TOPICS
At rational arguments 
, the digamma function 
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)]](/images/equations/GausssDigammaTheorem/NumberedEquation1.svg) |
(1)
|
for 
(Knuth 1997, p. 94). These give the special values
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(2)
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(3)
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(4)
|
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(5)
|
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(6)
|
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(7)
|
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(8)
|
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(9)
|
where 
is the Euler-Mascheroni constant.
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.Weisstein, Eric W. "Gauss's Digamma Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssDigammaTheorem.html
