Digamma function

For Barnes' gamma function of two variables, see double gamma function.

The digamma function ${\displaystyle \psi (z)}$,
visualized using domain coloring

Plots of the digamma and the next three polygamma functions along the real line (they are real-valued on the real line)

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ^{[1] [2] [3]}

${\displaystyle \psi (z)={\frac {\mathrm {d} }{\mathrm {d} z}}\ln \Gamma (z)={\frac {\Gamma '(z)}{\Gamma (z)}}.}$

It is the first of the polygamma functions. This function is strictly increasing and strictly concave on ${\displaystyle (0,\infty )}$, ^[4] and it asymptotically behaves as ^[5]

${\displaystyle \psi (z)\sim \ln {z}-{\frac {1}{2z}},}$

for complex numbers with large modulus (${\displaystyle |z|\rightarrow \infty }$) in the sector ${\displaystyle |\arg z|<\pi -\varepsilon }$ with some infinitesimally small positive constant ${\displaystyle \varepsilon }$.

The digamma function is often denoted as ${\displaystyle \psi _{0}(x),\psi ^{(0)}(x)}$ or Ϝ ^[6] (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).

Relation to harmonic numbers

The gamma function obeys the equation

${\displaystyle \Gamma (z+1)=z\Gamma (z).\,}$

Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives:

${\displaystyle \log \Gamma (z+1)=\log(z)+\log \Gamma (z),}$

Differentiating both sides with respect to z gives:

${\displaystyle \psi (z+1)=\psi (z)+{\frac {1}{z}}}$

Since the harmonic numbers are defined for positive integers n as

${\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k}},}$

the digamma function is related to them by

${\displaystyle \psi (n)=H_{n-1}-\gamma ,}$

where H₀ = 0, and γ is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values

${\displaystyle \psi \left(n+{\tfrac {1}{2}}\right)=-\gamma -2\ln 2+\sum _{k=1}^{n}{\frac {2}{2k-1}}=-\gamma -2\ln 2+2H_{2n}-H_{n}.}$

Integral representations

If the real part of z is positive then the digamma function has the following integral representation due to Gauss: ^[7]

${\displaystyle \psi (z)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-zt}}{1-e^{-t}}}\right)\,dt.}$

Combining this expression with an integral identity for the Euler–Mascheroni constant ${\displaystyle \gamma }$ gives:

${\displaystyle \psi (z+1)=-\gamma +\int _{0}^{1}\left({\frac {1-t^{z}}{1-t}}\right)\,dt.}$

The integral is Euler's harmonic number ${\displaystyle H_{z}}$, so the previous formula may also be written

${\displaystyle \psi (z+1)=\psi (1)+H_{z}.}$

A consequence is the following generalization of the recurrence relation:

${\displaystyle \psi (w+1)-\psi (z+1)=H_{w}-H_{z}.}$

An integral representation due to Dirichlet is: ^[7]

${\displaystyle \psi (z)=\int _{0}^{\infty }\left(e^{-t}-{\frac {1}{(1+t)^{z}}}\right)\,{\frac {dt}{t}}.}$

Gauss's integral representation can be manipulated to give the start of the asymptotic expansion of ${\displaystyle \psi }$. ^[8]

${\displaystyle \psi (z)=\log z-{\frac {1}{2z}}-\int _{0}^{\infty }\left({\frac {1}{2}}-{\frac {1}{t}}+{\frac {1}{e^{t}-1}}\right)e^{-tz}\,dt.}$

This formula is also a consequence of Binet's first integral for the gamma function. The integral may be recognized as a Laplace transform.

Binet's second integral for the gamma function gives a different formula for ${\displaystyle \psi }$ which also gives the first few terms of the asymptotic expansion: ^[9]

${\displaystyle \psi (z)=\log z-{\frac {1}{2z}}-2\int _{0}^{\infty }{\frac {t\,dt}{(t^{2}+z^{2})(e^{2\pi t}-1)}}.}$

From the definition of ${\displaystyle \psi }$ and the integral representation of the gamma function, one obtains

${\displaystyle \psi (z)={\frac {1}{\Gamma (z)}}\int _{0}^{\infty }t^{z-1}\ln(t)e^{-t}\,dt,}$

with ${\displaystyle \Re z>0}$. ^[10]

Infinite product representation

The function ${\displaystyle \psi (z)/\Gamma (z)}$ is an entire function, ^[11] and it can be represented by the infinite product

${\displaystyle {\frac {\psi (z)}{\Gamma (z)}}=-e^{2\gamma z}\prod _{k=0}^{\infty }\left(1-{\frac {z}{x_{k}}}\right)e^{\frac {z}{x_{k}}}.}$

Here ${\displaystyle x_{k}}$ is the kth zero of ${\displaystyle \psi }$ (see below), and ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.

Note: This is also equal to ${\displaystyle -{\frac {d}{dz}}{\frac {1}{\Gamma (z)}}}$ due to the definition of the digamma function: ${\displaystyle {\frac {\Gamma '(z)}{\Gamma (z)}}=\psi (z)}$.

