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Digamma function

Table of contents: Definitions - Illustrations - Domain and singularities - Specific values - Zeros - Derivatives and differential equations - Series representations - Representation by other functions - Functional equations - Integral representations - Generating functions - Finite sums - Bounds and inequalities - Summation of rational functions

Definitions

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Symbol: DigammaFunction ψ ⁣(z)\psi\!\left(z\right) Digamma function
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Symbol: DigammaFunctionZero xnx_{n} Zero of the digamma function

Illustrations

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Image: X-ray of ψ ⁣(z)\psi\!\left(z\right) on z[5,5]+[5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i
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Image: X-ray of ψ ⁣(z)\psi'\!\left(z\right) on z[5,5]+[5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i

Domain and singularities

Digamma function

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xR{0,1,}        ψ ⁣(x)Rx \in \mathbb{R} \setminus \{0, -1, \ldots\} \;\implies\; \psi\!\left(x\right) \in \mathbb{R}
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zC{0,1,}        ψ ⁣(z)Cz \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \psi\!\left(z\right) \in \mathbb{C}
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ψ ⁣(z) is holomorphic on zC{0,1,}\psi\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \{0, -1, \ldots\}
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ψ ⁣(z) is meromorphic on zC\psi\!\left(z\right) \text{ is meromorphic on } z \in \mathbb{C}
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poleszCψ ⁣(z)={0,1,}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C}} \psi\!\left(z\right) = \{0, -1, \ldots\}

Polygamma functions

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ψ(0) ⁣(z)=ψ ⁣(z)\psi^{(0)}\!\left(z\right) = \psi\!\left(z\right)
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(mZ0andxR{0,1,})    (ψ(m) ⁣(x)R)\left(m \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{R} \setminus \{0, -1, \ldots\}\right) \implies \left(\psi^{(m)}\!\left(x\right) \in \mathbb{R}\right)
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(mZ0andzC{0,1,})    (ψ(m) ⁣(z)C)\left(m \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \implies \left(\psi^{(m)}\!\left(z\right) \in \mathbb{C}\right)
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ψ(m) ⁣(z) is holomorphic on zC{0,1,}\psi^{(m)}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \{0, -1, \ldots\}
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ψ(m) ⁣(z) is meromorphic on zC\psi^{(m)}\!\left(z\right) \text{ is meromorphic on } z \in \mathbb{C}
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poleszCψ(m) ⁣(z)={0,1,}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C}} \psi^{(m)}\!\left(z\right) = \{0, -1, \ldots\}

Specific values

Main topic: Specific values of the digamma function

Zeros

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Table of xnx_{n} to 50 digits for 0n100 \le n \le 10
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ψ ⁣(xn)=0\psi\!\left(x_{n}\right) = 0

Values at integers and simple fractions

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ψ ⁣(1)=γ\psi\!\left(1\right) = -\gamma
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ψ ⁣(2)=1γ\psi\!\left(2\right) = 1 - \gamma
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ψ ⁣(n)=Hn1γ\psi\!\left(n\right) = H_{n - 1} - \gamma
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ψ ⁣(12)=2log(2)γ\psi\!\left(\frac{1}{2}\right) = -2 \log(2) - \gamma
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ψ ⁣(13)=3π6γ3log(3)2\psi\!\left(\frac{1}{3}\right) = -\frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2}
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ψ ⁣(14)=π2γ3log(2)\psi\!\left(\frac{1}{4}\right) = -\frac{\pi}{2} - \gamma - 3 \log(2)
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ψ ⁣(pq)=γlog ⁣(2q)π2cot ⁣(πpq)+2k=1(q1)/2cos ⁣(2πkpq)log ⁣(sin ⁣(πkq))\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)

Values of polygamma functions at integers and simple fractions

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ψ ⁣(1)=π26\psi'\!\left(1\right) = \frac{{\pi}^{2}}{6}
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ψ(m) ⁣(n)=(1)m+1m!ζ ⁣(m+1,n)\psi^{(m)}\!\left(n\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, n\right)
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ψ(m) ⁣(12)=(1)m+1(2m+11)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)
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ψ ⁣(14)=π2+8G\psi'\!\left(\frac{1}{4}\right) = {\pi}^{2} + 8 G

