# Digamma function

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File:Complex Polygamma 0.jpg
Digamma function ${\displaystyle \psi (s)}$ in the complex plane. The color of a point ${\displaystyle s}$ encodes the value of ${\displaystyle \psi (s)}$. Strong colors denote values close to zero and hue encodes the value's argument.

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

${\displaystyle \psi (x)={\frac {d}{dx}}\ln {\Gamma (x)}={\frac {\Gamma '(x)}{\Gamma (x)}}.}$

It is the first of the polygamma functions.

## Relation to harmonic numbers

The digamma function, often denoted also as ψ0(x), ψ0(x) or ${\displaystyle \digamma }$ (after the shape of the obsolete Greek letter Ϝ digamma), is related to the harmonic numbers in that

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

where Hn is the n 'th harmonic number, and γ is the Euler-Mascheroni constant. For half-integer values, it may be expressed as

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

## Integral representations

It has the integral representation

${\displaystyle \psi (x)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-xt}}{1-e^{-t}}}\right)\,dt}$

This may be written as

${\displaystyle \psi (s+1)=-\gamma +\int _{0}^{1}{\frac {1-x^{s}}{1-x}}dx}$

which follows from Euler's integral formula for the harmonic numbers.

## Taylor series

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

${\displaystyle \psi (z+1)=-\gamma -\sum _{k=1}^{\infty }\zeta (k+3)\;(-z)^{k}}$,

which converges for |z|<1. Here, ${\displaystyle \zeta (n)}$ 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 follows from Euler's integral formula:

${\displaystyle \psi (s+1)=-\gamma -\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{s \choose k}}$

where ${\displaystyle \textstyle {s \choose k}}$ is the binomial coefficient.

## Reflection formula

The digamma function satisfies a reflection formula similar to that of the Gamma function,

${\displaystyle \psi (1-x)-\psi (x)=\pi \,\!\cot {\left(\pi x\right)}}$

## Recurrence formula

The digamma function satisfies the recurrence relation

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

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

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

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

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

where ${\displaystyle \gamma }$ is the Euler-Mascheroni constant.

More generally, one has

${\displaystyle \psi (x+1)=-\gamma +\sum _{k=1}^{\infty }\left({\frac {1}{k}}-{\frac {1}{x+k}}\right)}$

## Gaussian sum

The digamma has a Gaussian sum of the form

${\displaystyle {\frac {-1}{\pi k}}\sum _{n=1}^{k}\sin \left({\frac {2\pi nm}{k}}\right)\psi \left({\frac {n}{k}}\right)=\zeta \left(0,{\frac {m}{k}}\right)=-B_{1}\left({\frac {m}{k}}\right)={\frac {1}{2}}-{\frac {m}{k}}}$

for integers ${\displaystyle 0. Here, ζ(s,q) is the Hurwitz zeta function and ${\displaystyle B_{n}(x)}$ is a Bernoulli polynomial. A special case of the multiplication theorem is

${\displaystyle \sum _{n=1}^{k}\psi \left({\frac {n}{k}}\right)=-k(\gamma +\log k),}$

and a neat generalization of this is

${\displaystyle \sum _{p=0}^{q-1}\psi (a+p/q)=q(\psi (qa)-\ln(q)),}$

in which it is assumed that q is a natural number, and that 1-qa is not.

## Gauss's digamma theorem

For positive integers m and k (with m < k), the digamma function may be expressed in terms of elementary functions as

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

## Special values

The digamma function has the following special values:

${\displaystyle \psi (1)=-\gamma \,\!}$
${\displaystyle \psi \left({\frac {1}{2}}\right)=-2\ln {2}-\gamma }$
${\displaystyle \psi \left({\frac {1}{3}}\right)=-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln {3}-\gamma }$
${\displaystyle \psi \left({\frac {1}{4}}\right)=-{\frac {\pi }{2}}-3\ln {2}-\gamma }$
${\displaystyle \psi \left({\frac {1}{6}}\right)=-{\frac {\pi }{2}}{\sqrt {3}}-2\ln {2}-{\frac {3}{2}}\ln(3)-\gamma }$
${\displaystyle \psi \left({\frac {1}{8}}\right)=-{\frac {\pi }{2}}-4\ln {2}-{\frac {1}{\sqrt {2}}}\left\{\pi +\ln(2+{\sqrt {2}})-\ln(2-{\sqrt {2}})\right\}-\gamma }$