Series representation

Series formula

Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16): ^[1]

${\displaystyle {\begin{aligned}\psi (z+1)&=-\gamma +\sum _{n=1}^{\infty }\left({\frac {1}{n}}-{\frac {1}{n+z}}\right),\qquad z\neq -1,-2,-3,\ldots ,\\&=-\gamma +\sum _{n=1}^{\infty }\left({\frac {z}{n(n+z)}}\right),\qquad z\neq -1,-2,-3,\ldots .\end{aligned}}}$

Equivalently,

${\displaystyle {\begin{aligned}\psi (z)&=-\gamma +\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}-{\frac {1}{n+z}}\right),\qquad z\neq 0,-1,-2,\ldots ,\\&=-\gamma +\sum _{n=0}^{\infty }{\frac {z-1}{(n+1)(n+z)}},\qquad z\neq 0,-1,-2,\ldots .\end{aligned}}}$

Evaluation of sums of rational functions

The above identity can be used to evaluate sums of the form

${\displaystyle \sum _{n=0}^{\infty }u_{n}=\sum _{n=0}^{\infty }{\frac {p(n)}{q(n)}},}$

where p(n) and q(n) are polynomials of n.

Performing partial fraction on u_n in the complex field, in the case when all roots of q(n) are simple roots,

${\displaystyle u_{n}={\frac {p(n)}{q(n)}}=\sum _{k=1}^{m}{\frac {a_{k}}{n+b_{k}}}.}$

For the series to converge,

${\displaystyle \lim _{n\to \infty }nu_{n}=0,}$

otherwise the series will be greater than the harmonic series and thus diverge. Hence

${\displaystyle \sum _{k=1}^{m}a_{k}=0,}$

and

${\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }u_{n}&=\sum _{n=0}^{\infty }\sum _{k=1}^{m}{\frac {a_{k}}{n+b_{k}}}\\&=\sum _{n=0}^{\infty }\sum _{k=1}^{m}a_{k}\left({\frac {1}{n+b_{k}}}-{\frac {1}{n+1}}\right)\\&=\sum _{k=1}^{m}\left(a_{k}\sum _{n=0}^{\infty }\left({\frac {1}{n+b_{k}}}-{\frac {1}{n+1}}\right)\right)\\&=-\sum _{k=1}^{m}a_{k}{\big (}\psi (b_{k})+\gamma {\big )}\\&=-\sum _{k=1}^{m}a_{k}\psi (b_{k}).\end{aligned}}}$

With the series expansion of higher rank polygamma function a generalized formula can be given as

${\displaystyle \sum _{n=0}^{\infty }u_{n}=\sum _{n=0}^{\infty }\sum _{k=1}^{m}{\frac {a_{k}}{(n+b_{k})^{r_{k}}}}=\sum _{k=1}^{m}{\frac {(-1)^{r_{k}}}{(r_{k}-1)!}}a_{k}\psi ^{(r_{k}-1)}(b_{k}),}$

provided the series on the left converges.

Taylor series

The digamma has a rational zeta series, given by the Taylor series at z = 1. This is

${\displaystyle \psi (z+1)=-\gamma -\sum _{k=1}^{\infty }(-1)^{k}\,\zeta (k+1)\,z^{k},}$

which converges for |z| < 1. Here, ζ(n) is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function.

Newton series

The Newton series for the digamma, sometimes referred to as Stern series, derived by Moritz Abraham Stern in 1847, ^{[12] [13] [14]} reads

${\displaystyle {\begin{aligned}\psi (s)&=-\gamma +(s-1)-{\frac {(s-1)(s-2)}{2\cdot 2!}}+{\frac {(s-1)(s-2)(s-3)}{3\cdot 3!}}\cdots ,\quad \Re (s)>0,\\&=-\gamma -\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{\binom {s-1}{k}}\cdots ,\quad \Re (s)>0.\end{aligned}}}$

where (^s
_k) is the binomial coefficient. It may also be generalized to

${\displaystyle \psi (s+1)=-\gamma -{\frac {1}{m}}\sum _{k=1}^{m-1}{\frac {m-k}{s+k}}-{\frac {1}{m}}\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}\left\{{\binom {s+m}{k+1}}-{\binom {s}{k+1}}\right\},\qquad \Re (s)>-1,}$

where m = 2, 3, 4, ... ^[13]

Series with Gregory's coefficients, Cauchy numbers and Bernoulli polynomials of the second kind

There exist various series for the digamma containing rational coefficients only for the rational arguments. In particular, the series with Gregory's coefficients G_n is

${\displaystyle \psi (v)=\ln v-\sum _{n=1}^{\infty }{\frac {{\big |}G_{n}{\big |}(n-1)!}{(v)_{n}}},\qquad \Re (v)>0,}$

