A special function which is given by the logarithmic derivative of the gamma function (or, depending
on the definition, the logarithmic derivative
of the factorial ).
Because of this ambiguity, two different notations are sometimes (but not always) used, with
(1)
defined as the logarithmic derivative of the gamma function , and
(2)
defined as the logarithmic derivative of the factorial function. The two are connected by the
relationship
(3)
The th
derivative of is called the polygamma
function , denoted .
The notation
(4)
is therefore frequently used for the digamma function itself, and Erdélyi et al. (1981) use the notation for . The digamma function is returned by the function PolyGamma [z ]
or PolyGamma [0,
z ] in the Wolfram Language ,
and typeset using the notation .
The digamma function arises in simple sums such as
where
is a Lerch transcendent .
Special cases are given by
Gauss's digamma theorem states that
(11)
(Allouche 1992, Knuth 1997, p. 94).
An asymptotic series for the digamma function
is given by
where
is the Euler-Mascheroni constant and
are Bernoulli
numbers .
The digamma function satisfies
(17)
For integer ,
(18)
where
is the Euler-Mascheroni constant and
is a harmonic
number .
Other identities include
(19)
(20)
(21)
(22)
Special values are
At integer values,
(Derbyshire 2004, p. 58), and at half-integral values,
where
is a harmonic number .
It is given by the unit square integral
(29)
for
(Guillera and Sondow 2005). Plugging in gives a special case involving the Euler-Mascheroni
constant .
The series for
is given by
(30)
A logarithmic series is given by
(31)
(Guillera and Sondow 2005).
A surprising identity that arises from the FoxTrot
series is given by
(32)
See also Barnes G-Function ,
G-Function ,
Gamma Function ,
Gauss's
Digamma Theorem ,
Harmonic Number ,
Hurwitz
Zeta Function ,
Logarithmic Derivative ,
Mellin's Formula ,
Polygamma
Function ,
Ramanujan phi -Function ,
Trigamma Function
Related Wolfram sites http://functions.wolfram.com/GammaBetaErf/PolyGamma/
Explore with Wolfram|Alpha
References Abramowitz, M. and Stegun, I. A. (Eds.). "Psi (Digamma) Function." §6.3 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 258-259, 1972. Allouche, J.-P. "Series
and Infinite Products related to Binary Expansions of Integers." 1992. http://algo.inria.fr/seminars/sem92-93/allouche.ps . Arfken,
G. "Digamma and Polygamma Functions." §10.2 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 549-555,
1985. Boros, G. and Moll, V. "The Psi Function." §10.11
in Irresistible
Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals.
Cambridge, England: Cambridge University Press, pp. 212-215, 2004. Derbyshire,
J. Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.
New York: Penguin, 2004. Erdélyi, A.; Magnus, W.; Oberhettinger,
F.; and Tricomi, F. G. "The Function." §1.7 in Higher
Transcendental Functions, Vol. 1. New York: Krieger, pp. 15-20,
1981. Guillera, J. and Sondow, J. "Double Integrals and Infinite
Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent."
16 June 2005 http://arxiv.org/abs/math.NT/0506319 . Havil,
J. Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003. Jeffreys,
H. and Jeffreys, B. S. "The Digamma ( ) and Trigamma ( ) Functions." Methods
of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University
Press, pp. 465-466, 1988. Knuth, D. E. The
Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed.
Reading, MA: Addison-Wesley, 1997. Spanier, J. and Oldham, K. B.
"The Digamma Function ." Ch. 44 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 423-434, 1987. Referenced
on Wolfram|Alpha Digamma Function
Cite this as:
Weisstein, Eric W. "Digamma Function."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/DigammaFunction.html
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