Digamma function

File:Digamma.png

The digamma function <math>\psi(z)</math>,
visualized using domain coloring

File:Mplwp polygamma03.svg

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:<ref name="AbramowitzStegun"/><ref name="DLMF5"/><ref name="Weissstein"/>

<math>\psi(z) = \frac{\mathrm{d}}{\mathrm{d}z}\ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}.</math>

It is the first of the polygamma functions. This function is strictly increasing and strictly concave on <math>(0,\infty)</math>,<ref>Template:Cite journal</ref> and it asymptotically behaves as<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>\psi(z) \sim \ln{z} - \frac{1}{2z},</math>

for complex numbers with large modulus (<math>|z|\rightarrow\infty</math>) in the sector <math>|\arg z|<\pi-\varepsilon</math> for any <math>\varepsilon > 0</math>.

The digamma function is often denoted as <math>\psi_0(x), \psi^{(0)}(x) </math> or Template:Math<ref>Template:Cite book</ref> (the uppercase form of the archaic Greek consonant digamma meaning double-gamma). Gamma.

Relation to harmonic numbers

The gamma function obeys the equation

<math>\Gamma(z+1)=z\Gamma(z). \, </math>

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

<math>\log \Gamma(z+1)=\log(z)+\log \Gamma(z), </math>

Differentiating both sides with respect to Template:Mvar gives:

<math>\psi(z+1)=\psi(z)+\frac{1}{z}</math>

Since the harmonic numbers are defined for positive integers Template:Mvar as

<math>H_n=\sum_{k=1}^n \frac 1 k, </math>

the digamma function is related to them by

<math>\psi(n)=H_{n-1}-\gamma,</math>

where Template:Math and Template:Mvar is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values

<math> \psi \left(n+\tfrac12\right)=-\gamma-2\ln 2 +\sum_{k=1}^n \frac 2 {2k-1} = -\gamma-2\ln 2 + 2H_{2n}-H_n.</math>

Integral representations

If the real part of Template:Mvar is positive then the digamma function has the following integral representation due to Gauss:<ref name="Whittaker and Watson, 12.3">Whittaker and Watson, 12.3.</ref>

<math>\psi(z) = \int_0^\infty \left(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1-e^{-t}}\right)\,dt.</math>

Combining this expression with an integral identity for the Euler–Mascheroni constant <math>\gamma</math> gives:

<math>\psi(z + 1) = -\gamma + \int_0^1 \left(\frac{1-t^z}{1-t}\right)\,dt.</math>

The integral is Euler's harmonic number <math>H_z</math>, so the previous formula may also be written

<math>\psi(z + 1) = \psi(1) + H_z.</math>

A consequence is the following generalization of the recurrence relation:

<math>\psi(w + 1) - \psi(z + 1) = H_w - H_z.</math>

An integral representation due to Dirichlet is:<ref name="Whittaker and Watson, 12.3"/>

<math>\psi(z) = \int_0^\infty \left(e^{-t} - \frac{1}{(1 + t)^z}\right)\,\frac{dt}{t}.</math>

Gauss's integral representation can be manipulated to give the start of the asymptotic expansion of <math>\psi</math>.<ref>Whittaker and Watson, 12.31.</ref>

<math>\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.</math>

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 <math>\psi</math> which also gives the first few terms of the asymptotic expansion:<ref>Whittaker and Watson, 12.32, example.</ref>

<math>\psi(z) = \log z - \frac{1}{2z} - 2\int_0^\infty \frac{t\,dt}{(t^2 + z^2)(e^{2\pi t} - 1)}.</math>

From the definition of <math>\psi</math> and the integral representation of the gamma function, one obtains

<math>\psi(z) = \frac{1}{\Gamma(z)} \int_0^\infty t^{z-1} \ln (t) e^{-t}\,dt,</math>

with <math>\Re z > 0</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Infinite product representation

The function <math>\psi(z)/\Gamma(z)</math> is an entire function,<ref name=MezoHoffman/> and it can be represented by the infinite product

<math>

\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}}. </math>

Here <math>x_k</math> is the kth zero of <math>\psi</math> (see below), and <math>\gamma</math> is the Euler–Mascheroni constant.

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

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):<ref name="AbramowitzStegun"/>

<math>\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}</math> Equivalently,

<math>\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}</math>

Evaluation of sums of rational functions

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

<math>\sum_{n=0}^\infty u_n=\sum_{n=0}^\infty \frac{p(n)}{q(n)},</math>

where Template:Math and Template:Math are polynomials of Template:Mvar.

