Английская Википедия:Digamma function

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Файл:Digamma.png

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

Файл:Mplwp polygamma03.svg

Real part plots of the digamma and the next three polygamma functions along the real line

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

<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>, and it asymptotically behaves as

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

for large arguments (<math>|z|\rightarrow\infty</math>) in the sector <math>|\arg z|<\pi-\varepsilon</math> with some infinitesimally small positive constant <math>\varepsilon</math>.

The digamma function is often denoted as <math>\psi_0(x), \psi^{(0)}(x) </math> or Шаблон:Math (the uppercase form of the archaic Greek consonant digamma meaning double-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 gives:

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

Differentiating both sides with respect to Шаблон:Mvar gives:

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

Since the harmonic numbers are defined for positive integers Шаблон: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 Шаблон:Math and Шаблон: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}.</math>

Integral representations

If the real part of Шаблон:Mvar is positive then the digamma function has the following integral representation due to Gauss:

<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:

<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>.

<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:

<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>.

Infinite product representation

The function <math>\psi(z)/\Gamma(z)</math> is an entire function, 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):

<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 Шаблон:Math and Шаблон:Math are polynomials of Шаблон:Mvar.

Performing partial fraction on Шаблон:Mvar in the complex field, in the case when all roots of Шаблон: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 Шаблон:Math. This is

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

which converges for Шаблон:Math. Here, Шаблон: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, reads

<math>\psi(s+1)=-\gamma-\sum_{k=1}^\infty \frac{(-1)^k}{k} \binom{s}{k}</math>

where Шаблон: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 Шаблон:Math

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 Шаблон: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 Шаблон:Math is the rising factorial Шаблон:Math, Шаблон:Math are the Gregory coefficients of higher order with Шаблон:Math, Шаблон:Math is the gamma function and Шаблон:Math is the Hurwitz zeta function. Similar series with the Cauchy numbers of the second kind Шаблон:Math reads

<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

<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 Шаблон: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 Шаблон: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 Шаблон:Math. Similar expressions with the logarithm of the gamma function involve these formulas

<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 function satisfies a reflection formula similar to that of the gamma function:

<math>\psi(1-x)-\psi(x)=\pi \cot \pi x</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" Шаблон:Math, for one has

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

where Шаблон: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 Шаблон:Mvar is the Euler–Mascheroni constant.

More generally, one has

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

for <math> \mathrm{Re}(z)>0</math>. Another series expansion is:

<math> \psi(1+z)=\ln(z)+\frac{1}{2z}-\displaystyle\sum_{j=1}^{\infty} \frac{B_{2j}}{2jz^{2j}} </math>,

where <math>B_{2j}</math> are the Bernoulli numbers. This series diverges for all Шаблон:Math and is known as the Stirling series.

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

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

that is monotonic on Шаблон:Math and satisfies Шаблон:Math. This fact follows immediately from the uniqueness of the Шаблон: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. 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)).

We also have

<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 Шаблон:Mvar and Шаблон:Mvar (Шаблон:Math), the digamma function may be expressed in terms of Euler's constant and a finite number of elementary functions

<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.

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 Шаблон:Mvar is the Шаблон:Mvarth Bernoulli number and Шаблон: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 Шаблон:Mvar, any finite partial sum becomes increasingly accurate as Шаблон:Mvar increases.

The expansion can be found by applying the Euler–Maclaurin formula to the sum

<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 Шаблон: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>. Шаблон:Not a typo

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

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

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

This recovers a theorem of Horst Alzer. Alzer also proved that, for Шаблон: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 Шаблон: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. The constants (<math>0.5</math> and <math>e^{-\gamma}\approx0.56</math>) appearing in these bounds are the best possible.

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

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>.

Computation and approximation

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

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

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

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

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

or, for any positive Шаблон:Mvar,

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

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

For Шаблон:Math, we can calculate limits based on the fact that between 1 and 2, Шаблон: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 Шаблон:Mvar, one can derive an asymptotic series for Шаблон: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 Шаблон:Math at Шаблон:Math, but it does not converge. (The function is not analytic at infinity.) A similar series exists for Шаблон:Math which starts with <math>\exp \psi(x) \sim x- \frac 12.</math>

If one calculates the asymptotic series for Шаблон:Math it turns out that there are no odd powers of Шаблон:Mvar (there is no Шаблон: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>

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

Шаблон:Math

Шаблон:Math

Шаблон:Math

Шаблон:Math

<math>\vdots</math>

Already in 1881, Charles Hermite observed 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 Шаблон:Math by its not convergent asymptotic expansion. The correct second term of this expansion is Шаблон:Math, where the given one works well to approximate roots with small Шаблон:Mvar.

Another improvement of Hermite's formula can be given:

<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

<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

<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>

See also

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

External links

• Шаблон:OEIS el—psi(1/2)

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