Zeros

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nZ0        xnRn \in \mathbb{Z}_{\ge 0} \;\implies\; x_{n} \in \mathbb{R}
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zeroszCψ ⁣(z)={xn:nZ0}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \psi\!\left(z\right) = \left\{ x_{n} : n \in \mathbb{Z}_{\ge 0} \right\}
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xn=zero*xSψ ⁣(x)   where S={(0,),n=0(n,n+1),n<0x_{n} = \mathop{\operatorname{zero*}\,}\limits_{x \in S} \psi\!\left(x\right)\; \text{ where } S = \begin{cases} \left(0, \infty\right), & n = 0\\\left(-n, -n + 1\right), & n < 0\\ \end{cases}
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xnn+1πatan ⁣(πlog(n)),  nx_{n} \sim -n + \frac{1}{\pi} \operatorname{atan}\!\left(\frac{\pi}{\log(n)}\right), \; n \to \infty

Derivatives and differential equations

Related topic: Gamma function
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ψ ⁣(z)=Γ(z)Γ(z)\psi\!\left(z\right) = \frac{\Gamma'(z)}{\Gamma(z)}
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ψ ⁣(z)=ddz[logΓ(z)]\psi\!\left(z\right) = \frac{d}{d z}\, \left[\log \Gamma(z)\right]
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ψ(m) ⁣(z)=dm+1dzm+1[logΓ(z)]\psi^{(m)}\!\left(z\right) = \frac{d^{m + 1}}{{d z}^{m + 1}} \left[\log \Gamma(z)\right]
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dndzn[ψ ⁣(z)]=ψ(n) ⁣(z)\frac{d^{n}}{{d z}^{n}} \left[\psi\!\left(z\right)\right] = \psi^{(n)}\!\left(z\right)
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dndzn[ψ(m) ⁣(z)]=ψ(m+n) ⁣(z)\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)

Series representations

Series of rational functions

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ψ ⁣(z)=γ+n=0(1n+11n+z)\psi\!\left(z\right) = -\gamma + \sum_{n=0}^{\infty} \left(\frac{1}{n + 1} - \frac{1}{n + z}\right)
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ψ(m) ⁣(z)=(1)m+1m!n=01(n+z)m+1\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \sum_{n=0}^{\infty} \frac{1}{{\left(n + z\right)}^{m + 1}}

Taylor series

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ψ ⁣(1+z)=γ+n=1(1)n+1ζ ⁣(n+1)zn\psi\!\left(1 + z\right) = -\gamma + \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} \zeta\!\left(n + 1\right) {z}^{n}

Laurent series

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ψ ⁣(z)=1zγ+n=1(1)n+1ζ ⁣(n+1)zn\psi\!\left(z\right) = -\frac{1}{z} - \gamma + \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} \zeta\!\left(n + 1\right) {z}^{n}
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ψ ⁣(n+z)=1z+ψ ⁣(n+1)+k=1((1)k+1ζ ⁣(k+1)+j=1n1jk+1)zk\psi\!\left(-n + z\right) = -\frac{1}{z} + \psi\!\left(n + 1\right) + \sum_{k=1}^{\infty} \left({\left(-1\right)}^{k + 1} \zeta\!\left(k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + 1}}\right) {z}^{k}
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ψ(m) ⁣(n+z)=(1)m+1m!zm+1+k=0(k+1)m((1)m+k+1ζ ⁣(m+k+1)+j=1n1jk+m+1)zk\psi^{(m)}\!\left(-n + z\right) = \frac{{\left(-1\right)}^{m + 1} m !}{{z}^{m + 1}} + \sum_{k=0}^{\infty} \left(k + 1\right)_{m} \left({\left(-1\right)}^{m + k + 1} \zeta\!\left(m + k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + m + 1}}\right) {z}^{k}

