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)]](https://mathworld.wolfram.com/images/equations/GausssDigammaTheorem/NumberedEquation1.svg){kind=link}
for 👁 0<p<q
(Knuth 1997, p. 94). These give the special values
where 👁 gamma
is the Euler-Mascheroni constant.
See also
Digamma Function, Polygamma FunctionExplore with Wolfram|Alpha
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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 TheoremCite this as:
Weisstein, Eric W. "Gauss's Digamma Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GausssDigammaTheorem.html