Digamma function

The gamma function obeys the equation

:$\backslash Gamma(z+1)=z\backslash Gamma(z).\; \backslash ,$

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

:$\backslash log\; \backslash Gamma(z+1)=\backslash log(z)+\backslash log\; \backslash Gamma(z),$

Differentiating both sides with respect to {{mvar|z}} gives:

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

Since the harmonic numbers are defined for positive integers {{mvar|n}} as

:$H\_n=\backslash sum\_\{k=1\}^n\; \backslash frac\; 1\; k,$

the digamma function is related to them by

:$\backslash psi(n)=H\_\{n-1\}-\backslash gamma,$

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

:$\backslash psi\; \backslash left(n+\backslash tfrac12\backslash right)=-\backslash gamma-2\backslash ln\; 2\; +\backslash sum\_\{k=1\}^n\; \backslash frac\; 2\; \{2k-1\}\; =\; -\backslash gamma-2\backslash ln\; 2\; +\; 2H\_\{2n\}-H\_n.$

If the real part of {{mvar|z}} is positive then the digamma function has the following integral representation due to Gauss:Whittaker and Watson, 12.3.

:$\backslash psi(z)\; =\; \backslash int\_0^\backslash infty\; \backslash left(\backslash frac\{e^\{-t\}\}\{t\}\; -\; \backslash frac\{e^\{-zt\}\}\{1-e^\{-t\}\}\backslash right)\backslash ,dt.$

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

:$\backslash psi(z\; +\; 1)\; =\; -\backslash gamma\; +\; \backslash int\_0^1\; \backslash left(\backslash frac\{1-t^z\}\{1-t\}\backslash right)\backslash ,dt.$

The integral is Euler's harmonic number $H\_z$, so the previous formula may also be written

:$\backslash psi(z\; +\; 1)\; =\; \backslash psi(1)\; +\; H\_z.$

A consequence is the following generalization of the recurrence relation:

:$\backslash psi(w\; +\; 1)\; -\; \backslash psi(z\; +\; 1)\; =\; H\_w\; -\; H\_z.$

An integral representation due to Dirichlet is:

:$\backslash psi(z)\; =\; \backslash int\_0^\backslash infty\; \backslash left(e^\{-t\}\; -\; \backslash frac\{1\}\{(1\; +\; t)^z\}\backslash right)\backslash ,\backslash frac\{dt\}\{t\}.$

Gauss's integral representation can be manipulated to give the start of the asymptotic expansion of $\backslash psi$.Whittaker and Watson, 12.31.

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

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 $\backslash psi$ which also gives the first few terms of the asymptotic expansion:Whittaker and Watson, 12.32, example.

:$\backslash psi(z)\; =\; \backslash log\; z\; -\; \backslash frac\{1\}\{2z\}\; -\; 2\backslash int\_0^\backslash infty\; \backslash frac\{t\backslash ,dt\}\{(t^2\; +\; z^2)(e^\{2\backslash pi\; t\}\; -\; 1)\}.$

From the definition of $\backslash psi$ and the integral representation of the gamma function, one obtains

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

with $\backslash Re\; z\; >\; 0$.{{cite web | url=https://dlmf.nist.gov/5.9 | title=NIST. Digital Library of Mathematical Functions (DLMF), 5.9.}}

The function $\backslash psi(z)/\backslash Gamma(z)$ is an entire function, and it can be represented by the infinite product

:$$

\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 $x\_k$ is the kth zero of $\backslash psi$ (see below), and $\backslash gamma$ is the Euler–Mascheroni constant.

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

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):

:$\backslash begin\{align\}$

\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{align}

Equivalently,

:$\backslash begin\{align\}$

\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{align}

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

: $\backslash sum\_\{n=0\}^\backslash infty\; u\_n=\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{p(n)\}\{q(n)\},$

where {{math|p(n)}} and {{math|q(n)}} are polynomials of {{mvar|n}}.