Performing partial fraction on Template:Mvar in the complex field, in the case when all roots of Template:Math are simple roots,

<math>u_n=\frac{p(n)}{q(n)}=\sum_{k=1}^m \frac{a_k}{n+b_k}.</math>

For the series to converge,

<math>\lim_{n\to\infty} nu_n=0,</math>

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

<math>\sum_{k=1}^m a_k=0,</math>

and

<math>\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}</math>

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

<math>\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),</math>

provided the series on the left converges.

Taylor series

The digamma has a rational zeta series, given by the Taylor series at Template:Math. This is

<math>\psi(z+1)= -\gamma -\sum_{k=1}^\infty (-1)^k\,\zeta (k+1) \, z^k,</math>

which converges for Template:Math. Here, Template:Math 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,<ref>Template:Cite book</ref><ref name="blag2018">Template:Cite journal</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> reads

<math>\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}</math>

where Template:Math is the binomial coefficient. It may also be generalized to

<math>\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,</math>

where Template:Math<ref name="blag2018" />

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 Template:Math is

<math>

\psi(v) =\ln v- \sum_{n=1}^\infty\frac{\big| G_{n}\big|(n-1)!}{(v)_{n}},\qquad \Re (v) >0, </math>

<math>

\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, </math>

<math>

\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, </math> where Template:Math is the rising factorial Template:Math, Template:Math are the Gregory coefficients of higher order with Template:Math, Template:Math is the gamma function and Template:Math is the Hurwitz zeta function.<ref name="blag2016">Template:Cite journal</ref><ref name="blag2018" /> Similar series with the Cauchy numbers of the second kind Template:Math reads<ref name="blag2016" /><ref name="blag2018" />

<math>

\psi(v)=\ln(v-1) + \sum_{n=1}^\infty\frac{C_{n}(n-1)!}{(v)_{n}},\qquad \Re(v) >1, </math> A series with the Bernoulli polynomials of the second kind has the following form<ref name="blag2018" />

<math>

\psi(v)=\ln(v+a) + \sum_{n=1}^\infty\frac{(-1)^n\psi_{n}(a)\,(n-1)!}{(v)_{n}},\qquad \Re(v)>-a, </math> where Template:Math are the Bernoulli polynomials of the second kind defined by the generating equation

<math>

\frac{z(1+z)^a}{\ln(1+z)}= \sum_{n=0}^\infty z^n \psi_n(a) \,,\qquad |z|<1\,, </math> It may be generalized to

<math>

\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 </math> where the polynomials Template:Math are given by the following generating equation

<math>

\frac{(1+z)^{a+m}-(1+z)^{a}}{\ln(1+z)}=\sum_{n=0}^\infty N_{n,m}(a) z^n , \qquad |z|<1, </math> so that Template:Math.<ref name="blag2018" /> Similar expressions with the logarithm of the gamma function involve these formulas<ref name="blag2018" />

<math>

\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, </math> and

<math>

\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\}, </math> where <math>\Re(v)>-a</math> and <math>r=2,3,4,\ldots</math>.

Reflection formula

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

<math>\psi(1-x)-\psi(x)=\pi \cot \pi x</math>.

<math>\psi'(-x)+\psi'(x) = \frac{\pi^2}{\sin^2(\pi x)}+\frac{1}{x^2}</math>.

Recurrence formula and characterization

The digamma function satisfies the recurrence relation

<math>\psi(x+1)=\psi(x)+\frac{1}{x}.</math>

Thus, it can be said to "telescope" Template:Math, for one has

<math>\Delta [\psi](x)=\frac{1}{x}</math>

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

<math>\psi(n)=H_{n-1}-\gamma</math>

where Template:Mvar is the Euler–Mascheroni constant.

Actually, Template:Mvar is the only solution of the functional equation

<math>F(x+1)=F(x)+\frac{1}{x}</math>

that is monotonic on Template:Math and satisfies Template:Math. This fact follows immediately from the uniqueness of the Template:Math function given its recurrence equation and convexity restriction. This implies the useful difference equation:

<math> \psi(x+N)-\psi(x)=\sum_{k=0}^{N-1} \frac{1}{x+k}</math>

Some finite sums involving the digamma function

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

<math>\sum_{r=1}^m \psi\left(\frac{r}{m}\right)=-m(\gamma+\ln m),</math>

<math>\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\Z,\quad m\in\N,\ k\ne m</math>

<math>\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 </math>

<math>\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 </math>

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

<math>\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</math>

<math>\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</math>

<math>\sum_{r=1}^{m-1} \psi\left(\frac{r}{m}\right)\cdot\cot\frac{\pi r}{m}= -\frac{\pi(m-1)(m-2)}{6}</math>

<math>\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} </math>

<math>\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} </math>

<math>\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right) \cdot\sin\dfrac{(2\ell+1)\pi r}{m}=-(\gamma+\ln2m)\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}</math>

<math>\sum_{r=1}^{m-1} \psi^2\left(\frac{r}{m}\right)= (m-1)\gamma^2 + m(2\gamma+\ln4m)\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}</math>

are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014)<ref name="iaroslav_07">Template:Cite journal</ref>).

