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variants of this functions
Zeta




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s,a] > Differentiation > Low-order differentiation > With respect to s > For zeta(s,a) > Derivatives at negarive integer points





http://functions.wolfram.com/10.02.20.0032.01









  


  








Input Form





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








Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02