Inmathematics, thedigamma function is defined as thelogarithmic derivative of thegamma function:^[1][2][3]

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

It is the first of thepolygamma functions. This function isstrictly increasing andstrictly concave on${\displaystyle (0,\mathrm{\infty })}$,^[4] and itasymptotically behaves as^[5]

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

for complex numbers with large modulus (${\displaystyle |z|\to \mathrm{\infty }}$) in thesector for any${\displaystyle \epsilon >0}$.

The digamma function is often denoted as${\displaystyle {\psi }_0(x),{\psi }^{(0)}(x)}$ orϜ^[6] (the uppercase form of the archaic Greek letterdigamma meaningdouble-gamma).

The gamma function obeys the equation

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

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

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

Differentiating both sides with respect toz gives:

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

Since theharmonic numbers are defined for positive integersn as

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

the digamma function is related to them by

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

whereH₀ = 0, andγ is theEuler–Mascheroni constant. For half-integer arguments the digamma function takes the values

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

If the real part ofz is positive then the digamma function has the followingintegral representation due to Gauss:^[7]

${\displaystyle \psi (z)={\int }_0^{\mathrm{\infty }}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)dt.}$

Combining this expression with an integral identity for theEuler–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'sharmonic 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]

${\displaystyle \psi (z)={\int }_0^{\mathrm{\infty }}\left(e^{-t}-\frac{1}{(1+t{)}^z}\right)\frac{dt}{t}.}$

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

${\displaystyle \psi (z)=\log z-\frac{1}{2z}-{\int }_0^{\mathrm{\infty }}\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}\right)e^{-tz}dt.}$

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

${\displaystyle \psi (z)=\log z-\frac{1}{2z}-2{\int }_0^{\mathrm{\infty }}\frac{tdt}{(t^2+z^2)(e^{2\pi t}-1)}.}$

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

${\displaystyle \psi (z)=\frac{1}{\mathrm{\Gamma }(z)}{\int }_0^{\mathrm{\infty }}{t}^{z-1}\ln (t)e^{-t}dt,}$

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

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

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

Here${\displaystyle x_k}$ is thekth zero of${\displaystyle \psi }$ (see below), and${\displaystyle \gamma }$ is theEuler–Mascheroni constant.

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

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]

Equivalently,

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

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

wherep(n) andq(n) are polynomials ofn.

Performingpartial fraction decomposition onu_n in the complex field, in the case when all roots ofq(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,

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}n{u}_n=0,}$

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

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

and

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

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{u}_n=\sum _{n=0}^{\mathrm{\infty }}\sum _{k=1}^m\frac{a_k}{(n+b_k{)}^{r_k}}=\sum _{k=1}^m\frac{(-1{)}^{r_k}}{(r_k-1)!}{a}_k{\psi }^{(r_k-1)}(b_k),}$

provided the series on the left converges.

The digamma has arational zeta series, given by theTaylor series atz = 1. This is

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

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

TheNewton series for the digamma, sometimes referred to asStern series, derived byMoritz Abraham Stern in 1847,^[12][13][14] reads

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

${\displaystyle \psi (s+1)=-\gamma -\frac{1}{m}\sum _{k=1}^{m-1}\frac{m-k}{s+k}-\frac{1}{m}\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^k}{k}\left\{(\genfrac{}{}{0ex}{}{s+m}{k+1})-(\genfrac{}{}{0ex}{}{s}{k+1})\right\},\mathrm{\Re }(s)>-1,}$

wherem = 2, 3, 4, ...^[13]

There exist various series for the digamma containing rational coefficients only for the rational arguments. In particular, the series withGregory's coefficientsG_n is

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

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

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

where(v)_n is therising factorial(v)_n =v(v+1)(v+2) ... (v+n-1),G_n(k) are theGregory coefficients of higher order withG_n(1) =G_n,Γ is thegamma function andζ is theHurwitz zeta function.^[15][13]Similar series with the Cauchy numbers of the second kindC_n reads^[15][13]

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

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

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

whereψ_n(a) are theBernoulli polynomials of the second kind defined by the generatingequation

It may be generalized to

${\displaystyle \psi (v)=\frac{1}{r}\sum _{l=0}^{r-1}\ln (v+a+l)+\frac{1}{r}\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{N}_{n,r}(a)(n-1)!}{(v{)}_n},\mathrm{\Re }(v)>-a,r=1,2,3,\dots }$

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

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

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

and

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

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

The digamma and polygamma functions satisfyreflection formulas similar to that of thegamma function:

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

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

${\displaystyle {\psi }^{″}(-x)-{\psi }^{″}(x)=\frac{2{\pi }^3\cot (\pi x)}{{\sin }^2(\pi x)}+\frac{2}{x^3}}$.

The digamma function satisfies therecurrence relation

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

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

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

whereΔ is theforward difference operator. This satisfies the recurrence relation of a partial sum of theharmonic series, thus implying the formula

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

whereγ is theEuler–Mascheroni constant.