Asymptotic expansions

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ψ ⁣(z)=log(z)12zn=1N1B2n2nz2n+RN(z)\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \sum_{n=1}^{N - 1} \frac{B_{2 n}}{2 n {z}^{2 n}} + R'_{N}(z)
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ψ(m) ⁣(z)=(1)m+1m!(1mzm+12zm+1+n=1N1(m+1)2n1(2n)!B2nzm+2n)+RN(m+1)(z)\psi^{(m)}\!\left(z\right) = \frac{{\left(-1\right)}^{m + 1}}{m !} \left(\frac{1}{m {z}^{m}} + \frac{1}{2 {z}^{m + 1}} + \sum_{n=1}^{N - 1} \frac{\left(m + 1\right)_{2 n - 1}}{\left(2 n\right)!} \frac{B_{2 n}}{{z}^{m + 2 n}}\right) + {R}^{(m + 1)}_{N}(z)

Weierstrass product

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ψ ⁣(z)Γ(z)=e2γzn=0(1zxn)exp ⁣(zxn)\frac{\psi\!\left(z\right)}{\Gamma(z)} = -{e}^{2 \gamma z} \prod_{n=0}^{\infty} \left(1 - \frac{z}{x_{n}}\right) \exp\!\left(\frac{z}{x_{n}}\right)

Representation by other functions

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ψ(m) ⁣(z)=(1)m+1m!ζ ⁣(m+1,z)\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, z\right)
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ψ(m) ⁣(z)=(1)m+1m!Φ ⁣(1,m+1,z)\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \Phi\!\left(1, m + 1, z\right)
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ψ ⁣(z)=(z1)3F2 ⁣(1,1,2z,2,2,1)γ\psi\!\left(z\right) = \left(z - 1\right) \,{}_3F_2\!\left(1, 1, 2 - z, 2, 2, 1\right) - \gamma
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ψ ⁣(z)=γ0 ⁣(z)\psi\!\left(z\right) = -\gamma_{0}\!\left(z\right)

Functional equations

Recurrence relations

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ψ ⁣(z+1)=ψ ⁣(z)+1z\psi\!\left(z + 1\right) = \psi\!\left(z\right) + \frac{1}{z}
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ψ ⁣(z+n)=ψ ⁣(z)+k=0n11z+k\psi\!\left(z + n\right) = \psi\!\left(z\right) + \sum_{k=0}^{n - 1} \frac{1}{z + k}
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ψ ⁣(zn)=ψ ⁣(z)k=1n1zk\psi\!\left(z - n\right) = \psi\!\left(z\right) - \sum_{k=1}^{n} \frac{1}{z - k}
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ψ(m) ⁣(z+1)=ψ(m) ⁣(z)+(1)mm!zm+1\psi^{(m)}\!\left(z + 1\right) = \psi^{(m)}\!\left(z\right) + \frac{{\left(-1\right)}^{m} m !}{{z}^{m + 1}}
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ψ(m) ⁣(z+n)=ψ(m) ⁣(z)+(1)mm!k=0n11(z+k)m+1\psi^{(m)}\!\left(z + n\right) = \psi^{(m)}\!\left(z\right) + {\left(-1\right)}^{m} m ! \sum_{k=0}^{n - 1} \frac{1}{{\left(z + k\right)}^{m + 1}}
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ψ(m) ⁣(zn)=ψ(m) ⁣(z)(1)mm!k=1n1(zk)m+1\psi^{(m)}\!\left(z - n\right) = \psi^{(m)}\!\left(z\right) - {\left(-1\right)}^{m} m ! \sum_{k=1}^{n} \frac{1}{{\left(z - k\right)}^{m + 1}}

Reflection formula

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ψ ⁣(1z)=ψ ⁣(z)+πcot ⁣(πz)\psi\!\left(1 - z\right) = \psi\!\left(z\right) + \pi \cot\!\left(\pi z\right)
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ψ(m) ⁣(1z)=(1)m(ψ(m) ⁣(z)+πdmdzmcot ⁣(πz))\psi^{(m)}\!\left(1 - z\right) = {\left(-1\right)}^{m} \left(\psi^{(m)}\!\left(z\right) + \pi \frac{d^{m}}{{d z}^{m}} \cot\!\left(\pi z\right)\right)

Multiplication theorem

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ψ ⁣(nz)=log(n)+1nk=0n1ψ ⁣(z+kn)\psi\!\left(n z\right) = \log(n) + \frac{1}{n} \sum_{k=0}^{n - 1} \psi\!\left(z + \frac{k}{n}\right)
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ψ(m) ⁣(nz)=1nm+1k=0n1ψ(m) ⁣(z+kn)\psi^{(m)}\!\left(n z\right) = \frac{1}{{n}^{m + 1}} \sum_{k=0}^{n - 1} \psi^{(m)}\!\left(z + \frac{k}{n}\right)

