i y ) ) = π 2 coth ( π y ) + 1 2 y \operatorname{Im}\!\left(\psi\!\left(i y\right)\right) = \frac{\pi}{2} \coth\!\left(\pi y\right) + \frac{1}{2 y} I m ( ψ ( i y ) ) = 2 π coth ( π y ) + 2 y 1 Assumptions: y ∈ R and y ≠ 0 y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0 y ∈ R a n d y = 0 TeX:
\operatorname{Im}\!\left(\psi\!\left(i y\right)\right) = \frac{\pi}{2} \coth\!\left(\pi y\right) + \frac{1}{2 y}
y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0 Definitions:
Fungrim symbol Notation Short description Im Im ( z ) \operatorname{Im}(z) I m ( z ) Imaginary part DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ConstI i i i Imaginary unit Pi π \pi π The constant pi (3.14...) RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("03e2a6"),
Formula(Equal(Im(DigammaFunction(Mul(ConstI, y))), Add(Mul(Div(Pi, 2), Coth(Mul(Pi, y))), Div(1, Mul(2, y))))),
Variables(y),
Assumptions(And(Element(y, RR), NotEqual(y, 0)))) Im ( ψ ( 1 + i y ) ) = π 2 coth ( π y ) − 1 2 y \operatorname{Im}\!\left(\psi\!\left(1 + i y\right)\right) = \frac{\pi}{2} \coth\!\left(\pi y\right) - \frac{1}{2 y} I m ( ψ ( 1 + i y ) ) = 2 π coth ( π y ) − 2 y 1 Assumptions: y ∈ R and y ≠ 0 y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0 y ∈ R a n d y = 0 TeX:
\operatorname{Im}\!\left(\psi\!\left(1 + i y\right)\right) = \frac{\pi}{2} \coth\!\left(\pi y\right) - \frac{1}{2 y}
y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0 Definitions:
Fungrim symbol Notation Short description Im Im ( z ) \operatorname{Im}(z) I m ( z ) Imaginary part DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ConstI i i i Imaginary unit Pi π \pi π The constant pi (3.14...) RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("22a9cd"),
Formula(Equal(Im(DigammaFunction(Add(1, Mul(ConstI, y)))), Sub(Mul(Div(Pi, 2), Coth(Mul(Pi, y))), Div(1, Mul(2, y))))),
Variables(y),
Assumptions(And(Element(y, RR), NotEqual(y, 0)))) Im ( ψ ( 1 2 + i y ) ) = π 2 tanh ( π y ) \operatorname{Im}\!\left(\psi\!\left(\frac{1}{2} + i y\right)\right) = \frac{\pi}{2} \tanh\!\left(\pi y\right) I m ( ψ ( 2 1 + i y ) ) = 2 π tanh ( π y ) Assumptions: y ∈ R and y ≠ 0 y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0 y ∈ R a n d y = 0 TeX:
\operatorname{Im}\!\left(\psi\!\left(\frac{1}{2} + i y\right)\right) = \frac{\pi}{2} \tanh\!\left(\pi y\right)
y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0 Definitions:
Fungrim symbol Notation Short description Im Im ( z ) \operatorname{Im}(z) I m ( z ) Imaginary part DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ConstI i i i Imaginary unit Pi π \pi π The constant pi (3.14...) RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("6f3fec"),
Formula(Equal(Im(DigammaFunction(Add(Div(1, 2), Mul(ConstI, y)))), Mul(Div(Pi, 2), Tanh(Mul(Pi, y))))),
Variables(y),
Assumptions(And(Element(y, RR), NotEqual(y, 0)))) ψ ( − n ) = ∞ ~ \psi\!\left(-n\right) = {\tilde \infty} ψ ( − n ) = ∞ ~ Assumptions: n ∈ Z ≥ 0 n \in \mathbb{Z}_{\ge 0} n ∈ Z ≥ 0 TeX:
\psi\!\left(-n\right) = {\tilde \infty}
n \in \mathbb{Z}_{\ge 0} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function UnsignedInfinity ∞ ~ {\tilde \infty} ∞ ~ Unsigned infinity ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("42c1f5"),
Formula(Equal(DigammaFunction(Neg(n)), UnsignedInfinity)),
Variables(n),