${\displaystyle \psi (v)=2\ln \Gamma (v)-2v\ln v+2v+2\ln v-\ln 2\pi -2\sum _{n=1}^{\infty }{\frac {{\big |}G_{n}(2){\big |}}{(v)_{n}}}\,(n-1)!,\qquad \Re (v)>0,}$

${\displaystyle \psi (v)=3\ln \Gamma (v)-6\zeta '(-1,v)+3v^{2}\ln {v}-{\frac {3}{2}}v^{2}-6v\ln(v)+3v+3\ln {v}-{\frac {3}{2}}\ln 2\pi +{\frac {1}{2}}-3\sum _{n=1}^{\infty }{\frac {{\big |}G_{n}(3){\big |}}{(v)_{n}}}\,(n-1)!,\qquad \Re (v)>0,}$

where (v)_n is the rising factorial (v)_n = v(v+1)(v+2) ... (v+n-1), G_n(k) are the Gregory coefficients of higher order with G_n(1) = G_n, Γ is the gamma function and ζ is the Hurwitz zeta function. ^{[15] [13]} Similar series with the Cauchy numbers of the second kind C_n reads ^{[15] [13]}

${\displaystyle \psi (v)=\ln(v-1)+\sum _{n=1}^{\infty }{\frac {C_{n}(n-1)!}{(v)_{n}}},\qquad \Re (v)>1,}$

A series with the Bernoulli polynomials of the second kind has the following form ^[13]

${\displaystyle \psi (v)=\ln(v+a)+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)\,(n-1)!}{(v)_{n}}},\qquad \Re (v)>-a,}$

where ψ_n(a) are the Bernoulli polynomials of the second kind defined by the generating equation

${\displaystyle {\frac {z(1+z)^{a}}{\ln(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(a)\,,\qquad |z|<1\,,}$

It may be generalized to

${\displaystyle \psi (v)={\frac {1}{r}}\sum _{l=0}^{r-1}\ln(v+a+l)+{\frac {1}{r}}\sum _{n=1}^{\infty }{\frac {(-1)^{n}N_{n,r}(a)(n-1)!}{(v)_{n}}},\qquad \Re (v)>-a,\quad r=1,2,3,\ldots }$

where the polynomials N_n,r(a) are given by the following generating equation

${\displaystyle {\frac {(1+z)^{a+m}-(1+z)^{a}}{\ln(1+z)}}=\sum _{n=0}^{\infty }N_{n,m}(a)z^{n},\qquad |z|<1,}$

so that N_n,1(a) = ψ_n(a). ^[13] Similar expressions with the logarithm of the gamma function involve these formulas ^[13]

${\displaystyle \psi (v)={\frac {1}{v+a-{\tfrac {1}{2}}}}\left\{\ln \Gamma (v+a)+v-{\frac {1}{2}}\ln 2\pi -{\frac {1}{2}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n+1}(a)}{(v)_{n}}}(n-1)!\right\},\qquad \Re (v)>-a,}$

and

${\displaystyle \psi (v)={\frac {1}{{\tfrac {1}{2}}r+v+a-1}}\left\{\ln \Gamma (v+a)+v-{\frac {1}{2}}\ln 2\pi -{\frac {1}{2}}+{\frac {1}{r}}\sum _{n=0}^{r-2}(r-n-1)\ln(v+a+n)+{\frac {1}{r}}\sum _{n=1}^{\infty }{\frac {(-1)^{n}N_{n+1,r}(a)}{(v)_{n}}}(n-1)!\right\},}$

where ${\displaystyle \Re (v)>-a}$ and ${\displaystyle r=2,3,4,\ldots }$.

Reflection formula

The digamma and polygamma functions satisfy reflection formulas similar to that of the gamma function:

${\displaystyle \psi (1-x)-\psi (x)=\pi \cot \pi x}$.

${\displaystyle \psi '(-x)+\psi '(x)={\frac {\pi ^{2}}{\sin ^{2}(\pi x)}}+{\frac {1}{x^{2}}}}$.

Recurrence formula and characterization

The digamma function satisfies the recurrence relation

${\displaystyle \psi (x+1)=\psi (x)+{\frac {1}{x}}.}$

Thus, it can be said to "telescope" ⁠1/x⁠, for one has

${\displaystyle \Delta [\psi ](x)={\frac {1}{x}}}$

where Δ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula

${\displaystyle \psi (n)=H_{n-1}-\gamma }$

where γ is the Euler–Mascheroni constant.