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

: $u\_n=\backslash frac\{p(n)\}\{q(n)\}=\backslash sum\_\{k=1\}^m\; \backslash frac\{a\_k\}\{n+b\_k\}.$

For the series to converge,

:$\backslash lim\_\{n\backslash to\backslash infty\}\; nu\_n=0,$

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

:$\backslash sum\_\{k=1\}^m\; a\_k=0,$

and

:$\backslash begin\{align\}$

\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{align}

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

:$\backslash sum\_\{n=0\}^\backslash infty\; u\_n=\backslash sum\_\{n=0\}^\backslash infty\backslash sum\_\{k=1\}^m\; \backslash frac\{a\_k\}\{(n+b\_k)^\{r\_k\}\}=\backslash sum\_\{k=1\}^m\; \backslash frac\{(-1)^\{r\_k\}\}\{(r\_k-1)!\}a\_k\backslash psi^\{(r\_k-1)\}(b\_k),$

provided the series on the left converges.

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

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

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

The Newton series for the digamma, sometimes referred to as Stern series, derived by Moritz Abraham Stern in 1847,{{cite book|first = N. E.|last = Nörlund|author-link = Niels Erik Nörlund|year = 1924|title = Vorlesungen über Differenzenrechnung|publisher =Springer|location = Berlin}}{{cite journal

| last = Blagouchine | first = Ia. V.

| arxiv = 1606.02044

| url = http://math.colgate.edu/~integers/sjs3/sjs3.pdf

| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory

| pages = 1–45

| title = Three Notes on Ser's and Hasse's Representations for the Zeta-functions

| volume = 18A

| year = 2018| doi = 10.5281/zenodo.10581385

| bibcode = 2016arXiv160602044B}}{{cite web |title=Leonhard Euler's Integral: An Historical Profile of the Gamma Function |url=https://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Davis.pdf |url-status=live |archive-url=https://web.archive.org/web/20140912213629/http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Davis.pdf |archive-date=2014-09-12 |access-date=11 April 2022}} reads

:$\backslash begin\{align\}$

\psi(s)

&= -\gamma + (s-1) - \frac{(s-1)(s-2)}{2\cdot2!} + \frac{(s-1)(s-2)(s-3)}{3\cdot3!}\cdots,\quad\Re(s)> 0, \\

&= -\gamma - \sum_{k=1}^\infty \frac{(-1)^k}{k} \binom{s-1}{k}\cdots,\quad\Re(s)> 0.

\end{align}

where {{math|({{su|p=s|b=k}})}} is the binomial coefficient. It may also be generalized to

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

where {{math|m {{=}} 2, 3, 4, ...}}

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

:$$

\psi(v) =\ln v- \sum_{n=1}^\infty\frac{\big| G_{n}\big|(n-1)!}{(v)_{n}},\qquad

\Re (v) >0,

: $$

\psi(v) =2\ln\Gamma(v) - 2v\ln v + 2v +2\ln v -\ln2\pi - 2\sum_{n=1}^\infty\frac{\big|G_{n}(2)\big|}{(v)_{n}}\,(n-1)! ,\qquad

\Re (v) >0,

: $$

\psi(v) =3\ln\Gamma(v) - 6\zeta'(-1,v) + 3v^2\ln{v} - \frac32 v^2 - 6v\ln(v)+ 3 v+3\ln{v} - \frac32\ln2\pi + \frac12

- 3\sum_{n=1}^\infty\frac{\big| G_{n}(3) \big|}{(v)_{n}}\,(n-1)! ,\qquad

\Re (v) >0,

where {{math|(v)_n}} is the rising factorial {{math|(v)_n {{=}}

v(v+1)(v+2) ... (v+n-1)}}, {{math|G_n(k)}} are the Gregory coefficients of higher order with {{math|G_n(1) {{=}} G_n}}, {{math|Γ}} is the gamma function and {{math|ζ}} is the Hurwitz zeta function.{{cite journal

| last = Blagouchine | first = Ia. V.

