TOPICS
Search

Gauss's Digamma Theorem


At rational arguments p/q, the digamma function psi_0(p/q) is given by

 psi_0(p/q)=-gamma-ln(2q)-1/2picot(p/qpi) 
 +2sum_(k=1)^([q/2]-1)cos((2pipk)/q)ln[sin((pik)/q)]
(1)

for 0<p<q (Knuth 1997, p. 94). These give the special values

psi_0(1/2)=-gamma-2ln2
(2)
psi_0(1/3)=1/6(-6gamma-pisqrt(3)-9ln3)
(3)
psi_0(2/3)=1/6(-6gamma+pisqrt(3)-9ln3)
(4)
psi_0(1/4)=1/2(-2gamma-pi-6ln2)
(5)
psi_0(3/4)=1/2(-2gamma+pi-6ln2)
(6)
psi_0(1/6)=-gamma-1/2sqrt(3)pi-2ln2-3/2ln3
(7)
psi_0(5/6)=-gamma+1/2sqrt(3)pi-2ln2-3/2ln3
(8)
psi_0(1)=-gamma,
(9)

where gamma is the Euler-Mascheroni constant.


See also

Digamma Function, Polygamma Function

Explore with Wolfram|Alpha

References

Allouche, J.-P. "Series and Infinite Products related to Binary Expansions of Integers." 1992. http://algo.inria.fr/seminars/sem92-93/allouche.ps.Böhmer, E. Differenzengleichungen und bestimmte Integrale. Leipzig, Germany: Teubner, p. 77, 1939.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.Gradshteyn, I. S. and Ryzhik, I. M. Formula 8.3636 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Jensen, J. L. W. V. "An Elementary Exposition of the Theory of the Gamma Function." Ann. Math. 17, 124-166, 1915.Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1997.Kölbig, K. S. "The Polygamma Function and the Derivatives of the Cotangent Function for Rational Arguments." Report CN/96/5. CERN Computing and Networks Division, 1996.Lösch, F. and Schoblik, F. Die Fakultät und verwandte Funktionen. Leipzig, Germany: Teubner, p. 12, 1951.

Referenced on Wolfram|Alpha

Gauss's Digamma Theorem

Cite this as:

Weisstein, Eric W. "Gauss's Digamma Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssDigammaTheorem.html

Subject classifications