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Digamma Function

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The digamma function is 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

Psi(z)=d/(dz)lnGamma(z)=(Gamma^'(z))/(Gamma(z))
(1)

defined as the logarithmic derivative of the gamma function Gamma(z), and

F(z)=d/(dz)lnz!
(2)

defined as the logarithmic derivative of the factorial function. The two are connected by the relationship

F(z)=Psi(z+1).
(3)

The nth derivative of Psi(z) is called the polygamma function, denoted psi_n(z). The notation

psi_0(z)=Psi(z)
(4)

is therefore frequently used for the digamma function itself, and Erdélyi et al. (1981) use the notation psi(z) for Psi(z). The digamma function psi_0(z) is returned by the function PolyGamma[z] or PolyGamma[0, z] in the Wolfram Language, and typeset using the notation psi^((0))(z).

The digamma function arises in simple sums such as

sum_(k=0)^(infty)((-1)^k)/(zk+1)=(Phi(-1,1,z^(-1)))/z
(5)
=1/(2z)[psi_0((z+1)/(2z))-psi_0(1/(2z))],
(6)

where Phi(z,s,a) is a Lerch transcendent.

Special cases are given by

sum_(k=0)^(infty)((-1)^k)/(k+1)=ln2
(7)
sum_(k=0)^(infty)((-1)^k)/(2k+1)=1/4pi
(8)
sum_(k=0)^(infty)((-1)^k)/(3k+1)=1/9(sqrt(3)pi+3ln2)
(9)
sum_(k=0)^(infty)((-1)^k)/(4k+1)=(pi+2coth^(-1)(sqrt(2)))/(4sqrt(2)).
(10)

Gauss's digamma theorem states that

(Gamma^'(p/q))/(Gamma(p/q))=-gamma-ln(2q)-1/2picot((pip)/q)+2sum_(0<n<q/2)cos((2pipn)/q)ln[sin((pin)/q)]
(11)

(Allouche 1992, Knuth 1997, p. 94).

An asymptotic series for the digamma function is given by

psi_0(z+1)∼d/(dz)lim_(n->infty)[lnn!+zlnn-ln(z+1)-ln(z+2)-...-ln(z+n)]
(12)
=lim_(n->infty)(lnn-1/(z+1)-1/(z+2)-...-1/(z+n))
(13)
=-gamma-sum_(n=1)^(infty)(1/(z+n)-1/n)
(14)
=-gamma+sum_(n=1)^(infty)z/(n(n+z))
(15)
=lnz+1/(2z)-sum_(n=1)^(infty)(B_(2n))/(2nz^(2n)),
(16)

where gamma is the Euler-Mascheroni constant and B_(2n) are Bernoulli numbers.

The digamma function satisfies

psi_0(z)=int_0^infty((e^(-t))/t-(e^(-zt))/(1-e^(-t)))dt.
(17)

For integer z=n,

psi_0(n)=-gamma+sum_(k=1)^(n-1)1/k=-gamma+H_(n-1),
(18)

where gamma is the Euler-Mascheroni constant and H_n is a harmonic number.

Other identities include

(dpsi_0)/(dz)=sum_(n=0)^infty1/((z+n)^2)
(19)
psi_0(1-z)-psi_0(z)=picot(piz)
(20)
psi_0(z+1)=psi_0(z)+1/z
(21)
psi_0(2z)=1/2psi_0(z)+1/2psi_0(z+1/2)+ln2.
(22)

Special values are

psi_0(1/2)=-gamma-2ln2
(23)
psi_0(1)=-gamma.
(24)

At integer values,

psi_0(n)=-gamma+sum_(k=1)^(n-1)1/k
(25)
=-gamma+H_(n-1)
(26)

(Derbyshire 2004, p. 58), and at half-integral values,

psi_0(1/2+n)=-gamma-2ln2+2sum_(k=1)^(n)1/(2k-1)
(27)
=-gamma+H_(n-1/2),
(28)

where H_n is a harmonic number.

It is given by the unit square integral

psi_0(u)=lnu-int_0^1int_0^1(1-x)/((1-xy)(-lnxy))(xy)^(u-1)dxdy
(29)

for u>0 (Guillera and Sondow 2005). Plugging in u=1 gives a special case involving the Euler-Mascheroni constant.

The series for psi_0(z) is given by

psi_0(z)=-1/z+sum_(n=0)^infty(psi_n(1))/(n!)z^n.
(30)

A logarithmic series is given by

psi_0(z)=sum_(n=0)^infty1/(n+1)sum_(k=0)^n(-1)^k(n; k)ln(z+k)
(31)

(Guillera and Sondow 2005).

A surprising identity that arises from the FoxTrot series is given by

-psi_0(1/2(-1)^(1/3))-psi_0(-1/2(-1)^(2/3))+psi_0(1/2(1+(-1)^(1/3)))+psi_0(1/2(1-1(-1)^(2/3)))=2pisech(1/2sqrt(3)pi).
(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/

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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 psi 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 (F) and Trigamma (F^') 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 psi(x)." Ch. 44 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 423-434, 1987.

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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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