| arxiv = 1408.3902

| journal = Journal of Mathematical Analysis and Applications

| pages = 404–434

| title = Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to {{math|π⁻¹}}

| volume = 442

| year = 2016| bibcode = 2014arXiv1408.3902B

| doi = 10.1016/J.JMAA.2016.04.032

| s2cid = 119661147

}}

Similar series with the Cauchy numbers of the second kind {{math|C_n}} reads

:$$

\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

:$$

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

where {{math|ψ_n(a)}} are the Bernoulli polynomials of the second kind defined by the generating

equation

: $$

\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

: $$

\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 {{math|N_n,r(a)}} are given by the following generating equation

: $$

\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 {{math|N_n,1(a) {{=}} ψ_n(a)}}. Similar expressions with the logarithm of the gamma function involve these formulas

: $$

\psi(v)= \frac{1}{v+a-\tfrac12}\left\{\ln\Gamma(v+a) + v - \frac12\ln2\pi - \frac12 + \sum_{n=1}^\infty\frac{(-1)^n \psi_{n+1}(a)}{(v)_{n}}(n-1)!\right\},\qquad \Re(v)>-a,

and

: $$

\psi(v)= \frac{1}{\tfrac{1}{2}r+v+a-1}\left\{\ln\Gamma(v+a) + v - \frac12\ln2\pi - \frac12 + \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 $\backslash Re(v)>-a$ and $r=2,3,4,\backslash ldots$.

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

:$\backslash psi(1-x)-\backslash psi(x)=\backslash pi\; \backslash cot\; \backslash pi\; x$.

:$\backslash psi\text{'}(-x)+\backslash psi\text{'}(x)\; =\; \backslash frac\{\backslash pi^2\}\{\backslash sin^2(\backslash pi\; x)\}+\backslash frac\{1\}\{x^2\}$.

The digamma function satisfies the recurrence relation

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

Thus, it can be said to "telescope" {{math|{{sfrac|1|x}}}}, for one has

:$\backslash Delta\; [\backslash psi](x)=\backslash frac\{1\}\{x\}$

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

:$\backslash psi(n)=H\_\{n-1\}-\backslash gamma$

where {{mvar|γ}} is the Euler–Mascheroni constant.

Actually, {{mvar|ψ}} is the only solution of the functional equation

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

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

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

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

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

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

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

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

are due to Gauss.R. Campbell. Les intégrales eulériennes et leurs applications, Dunod, Paris, 1966.H.M. Srivastava and J. Choi. Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, the Netherlands, 2001. More complicated formulas, such as

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

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

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

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

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

:$\backslash sum\_\{r=1\}^\{m-1\}\backslash psi\; \backslash left(\backslash frac\{r\}\{m\}\backslash right)\; \backslash cdot\backslash sin\backslash dfrac\{(2\backslash ell+1)\backslash pi\; r\}\{m\}=-(\backslash gamma+\backslash ln2m)\backslash cot\backslash frac\{(2\backslash ell+1)\backslash pi\}\{2m\}\; +\; \backslash sin\backslash dfrac\{(2\backslash ell+1)\backslash pi\; \}\{m\}\backslash sum\_\{r=1\}^\{m-1\}\; \backslash frac\{\backslash ln\backslash sin\backslash dfrac\{\backslash pi\; r\}\{m\}\}\; \{\backslash cos\backslash dfrac\{2\backslash pi\; r\}\{m\}\; -\backslash cos\backslash dfrac\{(2\backslash ell+1)\backslash pi\; \}\{m\}\; \}\; ,\; \backslash qquad\; \backslash ell\backslash in\backslash mathbb\{Z\}$

:$\backslash sum\_\{r=1\}^\{m-1\}\; \backslash psi^2\backslash left(\backslash frac\{r\}\{m\}\backslash right)=\; (m-1)\backslash gamma^2\; +\; m(2\backslash gamma+\backslash ln4m)\backslash ln\{m\}\; -m(m-1)\backslash ln^2\; 2\; +\backslash frac\{\backslash pi^2\; (m^2-3m+2)\}\{12\}\; +m\backslash sum\_\{\backslash ell=1\}^\{\; m-1\; \}\; \backslash ln^2\; \backslash sin\backslash frac\{\backslash pi\backslash ell\}\{m\}$

are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014){{cite journal|doi=10.1016/j.jnt.2014.08.009 |first=Iaroslav V. |last=Blagouchine |title=A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations |journal=Journal of Number Theory |volume=148 |pages=537–592 |date=2014 |arxiv=1401.3724}}).

