Editing Digamma function (section)

==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 line#In real algebras|real axis]]. The only one on the [[positive real axis]] is the unique minimum of the real-valued gamma function on {{math|'''[[Real number|R]]'''<sup>+</sup>}} at {{math|''x''<sub>0</sub> {{=}} {{val|1.46163214496836234126}}...}}. All others occur single between the poles on the negative axis:

:{{math|''x''<sub>1</sub> {{=}} {{val|-0.50408300826445540925}}...}}
:{{math|''x''<sub>2</sub> {{=}} {{val|-1.57349847316239045877}}...}}
:{{math|''x''<sub>3</sub> {{=}} {{val|-2.61072086844414465000}}...}}
:{{math|''x''<sub>4</sub> {{=}} {{val|-3.63529336643690109783}}...}}
:<math>\vdots</math>

Already in 1881, [[Charles Hermite]] observed<ref name="Hermite">{{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 }}</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 {{math|''ψ''(''x<sub>n</sub>'')}} by its not convergent asymptotic expansion. The correct second term of this expansion is {{math|{{sfrac|1|2''n''}}}}, where the given one works well to approximate roots with small {{mvar|n}}.

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">{{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 }}</ref><ref>
{{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
 }}
</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.