Actually,ψ is the only solution of the functional equation

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

that ismonotonic onR⁺ and satisfiesF(1) = −γ. This fact follows immediately from the uniqueness of theΓ function given its recurrence equation and convexity restriction^{[citation needed]}. This implies the useful difference equation:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We also have^[19]

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

For positive integersr andm (r <m), the digamma function may be expressed in terms ofEuler's constant and a finite number ofelementary functions^[20][21]

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

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

Themultiplication theorem of the${\displaystyle \mathrm{\Gamma }}$-function is equivalent to^[22]

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

The digamma function has the asymptotic expansion

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

whereB_k is thekthBernoulli number andζ is theRiemann 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 anyz, any finite partial sum becomes increasingly accurate asz increases.

The expansion can be found by applying theEuler–Maclaurin formula to the sum^[23]

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

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 ageometric 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:

${\displaystyle \psi (z)=\ln z-\frac{1}{2z}-\sum _{n=1}^N\frac{B_{2n}}{2n{z}^{2n}}+(-1{)}^{N+1}\frac{2}{z^{2N}}{\int }_0^{\mathrm{\infty }}\frac{t^{2N+1}dt}{(t^2+z^2)(e^{2\pi t}-1)}.}$

Whenx > 0, the function

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

is completely monotonic and in particular positive. This is a consequence ofBernstein'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${\displaystyle 1+t\le e^t}$, 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 allx > 0,

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

This recovers a theorem of Horst Alzer.^[24] Alzer also proved that, fors ∈ (0, 1),

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

where${\displaystyle \gamma =-\psi (1)}$ is theEuler–Mascheroni constant.^[25] The constants (${\displaystyle 0.5}$ and${\displaystyle e^{-\gamma }\approx 0.56}$) appearing in these bounds are the best possible.^[26]

Themean value theorem implies the following analog ofGautschi's inequality: Ifx >c, wherec ≈ 1.461 is the unique positive real root of the digamma function, and ifs > 0, then

${\displaystyle \exp \left((1-s)\frac{{\psi }^{\prime }(x+1)}{\psi (x+1)}\right)\le \frac{\psi (x+1)}{\psi (x+s)}\le \exp \left((1-s)\frac{{\psi }^{\prime }(x+s)}{\psi (x+s)}\right).}$

Moreover, equality holds if and only ifs = 1.^[27]

Inspired by theharmonic 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:

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

Equality holds if and only if${\displaystyle x=1}$.^[28]

The asymptotic expansion gives an easy way to computeψ(x) when the real part ofx is large. To computeψ(x) for smallx, the recurrence relation

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

can be used to shift the value ofx to a higher value. Beal^[29] suggests using the above recurrence to shiftx to a value greater than 6 and then applying the above expansion with terms abovex¹⁴ cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).

Asx goes to infinity,ψ(x) gets arbitrarily close to bothln(x −⁠1/2⁠) andlnx. Going down fromx + 1 tox,ψ decreases by⁠1/x⁠,ln(x −⁠1/2⁠) decreases byln(x +⁠1/2⁠) / (x −⁠1/2⁠), which is more than⁠1/x⁠, andlnx decreases byln(1 +⁠1/x⁠), which is less than⁠1/x⁠. From this we see that for any positivex greater than⁠1/2⁠,

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

or, for any positivex,

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

The exponentialexpψ(x) is approximatelyx −⁠1/2⁠ for largex, but gets closer tox at smallx, approaching 0 atx = 0.

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

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

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 forψ, one can derive an asymptotic series forexp(−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 ofexp(−ψ(1 /y)) aty = 0, but it does not converge.^[30] (The function is notanalytic at infinity.) A similar series exists forexp(ψ(x)) which starts with${\displaystyle \exp \psi (x)\sim x-\frac{1}{2}.}$

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

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

Similar in spirit to theLanczos approximation of the${\displaystyle \mathrm{\Gamma }}$-function isSpouge'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${\displaystyle 1\le x\le 3}$ and to evaluate the Chebyshev series there.^[31][32]

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

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

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

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

Apart from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at theimaginary unit the numerical approximationOEIS: A248177

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

The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on thereal axis. The only one on thepositive real axis is the unique minimum of the real-valued gamma function onR⁺ atx₀ =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...

${\displaystyle ⋮}$

Already in 1881,Charles Hermite observed^[33] 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

${\displaystyle x_n\approx -n+\frac{1}{\pi }\arctan \left(\frac{\pi }{\ln n}\right)n\ge 2}$

and using a further term it becomes still better

${\displaystyle x_n\approx -n+\frac{1}{\pi }\arctan \left(\frac{\pi }{\ln n+\frac{1}{8n}}\right)n\ge 1}$

which both spring off the reflection formula via

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

and substitutingψ(x_n) by its not convergent asymptotic expansion. The correct second term of this expansion is⁠1/2n⁠, where the given one works well to approximate roots with smalln.

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][34]

In general, the function

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

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

The following results^[11]

also hold true.

The digamma function appears in the regularization of divergent integrals

${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{dx}{x+a},}$

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

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

Many notable probability distributions use the gamma function in the definition of their probability density or mass functions. Then in statistics when doingmaximum likelihood estimation on models involving such distributions, the digamma function naturally appears when the derivative of the log-likelihood is taken for finding the maxima.

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

• OEISsequence A020759 (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 toOEIS: A200138 psi(1/5) to psi(4/5).

• Implementation in the GNU Scientific library
• C function with the Chebyshev series
• Java implementation with Taylor series

• Gamma and related functions

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