Hurwitz Zeta Function

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Hurwitz Zeta Function

Definition

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Representations by the Euler-Maclaurin Formula

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Series Representations

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&


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Special Values

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&


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integers, &


Derivatives

a-Derivative

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s-Derivatives

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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \opminus^k \HurwitzZeta^{(k)}@{s}{a} = \frac{(\ln@@{a})^k}{a^s} \left( \frac{1}{2} + \frac{a}{s-1} \right) + k! a^{1-s} \sum_{r=0}^{k-1} \frac{(\ln@@{a})^r}{r! (s-1)^{k-r+1}} - s (s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^k} {(x+a)^{s+2}} \diff{x} + k (2s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^{k-1}} {(x+a)^{s+2}} \diff{x} - k (k-1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^{k-2}} {(x+a)^{s+2}} \diff{x} }}

Constraint(s): &
&


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HurwitzZeta'@{1-2n}{\frac{h}{k}} = \frac{(\digamma@{2n} - \ln@{2 \cpi k}) \BernoulliB{2n}@{h/k}}{2n} - \frac{(\digamma@{2n} - \ln@{2 \cpi}) \BernoulliB{2n}}{2 n k^{2n}} + \frac{\opminus^{n+1} \cpi}{(2 \cpi k)^{2n}} \sum_{r=1}^{k-1} \sin@{\frac{2 \cpi r h}{k}} \digamma^{(2n-1)}@{\frac{r}{k}} + \frac{\opminus^{n+1} 2 \cdot (2n-1)!}{(2 \cpi k)^{2n}} \sum_{r=1}^{k-1} \cos@{\frac{2 \cpi r h}{k}} \HurwitzZeta'@{2n}{\frac{r}{k}} + \frac{\RiemannZeta'@{1-2n}}{k^{2n}} }}

Constraint(s): are integers with &


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Constraint(s):


Constraint(s): &


Integral Representations

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the integration contour is a loop around the negative real axis; it starts at , encircles the origin once in the positive direction without enclosing any of the points

, and returns to


Further Integral Representations

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Substitution(s):


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Further Series Representations

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or , ,


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is a Dirichlet character~


Sums

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a-Asymptotic Behavior

Constraint(s): with fixed


Constraint(s): with fixed &


Constraint(s): in the sector , with and fixed


Constraint(s): in the sector