Digamma function

The digamma function in the complex number plane . ${\displaystyle \psi (x)}$

The digamma function or psi function is a function in mathematics that is defined as:

${\displaystyle \psi (x)={\frac {\mathrm {d} }{\mathrm {d} x}}\ln {\big (}\Gamma (x){\big )}={\frac {\Gamma '(x)}{\Gamma (x)}}}$

So it is the logarithmic derivative of the gamma function . The digamma function is the first of the polygamma functions . Except for its first-order poles for negative whole arguments, it is (just like the gamma function) completely holomorphic . ${\displaystyle \mathbb {C} }$

calculation

The relationship to the harmonic series

The digamma function, which is usually represented as ψ ₀ ( x ), ψ ⁰ ( x ) or (according to the form of the pre-classical Greek letter Ϝ digamma ), has the following relationship with the harmonic series : ${\displaystyle \digamma }$

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

where H _{n is} the nth element of the harmonic series and γ is the Euler-Mascheroni constant . For half-integer values ​​it can be written as:

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

Integral representation

The digamma function can be represented as an integral as follows :

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

This can also be written as:

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

This follows from the formula for the Euler integral for the harmonic series.

Taylor series

By expanding the Taylor series around the point z = 1, the digamma function can be represented as follows:

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

It converges for | z | <1. Here is the Riemann ζ function . The series can easily be derived from the corresponding Taylor series for the Hurwitz ζ function . ${\displaystyle \zeta (n)}$

Binomial series

The binomial series for the digamma function follows from the Euler integral

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

where is the generalized binomial coefficient . ${\displaystyle {\tbinom {s}{k}}}$

Functional equation

The digamma function satisfies the following functional equation, which can be derived directly from the logarithmic derivation of the gamma function:

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

With this, however, ψ (1/2) cannot be calculated; this value is given below.

Recursion formula and sum expressions

The digamma function satisfies the recursion formula

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

or

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

where Δ is the right-hand difference operator. This satisfies the harmonic series recursion relationship . It follows

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

More generally applies:

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

The Gaussian product representation of the gamma function can be equivalent to this

${\displaystyle \psi (x)=\lim \limits _{n\to \infty }\left(\ln n-\sum \limits _{k=0}^{n}{\frac {1}{x+k}}\right)}$.

Conclude.

Ratio relation to the gamma function

The product representation provides the expression for the quotient of the digamma function and the gamma function

${\displaystyle {\frac {\psi (x)}{\Gamma (x)}}=\lim \limits _{n\to \infty }{\frac {\ln n\prod \limits _{k=0}^{n}(x+k)-\sum \limits _{j=0}^{n}\prod \limits _{k=0 \atop k\neq j}^{n}(x+k)}{n!\,n^{x}}}}$.

In the case of positive whole numbers , in the case of whose negative values ​​both the digamma- and the gamma-function diverge, then follows ${\displaystyle m\geq 0}$

${\displaystyle {\frac {\psi (-m)}{\Gamma (-m)}}=-\lim \limits _{n\to \infty }{\frac {\prod \limits _{k=0 \atop k\neq m}^{n}(k-m)}{n!\,n^{-m}}}=(-1)^{m-1}m!\lim \limits _{n\to \infty }{\frac {n^{m}}{\prod \limits _{k=n-m+1}^{n}k}}=(-1)^{m-1}m!}$.

With the help of the functional equation for the gamma function, one can even find out that the value of the quotient depends exclusively on the argument of the gamma function, so it ultimately applies to integer numbers ${\displaystyle m,n\geq 0}$

${\displaystyle {\frac {\psi (-m)}{\Gamma (-n)}}=(-1)^{n-1}n!}$.

Gaussian sum

The digamma function has a Gaussian sum of form

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

for natural numbers . Here ζ ( s , q ) is the Hurwitz ζ function and the Bernoulli polynomial . A special case of the multiplication theorem is ${\displaystyle 0<m<k}$ ${\displaystyle \mathrm {B} _{n}(x)}$

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

Gaussian Digamma theorem

For whole numbers and (with ) the digammafunction can be expressed with elementary functions ${\displaystyle m}$ ${\displaystyle k}$ ${\displaystyle m<k}$

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

Special values

The Digamma function has the following special values:

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

${\displaystyle \psi \left({\tfrac {1}{2}}\right)=-2\ln 2-\gamma }$

${\displaystyle \psi \left({\tfrac {1}{3}}\right)=-{\frac {\pi }{2{\sqrt {3}}}}-{\tfrac {3}{2}}\ln 3-\gamma }$

${\displaystyle \psi \left({\tfrac {1}{4}}\right)=-{\frac {\pi }{2}}-3\ln 2-\gamma }$

${\displaystyle \psi \left({\tfrac {1}{6}}\right)=-{\frac {\pi }{2}}{\sqrt {3}}-2\ln 2-{\tfrac {3}{2}}\ln 3-\gamma }$

Derivation

According to its definition, the derivative of the digamma function is the trigamma function

${\displaystyle \psi _{1}(x)={\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\ln \Gamma (x),}$

the second polygamma function .

literature

• Milton Abramowitz and Irene A. Stegun , Handbook of Mathematical Functions , (1964) Dover Publications, New York. ISBN 0-486-61272-4 . See §6.3

Web links

• Eric W. Weisstein : Digamma Function . In: MathWorld (English).