We also have {{cite book |title=Classical topi s in complex function theorey |pages=46}}

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

For positive integers {{mvar|r}} and {{mvar|m}} ({{math|r < m}}), the digamma function may be expressed in terms of Euler's constant and a finite number of elementary functions{{cite journal|first1=Junesang|last1=Choi|first2=Djurdje|last2=Cvijovic|title=Values of the polygamma functions at rational arguments|journal=Journal of Physics A|year=2007|volume=40|pages=15019|doi=10.1088/1751-8113/40/50/007|number=50|bibcode=2007JPhA...4015019C |s2cid=118527596 }}

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

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

The multiplication theorem of the $\backslash Gamma$-function is equivalent to{{cite book|title=Table of integrals, series and products|first1=I. S. |last1=Gradshteyn|first2=I. M. |last2=Ryzhik|year=2015|isbn=978-0-12-384933-5|chapter=8.365.5|publisher=Elsevier Science |lccn=2014010276}}

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

The digamma function has the asymptotic expansion

:$\backslash psi(z)\; \backslash sim\; \backslash ln\; z\; +\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash zeta(1-n)\}\{z^n\}\; =\; \backslash ln\; z\; -\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{B\_n\}\{nz^n\},$

where {{mvar|B_k}} is the {{mvar|k}}th Bernoulli number and {{mvar|ζ}} is the Riemann zeta function. The first few terms of this expansion are:

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

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

The expansion can be found by applying the Euler–Maclaurin formula to the sum{{cite journal| url=http://www.uv.es/~bernardo/1976AppStatist.pdf|first1=José M.|last1= Bernardo|title= Algorithm AS 103 psi(digamma function) computation|year=1976|journal=Applied Statistics|volume=25|pages=315–317|doi=10.2307/2347257|jstor=2347257}}

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

The expansion can also be derived from the integral representation coming from Binet's second integral formula for the gamma function. Expanding $t\; /\; (t^2\; +\; z^2)$ 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:

:$\backslash psi(z)\; =\; \backslash ln\; z\; -\; \backslash frac\{1\}\{2z\}\; -\; \backslash sum\_\{n=1\}^N\; \backslash frac\{B\_\{2n\}\}\{2nz^\{2n\}\}\; +\; (-1)^\{N+1\}\backslash frac\{2\}\{z^\{2N\}\}\; \backslash int\_0^\backslash infty\; \backslash frac\{t^\{2N+1\}\backslash ,dt\}\{(t^2\; +\; z^2)(e^\{2\backslash pi\; t\}\; -\; 1)\}.$

When {{math|x > 0}}, the function

:$\backslash ln\; x\; -\; \backslash frac\{1\}\{2x\}\; -\; \backslash psi(x)$

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 $1\; +\; t\; \backslash le\; e^t$, the integrand in this representation is bounded above by $e^\{-tz\}/2$. {{not a typo|Consequently}}

:$\backslash frac\{1\}\{x\}\; -\; \backslash ln\; x\; +\; \backslash psi(x)$

is also completely monotonic. It follows that, for all {{math|x > 0}},

:$\backslash ln\; x\; -\; \backslash frac\{1\}\{x\}\; \backslash le\; \backslash psi(x)\; \backslash le\; \backslash ln\; x\; -\; \backslash frac\{1\}\{2x\}.$

