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variants of this functions
PolyGamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Specific values > Values at fixed points





http://functions.wolfram.com/06.15.03.0062.01









  


  










Input Form





PolyGamma[-2 n, 1/2] == (2^(-1 - 2 n) (EulerGamma (-1 + 4 n) + I Pi (1 - 2 n) + 4 n Log[Pi] + 4 n PolyGamma[2 n]))/(n (-1 + 2 n)!) - (BernoulliB[2 n] Log[2])/ (2^(2 n) n (-1 + 2 n)!) - ((2^(2 n - 1) - 1) Derivative[1][Zeta][-2 n + 1])/ (2^(2 n - 1) (-1 + 2 n)!) - (2^(1 - 2 n)/(-1 + 2 n)!) Sum[((-1)^j Binomial[-1 + 2 n, j])/(-1 - j + 2 n), {j, 0, -2 + 2 n}] - 2^(1 - 2 n) Sum[Zeta[1 + j]/((I Pi)^j (-1 - j + 2 n)!), {j, 1, -1 + 2 n}] - (2^(1 - 2 n)/(-1 + 2 n)!) Sum[(-2)^k Binomial[-1 + 2 n, k] (Sum[((-1 + 2^(j - k)) Binomial[k, j] PolyGamma[1 - j + k] Zeta[j - k])/ 2^j, {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, -1 + 2 n}] + (2^(1 - 2 n)/(-1 + 2 n)!) Sum[(-1)^j Binomial[-1 + 2 n, j] (Log[2] + Sum[((1 - 2^(-k)) Binomial[-1 - j + 2 n, k] k! Zeta[1 + k])/ ((-I) Pi)^k, {k, 1, -1 - j + 2 n}]), {j, 0, -2 + 2 n}] /; Element[n, Integers] && n > 0










Standard Form





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MathML Form







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Date Added to functions.wolfram.com (modification date)





2007-05-02