Actually, ψ is the only solution of the functional equation

${\displaystyle F(x+1)=F(x)+{\frac {1}{x}}}$

that is monotonic on R ⁺ and satisfies F(1) = −γ. This fact follows immediately from the uniqueness of the Γ function given its recurrence equation and convexity restriction. This implies the useful difference equation:

${\displaystyle \psi (x+N)-\psi (x)=\sum _{k=0}^{N-1}{\frac {1}{x+k}}}$

Some finite sums involving the digamma function

There are numerous finite summation formulas for the digamma function. Basic summation formulas, such as

${\displaystyle \sum _{r=1}^{m}\psi \left({\frac {r}{m}}\right)=-m(\gamma +\ln m),}$

${\displaystyle \sum _{r=1}^{m}\psi \left({\frac {r}{m}}\right)\cdot \exp {\dfrac {2\pi rki}{m}}=m\ln \left(1-\exp {\frac {2\pi ki}{m}}\right),\qquad k\in \mathbb {Z} ,\quad m\in \mathbb {N} ,\ k\neq m}$

${\displaystyle \sum _{r=1}^{m-1}\psi \left({\frac {r}{m}}\right)\cdot \cos {\dfrac {2\pi rk}{m}}=m\ln \left(2\sin {\frac {k\pi }{m}}\right)+\gamma ,\qquad k=1,2,\ldots ,m-1}$

${\displaystyle \sum _{r=1}^{m-1}\psi \left({\frac {r}{m}}\right)\cdot \sin {\frac {2\pi rk}{m}}={\frac {\pi }{2}}(2k-m),\qquad k=1,2,\ldots ,m-1}$

are due to Gauss. ^{[16] [17]} More complicated formulas, such as

${\displaystyle \sum _{r=0}^{m-1}\psi \left({\frac {2r+1}{2m}}\right)\cdot \cos {\frac {(2r+1)k\pi }{m}}=m\ln \left(\tan {\frac {\pi k}{2m}}\right),\qquad k=1,2,\ldots ,m-1}$

${\displaystyle \sum _{r=0}^{m-1}\psi \left({\frac {2r+1}{2m}}\right)\cdot \sin {\dfrac {(2r+1)k\pi }{m}}=-{\frac {\pi m}{2}},\qquad k=1,2,\ldots ,m-1}$

${\displaystyle \sum _{r=1}^{m-1}\psi \left({\frac {r}{m}}\right)\cdot \cot {\frac {\pi r}{m}}=-{\frac {\pi (m-1)(m-2)}{6}}}$

${\displaystyle \sum _{r=1}^{m-1}\psi \left({\frac {r}{m}}\right)\cdot {\frac {r}{m}}=-{\frac {\gamma }{2}}(m-1)-{\frac {m}{2}}\ln m-{\frac {\pi }{2}}\sum _{r=1}^{m-1}{\frac {r}{m}}\cdot \cot {\frac {\pi r}{m}}}$

${\displaystyle \sum _{r=1}^{m-1}\psi \left({\frac {r}{m}}\right)\cdot \cos {\dfrac {(2\ell +1)\pi r}{m}}=-{\frac {\pi }{m}}\sum _{r=1}^{m-1}{\frac {r\cdot \sin {\dfrac {2\pi r}{m}}}{\cos {\dfrac {2\pi r}{m}}-\cos {\dfrac {(2\ell +1)\pi }{m}}}},\qquad \ell \in \mathbb {Z} }$

${\displaystyle \sum _{r=1}^{m-1}\psi \left({\frac {r}{m}}\right)\cdot \sin {\dfrac {(2\ell +1)\pi r}{m}}=-(\gamma +\ln 2m)\cot {\frac {(2\ell +1)\pi }{2m}}+\sin {\dfrac {(2\ell +1)\pi }{m}}\sum _{r=1}^{m-1}{\frac {\ln \sin {\dfrac {\pi r}{m}}}{\cos {\dfrac {2\pi r}{m}}-\cos {\dfrac {(2\ell +1)\pi }{m}}}},\qquad \ell \in \mathbb {Z} }$

${\displaystyle \sum _{r=1}^{m-1}\psi ^{2}\left({\frac {r}{m}}\right)=(m-1)\gamma ^{2}+m(2\gamma +\ln 4m)\ln {m}-m(m-1)\ln ^{2}2+{\frac {\pi ^{2}(m^{2}-3m+2)}{12}}+m\sum _{\ell =1}^{m-1}\ln ^{2}\sin {\frac {\pi \ell }{m}}}$

are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014) ^[18] ).

We also have ^[19]

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+...+{\frac {1}{k-1}}-\gamma ={\frac {1}{k}}\sum _{n=0}^{k-1}\psi \left(1+{\frac {n}{k}}\right),k=2,3,...}$

Gauss's digamma theorem

For positive integers r and m (r < m), the digamma function may be expressed in terms of Euler's constant and a finite number of elementary functions ^[20]

${\displaystyle \psi \left({\frac {r}{m}}\right)=-\gamma -\ln(2m)-{\frac {\pi }{2}}\cot \left({\frac {r\pi }{m}}\right)+2\sum _{n=1}^{\left\lfloor {\frac {m-1}{2}}\right\rfloor }\cos \left({\frac {2\pi nr}{m}}\right)\ln \sin \left({\frac {\pi n}{m}}\right)}$

which holds, because of its recurrence equation, for all rational arguments.

