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52 lines
1.5 KiB
ReStructuredText
52 lines
1.5 KiB
ReStructuredText
.. _algorithms_hurwitz:
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Algorithms for the Hurwitz zeta function
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===============================================================================
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Euler-Maclaurin summation
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-------------------------------------------------------------------------------
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The Euler-Maclaurin formula allows evaluating the Hurwitz zeta function and
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its derivatives for general complex input. The algorithm is described
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in [Joh2013]_.
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Parameter Taylor series
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-------------------------------------------------------------------------------
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To evaluate `\zeta(s,a)` for several nearby parameter values, the following
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Taylor expansion is useful:
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.. math ::
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\zeta(s,a+x) = \sum_{k=0}^{\infty} (-x)^k \frac{(s)_k}{k!} \zeta(s+k,a)
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We assume that `a \ge 1` is real and that `\sigma = \operatorname{re}(s)`
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with `K + \sigma > 1`. The tail is bounded by
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.. math ::
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\sum_{k=K}^{\infty} |x|^k \frac{|(s)_k|}{k!} \zeta(\sigma+k,a) \le
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\sum_{k=K}^{\infty}
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|x|^k \frac{|(s)_k|}{k!} \left[
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\frac{1}{a^{\sigma+k}} + \frac{1}{(\sigma+k-1) a^{\sigma+k-1}} \right].
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Denote the term on the right by `T(k)`. Then
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.. math ::
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\left|\frac{T(k+1)}{T(k)}\right| =
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\frac{|x|}{a}
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\frac{(k+\sigma-1)}{(k+\sigma)}
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\frac{(k+\sigma+a)}{(k+\sigma+a-1)}
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\frac{|k+s|}{(k+1)}
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\le
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\frac{|x|}{a}
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\left(1 + \frac{1}{K+\sigma+a-1}\right)
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\left(1 + \frac{|s-1|}{K+1}\right) = C
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and if `C < 1`,
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.. math ::
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\sum_{k=K}^{\infty} T(k) \le \frac{T(K)}{1-C}.
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