This recovers a theorem of Horst Alzer.{{cite journal |jstor=2153660 |url=https://www.ams.org/journals/mcom/1997-66-217/S0025-5718-97-00807-7/S0025-5718-97-00807-7.pdf|title=On Some Inequalities for the Gamma and Psi Functions |last1=Alzer |first1=Horst |journal=Mathematics of Computation |year=1997 |volume=66 |issue=217 |pages=373–389 |doi=10.1090/S0025-5718-97-00807-7 }} Alzer also proved that, for {{math|s ∈ (0, 1)}},

:

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

:

where $\backslash gamma=-\backslash psi(1)$ is the Euler–Mascheroni constant.{{cite journal |doi=10.7153/MIA-03-26|title=The best bounds in Gautschi's inequality |year=2000 |last1=Elezović |first1=Neven |last2=Giordano |first2=Carla |last3=Pečarić |first3=Josip |journal=Mathematical Inequalities & Applications |issue=2 |pages=239–252 |doi-access=free }} The constants ($0.5$ and $e^\{-\backslash gamma\}\backslash approx0.56$) appearing in these bounds are the best possible.{{cite journal | arxiv=0902.2524 | doi=10.1515/anly-2014-0001 | title=Sharp inequalities for the psi function and harmonic numbers | year=2014 | last1=Guo | first1=Bai-Ni | last2=Qi | first2=Feng | journal=Analysis | volume=34 | issue=2 | s2cid=16909853 }}

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

:$\backslash exp\backslash left((1\; -\; s)\backslash frac\{\backslash psi\text{'}(x\; +\; 1)\}\{\backslash psi(x\; +\; 1)\}\backslash right)\; \backslash le\; \backslash frac\{\backslash psi(x\; +\; 1)\}\{\backslash psi(x\; +\; s)\}\; \backslash le\; \backslash exp\backslash left((1\; -\; s)\backslash frac\{\backslash psi\text{'}(x\; +\; s)\}\{\backslash psi(x\; +\; s)\}\backslash right).$

Moreover, equality holds if and only if {{math|s {{=}} 1}}.{{cite journal |doi=10.1016/j.jmaa.2013.05.045 |doi-access=free|title=Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities |year=2013 |last1=Laforgia |first1=Andrea |last2=Natalini |first2=Pierpaolo |journal=Journal of Mathematical Analysis and Applications |volume=407 |issue=2 |pages=495–504 }}

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:

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

Equality holds if and only if $x=1$.{{cite journal |last1=Alzer |first1=Horst |last2=Jameson |first2=Graham |s2cid=41966777 |year=2017 |title=A harmonic mean inequality for the digamma function and related results |journal=Rendiconti del Seminario Matematico della Università di Padova |pages=203–209 |doi=10.4171/RSMUP/137-10 |volume=70 |issue=201 |issn=0041-8994 |lccn=50046633 |oclc=01761704|url=https://eprints.lancs.ac.uk/id/eprint/136736/1/5d0aee750965cd339d8a0965d80de4c18b68.pdf }}

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

:$\backslash psi(x+1)\; =\; \backslash frac\{1\}\{x\}\; +\; \backslash psi(x)$

can be used to shift the value of {{mvar|x}} to a higher value. Beal{{cite thesis |first1=Matthew J. |last1=Beal |title=Variational Algorithms for Approximate Bayesian Inference|year= 2003 |type=PhD thesis |publisher= The Gatsby Computational Neuroscience Unit, University College London |pages=265–266 |url=http://www.cse.buffalo.edu/faculty/mbeal/papers/beal03.pdf}} suggests using the above recurrence to shift {{mvar|x}} to a value greater than 6 and then applying the above expansion with terms above {{math|x¹⁴}} cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).

