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In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \digamma
(after the shape of the obsolete Greek letter Ϝ digamma), is related to the harmonic numbers in that
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(x) = \int_0^{\infty}\left(\frac{e^{-t}}{t} - \frac{e^{-xt}}{1 - e^{-t}}\right)\,dt
This may be written as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(z+1)= -\gamma -\sum_{k=1}^\infty \zeta (k+1)\;(-z)^k
,
which converges for |z|<1. Here, Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(s+1)=-\gamma-\sum_{k=1}^\infty \frac{(-1)^k}{k} {s \choose k}
where Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle{s \choose k}
is the binomial coefficient.
Reflection formula
The digamma function satisfies a reflection formula similar to that of the Gamma function,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(1 - x) - \psi(x) = \pi\,\!\cot{ \left ( \pi x \right ) }
Recurrence formula
The digamma function satisfies the recurrence relation
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(x + 1) = \psi(x) + \frac{1}{x}
Thus, it can be said to "telescope" 1/x, for one has
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(n)\ =\ H_{n-1} - \gamma
where Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma
is the Euler-Mascheroni constant.
More generally, one has
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): 0<m<k . Here, ζ(s,q) is the Hurwitz zeta function and Failed to parse (Missing texvc executable; please see math/README to configure.): B_n(x)
is a Bernoulli polynomial. A special case of the multiplication theorem is
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{n=1}^k \psi \left(\frac{n}{k}\right) =-k(\gamma+\log k),
and a neat generalization of this is
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(1) = -\gamma\,\!
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi\left(\frac{1}{2}\right) = -2\ln{2} - \gamma
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi\left(\frac{1}{3}\right) = -\frac{\pi}{2\sqrt{3}} -\frac{3}{2}\ln{3} - \gamma
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi\left(\frac{1}{4}\right) = -\frac{\pi}{2} - 3\ln{2} - \gamma
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi\left(\frac{1}{6}\right) = -\frac{\pi}{2}\sqrt{3} -2\ln{2} -\frac{3}{2}\ln(3) - \gamma
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
See also
References
External links
es:Función digamma fa:تابع دایگاما fr:Fonction digamma it:Funzione digamma fi:Digammafunktio th:ฟังก์ชันไดแกมมา
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