Multiplication theorem

The multiplication theorem of the ${\displaystyle \Gamma }$-function is equivalent to ^[21]

${\displaystyle \psi (nz)={\frac {1}{n}}\sum _{k=0}^{n-1}\psi \left(z+{\frac {k}{n}}\right)+\ln n.}$

Asymptotic expansion

The digamma function has the asymptotic expansion

${\displaystyle \psi (z)\sim \ln z+\sum _{n=1}^{\infty }{\frac {\zeta (1-n)}{z^{n}}}=\ln z-\sum _{n=1}^{\infty }{\frac {B_{n}}{nz^{n}}},}$

where B_k is the kth Bernoulli number and ζ is the Riemann zeta function. The first few terms of this expansion are:

${\displaystyle \psi (z)\sim \ln z-{\frac {1}{2z}}-{\frac {1}{12z^{2}}}+{\frac {1}{120z^{4}}}-{\frac {1}{252z^{6}}}+{\frac {1}{240z^{8}}}-{\frac {1}{132z^{10}}}+{\frac {691}{32760z^{12}}}-{\frac {1}{12z^{14}}}+\cdots .}$

Although the infinite sum does not converge for any z, any finite partial sum becomes increasingly accurate as z increases.

The expansion can be found by applying the Euler–Maclaurin formula to the sum ^[22]

${\displaystyle \sum _{n=1}^{\infty }\left({\frac {1}{n}}-{\frac {1}{z+n}}\right)}$

The expansion can also be derived from the integral representation coming from Binet's second integral formula for the gamma function. Expanding ${\displaystyle t/(t^{2}+z^{2})}$ as a geometric series and substituting an integral representation of the Bernoulli numbers leads to the same asymptotic series as above. Furthermore, expanding only finitely many terms of the series gives a formula with an explicit error term:

${\displaystyle \psi (z)=\ln z-{\frac {1}{2z}}-\sum _{n=1}^{N}{\frac {B_{2n}}{2nz^{2n}}}+(-1)^{N+1}{\frac {2}{z^{2N}}}\int _{0}^{\infty }{\frac {t^{2N+1}\,dt}{(t^{2}+z^{2})(e^{2\pi t}-1)}}.}$

Inequalities

When x > 0, the function

${\displaystyle \ln x-{\frac {1}{2x}}-\psi (x)}$

is completely monotonic and in particular positive. This is a consequence of Bernstein's theorem on monotone functions applied to the integral representation coming from Binet's first integral for the gamma function. Additionally, by the convexity inequality ${\displaystyle 1+t\leq e^{t}}$, the integrand in this representation is bounded above by ${\displaystyle e^{-tz}/2}$. Consequently

${\displaystyle {\frac {1}{x}}-\ln x+\psi (x)}$

is also completely monotonic. It follows that, for all x > 0,

${\displaystyle \ln x-{\frac {1}{x}}\leq \psi (x)\leq \ln x-{\frac {1}{2x}}.}$

This recovers a theorem of Horst Alzer. ^[23] Alzer also proved that, for s ∈ (0, 1),

${\displaystyle {\frac {1-s}{x+s}}<\psi (x+1)-\psi (x+s),}$

Related bounds were obtained by Elezovic, Giordano, and Pecaric, who proved that, for x > 0 ,

${\displaystyle \ln(x+{\tfrac {1}{2}})-{\frac {1}{x}}<\psi (x)<\ln(x+e^{-\gamma })-{\frac {1}{x}},}$

where ${\displaystyle \gamma =-\psi (1)}$ is the Euler–Mascheroni constant. ^[24] The constants (${\displaystyle 0.5}$ and ${\displaystyle e^{-\gamma }\approx 0.56}$) appearing in these bounds are the best possible. ^[25]

The mean value theorem implies the following analog of Gautschi's inequality: If x > c, where c ≈ 1.461 is the unique positive real root of the digamma function, and if s > 0, then

${\displaystyle \exp \left((1-s){\frac {\psi '(x+1)}{\psi (x+1)}}\right)\leq {\frac {\psi (x+1)}{\psi (x+s)}}\leq \exp \left((1-s){\frac {\psi '(x+s)}{\psi (x+s)}}\right).}$

Moreover, equality holds if and only if s = 1. ^[26]

Inspired by the harmonic mean value inequality for the classical gamma function, Horzt Alzer and Graham Jameson proved, among other things, a harmonic mean-value inequality for the digamma function:

${\displaystyle -\gamma \leq {\frac {2\psi (x)\psi ({\frac {1}{x}})}{\psi (x)+\psi ({\frac {1}{x}})}}}$ for ${\displaystyle x>0}$

Equality holds if and only if ${\displaystyle x=1}$. ^[27]

Computation and approximation

The asymptotic expansion gives an easy way to compute ψ(x) when the real part of x is large. To compute ψ(x) for small x, the recurrence relation

${\displaystyle \psi (x+1)={\frac {1}{x}}+\psi (x)}$

can be used to shift the value of x to a higher value. Beal ^[28] suggests using the above recurrence to shift x to a value greater than 6 and then applying the above expansion with terms above x¹⁴ cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).