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

:$\backslash psi(x)\backslash in\; \backslash left(\backslash ln\backslash left(x-\backslash tfrac12\backslash right),\; \backslash ln\; x\backslash right)$

or, for any positive {{mvar|x}},

:$\backslash exp\; \backslash psi(x)\backslash in\backslash left(x-\backslash tfrac12,x\backslash right).$

The exponential {{math|exp ψ(x)}} is approximately {{math|x − {{sfrac|1|2}}}} for large {{mvar|x}}, but gets closer to {{mvar|x}} at small {{mvar|x}}, approaching 0 at {{math|x {{=}} 0}}.

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

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

or

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

From the above asymptotic series for {{mvar|ψ}}, one can derive an asymptotic series for {{math|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.

:$\backslash frac\{1\}\{\backslash exp\; \backslash psi(x)\}\; \backslash sim\; \backslash frac\{1\}\{x\}+\backslash frac\{1\}\{2\backslash cdot\; x^2\}+\backslash frac\{5\}\{4\backslash cdot3!\backslash cdot\; x^3\}+\backslash frac\{3\}\{2\backslash cdot4!\backslash cdot\; x^4\}+\backslash frac\{47\}\{48\backslash cdot5!\backslash cdot\; x^5\}\; -\; \backslash frac\{5\}\{16\backslash cdot6!\backslash cdot\; x^6\}\; +\; \backslash cdots$

This is similar to a Taylor expansion of {{math|exp(−ψ(1 / y))}} at {{math|y {{=}} 0}}, but it does not converge.If it converged to a function {{math|f(y)}} then {{math|ln(f(y) / y)}} would have the same Maclaurin series as {{math|ln(1 / y) − φ(1 / y)}}. But this does not converge because the series given earlier for {{math|φ(x)}} does not converge. (The function is not analytic at infinity.) A similar series exists for {{math|exp(ψ(x))}} which starts with $\backslash exp\; \backslash psi(x)\; \backslash sim\; x-\; \backslash frac\; 12.$

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

:$\backslash exp\; \backslash psi\backslash left(x+\backslash tfrac\{1\}\{2\}\backslash right)\; \backslash sim\; x\; +\; \backslash frac\{1\}\{4!\backslash cdot\; x\}\; -\; \backslash frac\{37\}\{8\backslash cdot6!\backslash cdot\; x^3\}\; +\; \backslash frac\{10313\}\{72\backslash cdot8!\backslash cdot\; x^5\}\; -\; \backslash frac\{5509121\}\{384\backslash cdot10!\backslash cdot\; x^7\}\; +\; \backslash cdots$

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

Another alternative is to use the recurrence relation or the multiplication formula to shift the argument of $\backslash psi(x)$ into the range $1\backslash le\; x\backslash le\; 3$ and to evaluate the Chebyshev series there.{{cite journal|first1=Jet|last1=Wimp | title=Polynomial approximations to integral transforms|journal=Math. Comp. |year=1961|volume=15|issue=74 |pages=174–178| doi=10.1090/S0025-5718-61-99221-3|jstor=2004225}}{{cite journal|title=Chebyshev series expansion of inverse polynomials|first1=R. J.|last1=Mathar|journal=Journal of Computational and Applied Mathematics |year=2004|volume=196 |issue=2 |pages=596–607 |doi=10.1016/j.cam.2005.10.013 |arxiv=math/0403344}} App. E

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

:$\backslash begin\{align\}$

\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{align}

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

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

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

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

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

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 {{math|R⁺}} at {{math|x₀ {{=}} {{val|1.46163214496836234126}}...}}. All others occur single between the poles on the negative axis:

:{{math|x₁ {{=}} {{val|-0.50408300826445540925}}...}}

:{{math|x₂ {{=}} {{val|-1.57349847316239045877}}...}}

:{{math|x₃ {{=}} {{val|-2.61072086844414465000}}...}}

:{{math|x₄ {{=}} {{val|-3.63529336643690109783}}...}}

:$\backslash vdots$

Already in 1881, Charles Hermite observed{{cite journal |first=Charles |last=Hermite |title=Sur l'intégrale Eulérienne de seconde espéce | journal=Journal für die reine und angewandte Mathematik|issue=90|date=1881|pages=332–338 |doi=10.1515/crll.1881.90.332|s2cid=118866486 }} that