Assumptions(Element(n, ZZGreaterEqual(0)))) ψ ( m ) ( − n ) = ∞ ~ \psi^{(m)}\!\left(-n\right) = {\tilde \infty} ψ ( m ) ( − n ) = ∞ ~ Assumptions: n ∈ Z ≥ 0 and m ∈ Z ≥ 0 n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0} n ∈ Z ≥ 0 a n d m ∈ Z ≥ 0 TeX:
\psi^{(m)}\!\left(-n\right) = {\tilde \infty}
n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function UnsignedInfinity ∞ ~ {\tilde \infty} ∞ ~ Unsigned infinity ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("78c19c"),
Formula(Equal(DigammaFunction(Neg(n), m), UnsignedInfinity)),
Variables(n, m),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0))))) ψ ( m ) ( ∞ ) = lim x → ∞ ψ ( x ) = { ∞ , m = 0 0 , m > 0 \psi^{(m)}\!\left(\infty\right) = \lim_{x \to \infty} \psi\!\left(x\right) = \begin{cases} \infty, & m = 0\\0, & m > 0\\ \end{cases} ψ ( m ) ( ∞ ) = x → ∞ lim ψ ( x ) = { ∞ , 0 , m = 0 m > 0 Assumptions: m ∈ Z ≥ 0 m \in \mathbb{Z}_{\ge 0} m ∈ Z ≥ 0 TeX:
\psi^{(m)}\!\left(\infty\right) = \lim_{x \to \infty} \psi\!\left(x\right) = \begin{cases} \infty, & m = 0\\0, & m > 0\\ \end{cases}
m \in \mathbb{Z}_{\ge 0} Definitions:
Fungrim symbol Notation Short description DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Infinity ∞ \infty ∞ Positive infinity RealLimit lim x → a f ( x ) \lim_{x \to a} f(x) lim x → a f ( x ) Limiting value, real variable ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("1cbe83"),
Formula(Equal(DigammaFunction(Infinity, m), RealLimit(DigammaFunction(x), For(x, Infinity)), Cases(Tuple(Infinity, Equal(m, 0)), Tuple(0, Greater(m, 0))))),
Variables(m),
Assumptions(Element(m, ZZGreaterEqual(0)))) lim x → 0 + ψ ( x ) = ( − 1 ) m + 1 ∞ \lim_{x \to {0}^{+}} \psi\!\left(x\right) = {\left(-1\right)}^{m + 1} \infty x → 0 + lim ψ ( x ) = ( − 1 ) m + 1 ∞ Assumptions: m ∈ Z ≥ 0 m \in \mathbb{Z}_{\ge 0} m ∈ Z ≥ 0 TeX:
\lim_{x \to {0}^{+}} \psi\!\left(x\right) = {\left(-1\right)}^{m + 1} \infty
m \in \mathbb{Z}_{\ge 0} Definitions:
Fungrim symbol Notation Short description RightLimit lim x → a + f ( x ) \lim_{x \to {a}^{+}} f(x) lim x → a + f ( x ) Limiting value, from the right DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function Pow a b {a}^{b} a b Power Infinity ∞ \infty ∞ Positive infinity ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("dce62c"),
Formula(Equal(RightLimit(DigammaFunction(x), For(x, 0)), Mul(Pow(-1, Add(m, 1)), Infinity))),
Variables(m),
Assumptions(Element(m, ZZGreaterEqual(0)))) ∑ n = 0 ∞ 1 x n 2 = γ 2 + π 2 2 \sum_{n=0}^{\infty} \frac{1}{x_{n}^{2}} = {\gamma}^{2} + \frac{{\pi}^{2}}{2} n = 0 ∑ ∞ x n 2 1 = γ 2 + 2 π 2 TeX:
\sum_{n=0}^{\infty} \frac{1}{x_{n}^{2}} = {\gamma}^{2} + \frac{{\pi}^{2}}{2} Definitions:
Fungrim symbol Notation Short description Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum Pow a b {a}^{b} a b Power DigammaFunctionZero x n x_{n} x n Zero of the digamma function Infinity ∞ \infty ∞ Positive infinity ConstGamma γ \gamma γ The constant gamma (0.577...) Pi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("1165fc"),
Formula(Equal(Sum(Div(1, Pow(DigammaFunctionZero(n), 2)), For(n, 0, Infinity)), Add(Pow(ConstGamma, 2), Div(Pow(Pi, 2), 2))))) ∑ n = 0 ∞ 1 x n 3 = − γ 3 − γ π 2 2 − 4 ζ ( 3 ) \sum_{n=0}^{\infty} \frac{1}{x_{n}^{3}} = -{\gamma}^{3} - \frac{\gamma {\pi}^{2}}{2} - 4 \zeta\!\left(3\right) n = 0 ∑ ∞ x n 3 1 = − γ 3 − 2 γ π 2 − 4 ζ ( 3 ) TeX:
\sum_{n=0}^{\infty} \frac{1}{x_{n}^{3}} = -{\gamma}^{3} - \frac{\gamma {\pi}^{2}}{2} - 4 \zeta\!\left(3\right) Definitions:
Fungrim symbol Notation Short description Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum Pow a b {a}^{b} a b Power DigammaFunctionZero x n x_{n} x n Zero of the digamma function Infinity ∞ \infty ∞ Positive infinity ConstGamma γ \gamma γ The constant gamma (0.577...) Pi π \pi π The constant pi (3.14...) RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function
Source code for this entry:
Entry(ID("39ce44"),
Formula(Equal(Sum(Div(1, Pow(DigammaFunctionZero(n), 3)), For(n, 0, Infinity)), Sub(Sub(Neg(Pow(ConstGamma, 3)), Div(Mul(ConstGamma, Pow(Pi, 2)), 2)), Mul(4, RiemannZeta(3)))))) ∑ n = 0 ∞ 1 x n 4 = γ 4 + π 4 9 + 2 γ 2 π 2 3 + 4 γ ζ ( 3 ) \sum_{n=0}^{\infty} \frac{1}{x_{n}^{4}} = {\gamma}^{4} + \frac{{\pi}^{4}}{9} + \frac{2 {\gamma}^{2} {\pi}^{2}}{3} + 4 \gamma \zeta\!\left(3\right) n = 0 ∑ ∞ x n 4 1 = γ 4 + 9 π 4 + 3 2 γ 2 π 2 + 4 γ ζ ( 3 ) TeX:
\sum_{n=0}^{\infty} \frac{1}{x_{n}^{4}} = {\gamma}^{4} + \frac{{\pi}^{4}}{9} + \frac{2 {\gamma}^{2} {\pi}^{2}}{3} + 4 \gamma \zeta\!\left(3\right) Definitions:
Fungrim symbol Notation Short description Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum Pow a b {a}^{b} a b Power DigammaFunctionZero x n x_{n} x n Zero of the digamma function Infinity ∞ \infty ∞ Positive infinity ConstGamma γ \gamma γ The constant gamma (0.577...) Pi π \pi π The constant pi (3.14...) RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function
Source code for this entry:
Entry(ID("a4f9c9"),
Formula(Equal(Sum(Div(1, Pow(DigammaFunctionZero(n), 4)), For(n, 0, Infinity)), Add(Add(Add(Pow(ConstGamma, 4), Div(Pow(Pi, 4), 9)), Div(Mul(Mul(2, Pow(ConstGamma, 2)), Pow(Pi, 2)), 3)), Mul(4, Mul(ConstGamma, RiemannZeta(3))))))) ∑ n = 0 ∞ 1 x n r + 1 = f ( r ) ( 0 ) r ! where f ( z ) = lim t → z ( ψ ( t ) − ψ ′ ( t ) ψ ( t ) ) \sum_{n=0}^{\infty} \frac{1}{x_{n}^{r + 1}} = \frac{{f}^{(r)}(0)}{r !}\; \text{ where } f(z) = \lim_{t \to z} \left(\psi\!\left(t\right) - \frac{\psi'\!\left(t\right)}{\psi\!\left(t\right)}\right) n = 0 ∑ ∞ x n r + 1 1 = r ! f ( r ) ( 0 ) where f ( z ) = t → z lim ( ψ ( t ) − ψ ( t ) ψ ′ ( t ) ) Assumptions: r ∈ Z ≥ 1 r \in \mathbb{Z}_{\ge 1} r ∈ Z ≥ 1 References:
https://doi.org/10.1080%2F10652469.2017.1376193 TeX:
\sum_{n=0}^{\infty} \frac{1}{x_{n}^{r + 1}} = \frac{{f}^{(r)}(0)}{r !}\; \text{ where } f(z) = \lim_{t \to z} \left(\psi\!\left(t\right) - \frac{\psi'\!\left(t\right)}{\psi\!\left(t\right)}\right)
r \in \mathbb{Z}_{\ge 1} Definitions:
Fungrim symbol Notation Short description Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum Pow a b {a}^{b} a b Power DigammaFunctionZero x n x_{n} x n Zero of the digamma function Infinity ∞ \infty ∞ Positive infinity ComplexDerivative d d z f ( z ) \frac{d}{d z}\, f\!\left(z\right) d z d f ( z ) Complex derivative Factorial n ! n ! n ! Factorial ComplexLimit lim z → a f ( z ) \lim_{z \to a} f(z) lim z → a f ( z ) Limiting value, complex variable DigammaFunction ψ ( z ) \psi\!\left(z\right) ψ ( z ) Digamma function ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("6547da"),
Formula(Equal(Sum(Div(1, Pow(DigammaFunctionZero(n), Add(r, 1))), For(n, 0, Infinity)), Where(Div(ComplexDerivative(f(z), For(z, 0, r)), Factorial(r)), Equal(f(z), ComplexLimit(Parentheses(Sub(DigammaFunction(t), Div(DigammaFunction(t, 1), DigammaFunction(t)))), For(t, z)))))),
Variables(r),
Assumptions(Element(r, ZZGreaterEqual(1))),
References("https://doi.org/10.1080%2F10652469.2017.1376193"))