As x goes to infinity, ψ(x) gets arbitrarily close to both ln(x − ⁠1/2⁠) and ln x. Going down from x + 1 to x, ψ decreases by ⁠1/x⁠, ln(x − ⁠1/2⁠) decreases by ln(x + ⁠1/2⁠) / (x − ⁠1/2⁠), which is more than ⁠1/x⁠, and ln x decreases by ln(1 + ⁠1/x⁠), which is less than ⁠1/x⁠. From this we see that for any positive x greater than ⁠1/2⁠,

${\displaystyle \psi (x)\in \left(\ln \left(x-{\tfrac {1}{2}}\right),\ln x\right)}$

or, for any positive x,

${\displaystyle \exp \psi (x)\in \left(x-{\tfrac {1}{2}},x\right).}$

The exponential exp ψ(x) is approximately x − ⁠1/2⁠ for large x, but gets closer to x at small x, approaching 0 at x = 0.

For x < 1, we can calculate limits based on the fact that between 1 and 2, ψ(x) ∈ [−γ, 1 − γ], so

${\displaystyle \psi (x)\in \left(-{\frac {1}{x}}-\gamma ,1-{\frac {1}{x}}-\gamma \right),\quad x\in (0,1)}$

or

${\displaystyle \exp \psi (x)\in \left(\exp \left(-{\frac {1}{x}}-\gamma \right),e\exp \left(-{\frac {1}{x}}-\gamma \right)\right).}$

From the above asymptotic series for ψ, one can derive an asymptotic series for exp(−ψ(x)). The series matches the overall behaviour well, that is, it behaves asymptotically as it should for large arguments, and has a zero of unbounded multiplicity at the origin too.

${\displaystyle {\frac {1}{\exp \psi (x)}}\sim {\frac {1}{x}}+{\frac {1}{2\cdot x^{2}}}+{\frac {5}{4\cdot 3!\cdot x^{3}}}+{\frac {3}{2\cdot 4!\cdot x^{4}}}+{\frac {47}{48\cdot 5!\cdot x^{5}}}-{\frac {5}{16\cdot 6!\cdot x^{6}}}+\cdots }$

This is similar to a Taylor expansion of exp(−ψ(1 / y)) at y = 0, but it does not converge. ^[29] (The function is not analytic at infinity.) A similar series exists for exp(ψ(x)) which starts with ${\displaystyle \exp \psi (x)\sim x-{\frac {1}{2}}.}$

If one calculates the asymptotic series for ψ(x+1/2) it turns out that there are no odd powers of x (there is no x⁻¹ term). This leads to the following asymptotic expansion, which saves computing terms of even order.

${\displaystyle \exp \psi \left(x+{\tfrac {1}{2}}\right)\sim x+{\frac {1}{4!\cdot x}}-{\frac {37}{8\cdot 6!\cdot x^{3}}}+{\frac {10313}{72\cdot 8!\cdot x^{5}}}-{\frac {5509121}{384\cdot 10!\cdot x^{7}}}+\cdots }$

Similar in spirit to the Lanczos approximation of the ${\displaystyle \Gamma }$-function is Spouge's approximation.

Another alternative is to use the recurrence relation or the multiplication formula to shift the argument of ${\displaystyle \psi (x)}$ into the range ${\displaystyle 1\leq x\leq 3}$ and to evaluate the Chebyshev series there. ^{[30] [31]}

Special values

The digamma function has values in closed form for rational numbers, as a result of Gauss's digamma theorem. Some are listed below:

${\displaystyle {\begin{aligned}\psi (1)&=-\gamma \\\psi \left({\tfrac {1}{2}}\right)&=-2\ln {2}-\gamma \\\psi \left({\tfrac {1}{3}}\right)&=-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3\ln {3}}{2}}-\gamma \\\psi \left({\tfrac {1}{4}}\right)&=-{\frac {\pi }{2}}-3\ln {2}-\gamma \\\psi \left({\tfrac {1}{6}}\right)&=-{\frac {\pi {\sqrt {3}}}{2}}-2\ln {2}-{\frac {3\ln {3}}{2}}-\gamma \\\psi \left({\tfrac {1}{8}}\right)&=-{\frac {\pi }{2}}-4\ln {2}-{\frac {\pi +\ln \left({\sqrt {2}}+1\right)-\ln \left({\sqrt {2}}-1\right)}{\sqrt {2}}}-\gamma .\end{aligned}}}$

Moreover, by taking the logarithmic derivative of ${\displaystyle |\Gamma (bi)|^{2}}$ or ${\displaystyle |\Gamma ({\tfrac {1}{2}}+bi)|^{2}}$ where ${\displaystyle b}$ is real-valued, it can easily be deduced that

${\displaystyle \operatorname {Im} \psi (bi)={\frac {1}{2b}}+{\frac {\pi }{2}}\coth(\pi b),}$

${\displaystyle \operatorname {Im} \psi ({\tfrac {1}{2}}+bi)={\frac {\pi }{2}}\tanh(\pi b).}$

Apart from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at the imaginary unit the numerical approximation

${\displaystyle \operatorname {Re} \psi (i)=-\gamma -\sum _{n=0}^{\infty }{\frac {n-1}{n^{3}+n^{2}+n+1}}\approx 0.09465.}$

Roots of the digamma function

The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis. The only one on the positive real axis is the unique minimum of the real-valued gamma function on R ⁺ at x₀ = 1.46163214496836234126.... All others occur single between the poles on the negative axis:

x₁ = −0.50408300826445540925...

x₂ = −1.57349847316239045877...

x₃ = −2.61072086844414465000...

x₄ = −3.63529336643690109783...