:$x\_n\; =\; -n\; +\; \backslash frac\{1\}\{\backslash ln\; n\}\; +\; O\backslash left(\backslash frac\{1\}\{(\backslash ln\; n)^2\}\backslash right)$

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

:$x\_n\; \backslash approx\; -n\; +\; \backslash frac\{1\}\{\backslash pi\}\backslash arctan\backslash left(\backslash frac\{\backslash pi\}\{\backslash ln\; n\}\backslash right)\backslash qquad\; n\; \backslash ge\; 2$

and using a further term it becomes still better

:$x\_n\; \backslash approx\; -n\; +\; \backslash frac\{1\}\{\backslash pi\}\backslash arctan\backslash left(\backslash frac\{\backslash pi\}\{\backslash ln\; n\; +\; \backslash frac\{1\}\{8n\}\}\backslash right)\backslash qquad\; n\; \backslash ge\; 1$

which both spring off the reflection formula via

:$0\; =\; \backslash psi(1-x\_n)\; =\; \backslash psi(x\_n)\; +\; \backslash frac\{\backslash pi\}\{\backslash tan\; \backslash pi\; x\_n\}$

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

Another improvement of Hermite's formula can be given:

:$$

x_n=-n+\frac1{\log n}-\frac1{2n(\log n)^2}+O\left(\frac1{n^2(\log n)^2}\right).

Regarding the zeros, the following infinite sum identities were recently proved by István Mező and Michael Hoffman{{cite journal |first1=István |last1=Mező | first2=Michael E. | last2=Hoffman |title=Zeros of the digamma function and its Barnes G-function analogue |journal=Integral Transforms and Special Functions |volume=28 |

date=2017|issue=11|pages=846–858|doi=10.1080/10652469.2017.1376193|s2cid=126115156 }}

{{cite arXiv

| last = Mező

| first = István

| eprint = 1409.2971

| title = A note on the zeros and local extrema of Digamma related functions

| date = 2014

| class = math.CV

}}

:$\backslash begin\{align\}$

\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} + \frac23 \gamma^2 \pi^2 + 4\gamma\zeta(3).

\end{align}

In general, the function

:$$

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

:$\backslash begin\{align\}$

\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{align}

also hold true.

The digamma function appears in the regularization of divergent integrals

:$\backslash int\_0^\backslash infty\; \backslash frac\{dx\}\{x+a\},$

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

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

Many notable probability distributions use the gamma function in the definition of their probability density or mass functions. Then in statistics when doing maximum 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 in {{cite journal|first1=Jet|last1=Wimp | title=Polynomial approximations to integral transforms|journal=Math. Comp. |year=1961|volume=15|issue=74 |pages=174–178| doi=10.1090/S0025-5718-61-99221-3|doi-access=free}}

{{cite book | editor-last=Abramowitz | editor-first=M. | editor2-last=Stegun | editor2-first=I. A.| chapter=6.3 psi (Digamma) Function. | title= Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | edition=10th | location=New York | publisher=Dover | pages=258–259| year =1972 | chapter-url=http://www.math.sfu.ca/~cbm/aands/page_258.htm }}

{{cite web | title=NIST. Digital Library of Mathematical Functions (DLMF), Chapter 5 | url=https://dlmf.nist.gov/5}}

{{mathworld|urlname=DigammaFunction|title=Digamma function}}

• {{OEIS el|1=A020759|2=Decimal expansion of (-1)*Gamma'(1/2)/Gamma(1/2) where Gamma(x) denotes the Gamma function}}—psi(1/2)

:{{OEIS2C|A047787}} psi(1/3), {{OEIS2C|A200064}} psi(2/3), {{OEIS2C|A020777}} psi(1/4), {{OEIS2C|A200134}} psi(3/4), {{OEIS2C|A200135}} to {{OEIS2C|A200138}} psi(1/5) to psi(4/5).

Category:Gamma and related functions