We also have <ref>Template:Cite book</ref>

<math> 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, ...</math>

Gauss's digamma theorem

For positive integers Template:Mvar and Template:Mvar (Template:Math), the digamma function may be expressed in terms of Euler's constant and a finite number of elementary functions<ref>Template:Cite journal</ref>

<math>\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) </math>

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

Multiplication theorem

The multiplication theorem of the <math>\Gamma</math>-function is equivalent to<ref>Template:Cite book</ref>

<math>\psi(nz)=\frac{1}{n}\sum_{k=0}^{n-1} \psi\left(z+\frac{k}{n}\right) +\ln n .</math>

Asymptotic expansion

The digamma function has the asymptotic expansion

<math>\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},</math>

where Template:Mvar is the Template:Mvarth Bernoulli number and Template:Mvar is the Riemann zeta function. The first few terms of this expansion are:

<math>\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.</math>

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

The expansion can be found by applying the Euler–Maclaurin formula to the sum<ref>Template:Cite journal</ref>

<math>\sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{z + n}\right)</math>

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

<math>\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)}.</math>

Inequalities

When Template:Math, the function

<math>\ln x - \frac{1}{2x} - \psi(x)</math>

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 <math>1 + t \le e^t</math>, the integrand in this representation is bounded above by <math>e^{-tz}/2</math>. Template:Not a typo

<math>\frac{1}{x} - \ln x + \psi(x)</math>

is also completely monotonic. It follows that, for all Template:Math,

<math>\ln x - \frac{1}{x} \le \psi(x) \le \ln x - \frac{1}{2x}.</math>

This recovers a theorem of Horst Alzer.<ref>Template:Cite journal</ref> Alzer also proved that, for Template:Math,

<math>\frac{1 - s}{x + s} < \psi(x + 1) - \psi(x + s),</math>

Related bounds were obtained by Elezovic, Giordano, and Pecaric, who proved that, for Template:Math,

<math>\ln(x + \tfrac{1}{2}) - \frac{1}{x} < \psi(x) < \ln(x + e^{-\gamma}) - \frac{1}{x},</math>

where <math>\gamma=-\psi(1)</math> is the Euler–Mascheroni constant.<ref>Template:Cite journal</ref> The constants (<math>0.5</math> and <math>e^{-\gamma}\approx0.56</math>) appearing in these bounds are the best possible.<ref>Template:Cite journal</ref>

The mean value theorem implies the following analog of Gautschi's inequality: If Template:Math, where Template:Math is the unique positive real root of the digamma function, and if Template:Math, then

<math>\exp\left((1 - s)\frac{\psi'(x + 1)}{\psi(x + 1)}\right) \le \frac{\psi(x + 1)}{\psi(x + s)} \le \exp\left((1 - s)\frac{\psi'(x + s)}{\psi(x + s)}\right).</math>

Moreover, equality holds if and only if Template:Math.<ref>Template:Cite journal</ref>

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:

<math> -\gamma \leq \frac{2 \psi(x) \psi(\frac{1}{x})}{\psi(x)+\psi(\frac{1}{x})} </math> for <math>x>0</math>

Equality holds if and only if <math>x=1</math>.<ref>Template:Cite journal</ref>

Computation and approximation

The asymptotic expansion gives an easy way to compute Template:Math when the real part of Template:Mvar is large. To compute Template:Math for small Template:Mvar, the recurrence relation

<math> \psi(x+1) = \frac{1}{x} + \psi(x)</math>

can be used to shift the value of Template:Mvar to a higher value. Beal<ref>Template:Cite thesis</ref> suggests using the above recurrence to shift Template:Mvar to a value greater than 6 and then applying the above expansion with terms above Template:Math cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).

As Template:Mvar goes to infinity, Template:Math gets arbitrarily close to both Template:Math and Template:Math. Going down from Template:Math to Template:Mvar, Template:Mvar decreases by Template:Math, Template:Math decreases by Template:Math, which is more than Template:Math, and Template:Math decreases by Template:Math, which is less than Template:Math. From this we see that for any positive Template:Mvar greater than Template:Math,

<math>\psi(x)\in \left(\ln\left(x-\tfrac12\right), \ln x\right)</math>

or, for any positive Template:Mvar,

<math>\exp \psi(x)\in\left(x-\tfrac12,x\right).</math>

The exponential Template:Math is approximately Template:Math for large Template:Mvar, but gets closer to Template:Mvar at small Template:Mvar, approaching 0 at Template:Math.