${\displaystyle \vdots }$

Already in 1881, Charles Hermite observed ^[32] that

${\displaystyle x_{n}=-n+{\frac {1}{\ln n}}+O\left({\frac {1}{(\ln n)^{2}}}\right)}$

holds asymptotically. A better approximation of the location of the roots is given by

${\displaystyle x_{n}\approx -n+{\frac {1}{\pi }}\arctan \left({\frac {\pi }{\ln n}}\right)\qquad n\geq 2}$

and using a further term it becomes still better

${\displaystyle x_{n}\approx -n+{\frac {1}{\pi }}\arctan \left({\frac {\pi }{\ln n+{\frac {1}{8n}}}}\right)\qquad n\geq 1}$

which both spring off the reflection formula via

${\displaystyle 0=\psi (1-x_{n})=\psi (x_{n})+{\frac {\pi }{\tan \pi x_{n}}}}$

and substituting ψ(x_n) by its not convergent asymptotic expansion. The correct second term of this expansion is ⁠1/2n⁠, where the given one works well to approximate roots with small n.

Another improvement of Hermite's formula can be given: ^[11]

${\displaystyle x_{n}=-n+{\frac {1}{\log n}}-{\frac {1}{2n(\log n)^{2}}}+O\left({\frac {1}{n^{2}(\log n)^{2}}}\right).}$

Regarding the zeros, the following infinite sum identities were recently proved by István Mező and Michael Hoffman ^{[11] [33]}

${\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }{\frac {1}{x_{n}^{2}}}&=\gamma ^{2}+{\frac {\pi ^{2}}{2}},\\\sum _{n=0}^{\infty }{\frac {1}{x_{n}^{3}}}&=-4\zeta (3)-\gamma ^{3}-{\frac {\gamma \pi ^{2}}{2}},\\\sum _{n=0}^{\infty }{\frac {1}{x_{n}^{4}}}&=\gamma ^{4}+{\frac {\pi ^{4}}{9}}+{\frac {2}{3}}\gamma ^{2}\pi ^{2}+4\gamma \zeta (3).\end{aligned}}}$

In general, the function

${\displaystyle Z(k)=\sum _{n=0}^{\infty }{\frac {1}{x_{n}^{k}}}}$

can be determined and it is studied in detail by the cited authors.

The following results ^[11]

${\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }{\frac {1}{x_{n}^{2}+x_{n}}}&=-2,\\\sum _{n=0}^{\infty }{\frac {1}{x_{n}^{2}-x_{n}}}&=\gamma +{\frac {\pi ^{2}}{6\gamma }}\end{aligned}}}$

also hold true.

Regularization

The digamma function appears in the regularization of divergent integrals

${\displaystyle \int _{0}^{\infty }{\frac {dx}{x+a}},}$

this integral can be approximated by a divergent general Harmonic series, but the following value can be attached to the series

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n+a}}=-\psi (a).}$

See also

• Polygamma function
• Trigamma function
• Chebyshev expansions of the digamma function in Wimp, Jet (1961). "Polynomial approximations to integral transforms". Math. Comp. 15 (74): 174–178. doi: 10.1090/S0025-5718-61-99221-3 .