Conjugate symmetry

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ψ ⁣(z)=ψ ⁣(z)\psi\!\left(\overline{z}\right) = \overline{\psi\!\left(z\right)}
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ψ(m) ⁣(z)=ψ(m) ⁣(z)\psi^{(m)}\!\left(\overline{z}\right) = \overline{\psi^{(m)}\!\left(z\right)}

Integral representations

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ψ ⁣(z)=0(ettezt1et)dt\psi\!\left(z\right) = \int_{0}^{\infty} \left(\frac{{e}^{-t}}{t} - \frac{{e}^{-z t}}{1 - {e}^{-t}}\right) \, dt
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ψ ⁣(z)=0(et1(1+t)z)1tdt\psi\!\left(z\right) = \int_{0}^{\infty} \left({e}^{-t} - \frac{1}{{\left(1 + t\right)}^{z}}\right) \frac{1}{t} \, dt
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ψ ⁣(z)=γ+011tz11tdt\psi\!\left(z\right) = -\gamma + \int_{0}^{1} \frac{1 - {t}^{z - 1}}{1 - t} \, dt
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ψ ⁣(z)=γ+0etezt1etdt\psi\!\left(z\right) = -\gamma + \int_{0}^{\infty} \frac{{e}^{-t} - {e}^{-z t}}{1 - {e}^{-t}} \, dt
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ψ ⁣(z)=log(z)+0ezt(1t11et)dt\psi\!\left(z\right) = \log(z) + \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{t} - \frac{1}{1 - {e}^{-t}}\right) \, dt
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ψ ⁣(z)=log(z)12z20t(t2+z2)(e2πt1)dt\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - 2 \int_{0}^{\infty} \frac{t}{\left({t}^{2} + {z}^{2}\right) \left({e}^{2 \pi t} - 1\right)} \, dt
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ψ ⁣(z)=log(z)12z0ezt(121t+1et1)dt\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{{e}^{t} - 1}\right) \, dt
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ψ(m) ⁣(z)=(1)m+10tmezt1etdt\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} \int_{0}^{\infty} \frac{{t}^{m} {e}^{-z t}}{1 - {e}^{-t}} \, dt
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ψ(m) ⁣(z)=01tz11tlogm ⁣(t)dt\psi^{(m)}\!\left(z\right) = -\int_{0}^{1} \frac{{t}^{z - 1}}{1 - t} \log^{m}\!\left(t\right) \, dt

Generating functions

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n=1ψ ⁣(n)zn=z(γ+log ⁣(1z))z1\sum_{n=1}^{\infty} \psi\!\left(n\right) {z}^{n} = \frac{z \left(\gamma + \log\!\left(1 - z\right)\right)}{z - 1}
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n=1ψ ⁣(n)znn!=z[dda1F1 ⁣(a,2,z)]a=1γ(ez1)\sum_{n=1}^{\infty} \psi\!\left(n\right) \frac{{z}^{n}}{n !} = z \left[ \frac{d}{d a}\, \,{}_1F_1\!\left(a, 2, z\right) \right]_{a = 1} - \gamma \left({e}^{z} - 1\right)

Finite sums

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k=1nψ ⁣(k)=n(ψ ⁣(n+1)1)\sum_{k=1}^{n} \psi\!\left(k\right) = n \left(\psi\!\left(n + 1\right) - 1\right)
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k=1nψ ⁣(k)e2πrki/n=nlog ⁣(1e2πri/n)\sum_{k=1}^{n} \psi\!\left(k\right) {e}^{2 \pi r k i / n} = n \log\!\left(1 - {e}^{2 \pi r i / n}\right)
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k=1nψ ⁣(k)cos ⁣(2πrkn)=nlog ⁣(2sin ⁣(πrn))\sum_{k=1}^{n} \psi\!\left(k\right) \cos\!\left(\frac{2 \pi r k}{n}\right) = n \log\!\left(2 \sin\!\left(\frac{\pi r}{n}\right)\right)