For Template:Math, we can calculate limits based on the fact that between 1 and 2, Template:Math, so

<math>\psi(x)\in\left(-\frac{1}{x}-\gamma, 1-\frac{1}{x}-\gamma\right),\quad x\in(0, 1)</math>

or

<math>\exp \psi(x)\in\left(\exp\left(-\frac{1}{x}-\gamma\right),e\exp\left(-\frac{1}{x}-\gamma\right)\right).</math>

From the above asymptotic series for Template:Mvar, one can derive an asymptotic series for Template:Math. 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.

<math> \frac{1}{\exp \psi(x)} \sim \frac{1}{x}+\frac{1}{2\cdot x^2}+\frac{5}{4\cdot3!\cdot x^3}+\frac{3}{2\cdot4!\cdot x^4}+\frac{47}{48\cdot5!\cdot x^5} - \frac{5}{16\cdot6!\cdot x^6} + \cdots</math>

This is similar to a Taylor expansion of Template:Math at Template:Math, but it does not converge.<ref>If it converged to a function Template:Math then Template:Math would have the same Maclaurin series as Template:Math. But this does not converge because the series given earlier for Template:Math does not converge.</ref> (The function is not analytic at infinity.) A similar series exists for Template:Math which starts with <math>\exp \psi(x) \sim x- \frac 12.</math>

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

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

Similar in spirit to the Lanczos approximation of the <math>\Gamma</math>-function is Spouge's approximation.

Another alternative is to use the recurrence relation or the multiplication formula to shift the argument of <math>\psi(x)</math> into the range <math>1\le x\le 3</math> and to evaluate the Chebyshev series there.<ref>Template:Cite journal</ref><ref>Template:Cite journal App. E</ref>

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:

<math>\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}</math>

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

<math>\operatorname{Im} \psi(bi) = \frac{1}{2b}+\frac{\pi}{2}\coth (\pi b),</math>

<math>\operatorname{Im} \psi(\tfrac{1}{2}+bi) = \frac{\pi}{2}\tanh (\pi b).</math>

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

<math>\operatorname{Re} \psi(i) = -\gamma-\sum_{n=0}^\infty\frac{n-1}{n^3+n^2+n+1} \approx 0.09465.</math>

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 Template:Math at Template:Math. All others occur single between the poles on the negative axis:

Template:Math

Template:Math

Template:Math

Template:Math

<math>\vdots</math>

Already in 1881, Charles Hermite observed<ref name="Hermite">Template:Cite journal</ref> that

<math>x_n = -n + \frac{1}{\ln n} + O\left(\frac{1}{(\ln n)^2}\right)</math>

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

<math>x_n \approx -n + \frac{1}{\pi}\arctan\left(\frac{\pi}{\ln n}\right)\qquad n \ge 2</math>

and using a further term it becomes still better

<math>x_n \approx -n + \frac{1}{\pi}\arctan\left(\frac{\pi}{\ln n + \frac{1}{8n}}\right)\qquad n \ge 1</math>

which both spring off the reflection formula via

<math>0 = \psi(1-x_n) = \psi(x_n) + \frac{\pi}{\tan \pi x_n}</math>

and substituting Template:Math by its not convergent asymptotic expansion. The correct second term of this expansion is Template:Math, where the given one works well to approximate roots with small Template:Mvar.

Another improvement of Hermite's formula can be given:<ref name=MezoHoffman/>

<math>

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

Regarding the zeros, the following infinite sum identities were recently proved by István Mező and Michael Hoffman<ref name="MezoHoffman">Template:Cite journal</ref><ref> Template:Cite arXiv </ref>

<math>\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}</math>

In general, the function

<math>

Z(k)=\sum_{n=0}^\infty\frac{1}{x_n^k} </math> can be determined and it is studied in detail by the cited authors.

The following results<ref name=MezoHoffman/>

<math>\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}</math> also hold true.

Regularization

The digamma function appears in the regularization of divergent integrals

<math> \int_0^\infty \frac{dx}{x+a},</math>

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

<math> \sum_{n=0}^\infty \frac{1}{n+a}= - \psi (a).</math>

In applied mathematics

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.

See also

• Polygamma function
• Trigamma function
• Chebyshev expansions of the digamma function in Template:Cite journal

References

<references> <ref name="AbramowitzStegun"> Template:Cite book</ref> <ref name="DLMF5">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <ref name="Weissstein">Template:Mathworld</ref> </references>

External links

• Template:OEIS el—psi(1/2)

Template:OEIS2C psi(1/3), Template:OEIS2C psi(2/3), Template:OEIS2C psi(1/4), Template:OEIS2C psi(3/4), Template:OEIS2C to Template:OEIS2C psi(1/5) to psi(4/5).