References

1. Abramowitz, M.; Stegun, I. A., eds. (1972). "6.3 psi (Digamma) Function.". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (10th ed.). New York: Dover. pp. 258–259.
2. "NIST. Digital Library of Mathematical Functions (DLMF), Chapter 5".
3. Weisstein, Eric W. "Digamma function". MathWorld .
4. Alzer, Horst; Jameson, Graham (2017). "A harmonic mean inequality for the digamma function and related results" (PDF). Rendiconti del Seminario Matematico della Università di Padova. 137: 203–209. doi:10.4171/RSMUP/137-10.
5. "NIST. Digital Library of Mathematical Functions (DLMF), 5.11".
6. Pairman, Eleanor (1919). Tables of the Digamma and Trigamma Functions. Cambridge University Press. p. 5.
7. Whittaker and Watson, 12.3.
8. Whittaker and Watson, 12.31.
9. Whittaker and Watson, 12.32, example.
10. "NIST. Digital Library of Mathematical Functions (DLMF), 5.9".
11. Mező, István; Hoffman, Michael E. (2017). "Zeros of the digamma function and its Barnes G-function analogue". Integral Transforms and Special Functions. 28 (11): 846–858. doi:10.1080/10652469.2017.1376193. S2CID   126115156.
12. Nörlund, N. E. (1924). Vorlesungen über Differenzenrechnung. Berlin: Springer.
13. Blagouchine, Ia. V. (2018). "Three Notes on Ser's and Hasse's Representations for the Zeta-functions" (PDF). INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A: 1–45. arXiv: 1606.02044 . Bibcode:2016arXiv160602044B.
14. "Leonhard Euler's Integral: An Historical Profile of the Gamma Function" (PDF). Archived (PDF) from the original on 2014-09-12. Retrieved 11 April 2022.
15. Blagouchine, Ia. V. (2016). "Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π⁻¹". Journal of Mathematical Analysis and Applications. 442: 404–434. arXiv: 1408.3902 . Bibcode:2014arXiv1408.3902B. doi:10.1016/J.JMAA.2016.04.032. S2CID   119661147.
16. R. Campbell. Les intégrales eulériennes et leurs applications, Dunod, Paris, 1966.
17. H.M. Srivastava and J. Choi. Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, the Netherlands, 2001.
18. Blagouchine, Iaroslav V. (2014). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148: 537–592. arXiv: 1401.3724 . doi:10.1016/j.jnt.2014.08.009.
19. Classical topi s in complex function theorey. p. 46.
20. Choi, Junesang; Cvijovic, Djurdje (2007). "Values of the polygamma functions at rational arguments". Journal of Physics A. 40 (50): 15019. Bibcode:2007JPhA...4015019C. doi:10.1088/1751-8113/40/50/007. S2CID   118527596.
21. Gradshteyn, I. S.; Ryzhik, I. M. (2015). "8.365.5". Table of integrals, series and products. Elsevier Science. ISBN   978-0-12-384933-5. LCCN   2014010276.
22. Bernardo, José M. (1976). "Algorithm AS 103 psi(digamma function) computation" (PDF). Applied Statistics. 25: 315–317. doi:10.2307/2347257. JSTOR   2347257.
23. Alzer, Horst (1997). "On Some Inequalities for the Gamma and Psi Functions" (PDF). Mathematics of Computation. 66 (217): 373–389. doi:10.1090/S0025-5718-97-00807-7. JSTOR   2153660.
24. Elezović, Neven; Giordano, Carla; Pečarić, Josip (2000). "The best bounds in Gautschi's inequality". Mathematical Inequalities & Applications (2): 239–252. doi: 10.7153/MIA-03-26 .
25. Guo, Bai-Ni; Qi, Feng (2014). "Sharp inequalities for the psi function and harmonic numbers". Analysis. 34 (2). arXiv: 0902.2524 . doi:10.1515/anly-2014-0001. S2CID   16909853.
26. Laforgia, Andrea; Natalini, Pierpaolo (2013). "Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities". Journal of Mathematical Analysis and Applications. 407 (2): 495–504. doi: 10.1016/j.jmaa.2013.05.045 .
27. Alzer, Horst; Jameson, Graham (2017). "A harmonic mean inequality for the digamma function and related results" (PDF). Rendiconti del Seminario Matematico della Università di Padova . 70 (201): 203–209. doi:10.4171/RSMUP/137-10. ISSN   0041-8994. LCCN   50046633. OCLC   01761704. S2CID   41966777.
28. Beal, Matthew J. (2003). Variational Algorithms for Approximate Bayesian Inference (PDF) (PhD thesis). The Gatsby Computational Neuroscience Unit, University College London. pp. 265–266.
29. If it converged to a function f(y) then ln(f(y) / y) would have the same Maclaurin series as ln(1 / y) − φ(1 / y). But this does not converge because the series given earlier for φ(x) does not converge.
30. Wimp, Jet (1961). "Polynomial approximations to integral transforms". Math. Comp. 15 (74): 174–178. doi:10.1090/S0025-5718-61-99221-3. JSTOR   2004225.
31. Mathar, R. J. (2004). "Chebyshev series expansion of inverse polynomials". Journal of Computational and Applied Mathematics. 196 (2): 596–607. arXiv: math/0403344 . doi:10.1016/j.cam.2005.10.013. App. E
32. Hermite, Charles (1881). "Sur l'intégrale Eulérienne de seconde espéce". Journal für die reine und angewandte Mathematik (90): 332–338. doi:10.1515/crll.1881.90.332. S2CID   118866486.
33. Mező, István (2014). "A note on the zeros and local extrema of Digamma related functions". arXiv: 1409.2971 [math.CV].

External links

• OEIS sequenceA020759(Decimal expansion of (-1)*Gamma'(1/2)/Gamma(1/2) where Gamma(x) denotes the Gamma function)—psi(1/2)

OEIS:  A047787 psi(1/3), OEIS:  A200064 psi(2/3), OEIS:  A020777 psi(1/4), OEIS:  A200134 psi(3/4), OEIS:  A200135 to OEIS:  A200138 psi(1/5) to psi(4/5).