WOLFRAM

gives the Hurwitz zeta function TemplateBox[{s, a}, HurwitzZeta].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Hurwitz zeta function is defined as an analytic continuation of TemplateBox[{s, a}, HurwitzZeta]=sum_(k=0)^(infty)(k+a)^(-s).
  • HurwitzZeta is identical to Zeta for .
  • Unlike Zeta, HurwitzZeta has singularities at for non-negative integers .
  • HurwitzZeta has branch cut discontinuities in the complex plane running from to .
  • For certain special arguments, HurwitzZeta automatically evaluates to exact values.
  • HurwitzZeta can be evaluated to arbitrary numerical precision.
  • HurwitzZeta automatically threads over lists.

Examples

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Basic Examples  (6)Summary of the most common use cases

Evaluate numerically:

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Plot over a subset of the reals:

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Plot over a subset of the complexes:

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Series expansion at the origin:

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Series expansion at Infinity:

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Series expansion at a singular point:

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Scope  (35)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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Evaluate to high precision:

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The precision of the output tracks the precision of the input:

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Complex number input:

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Evaluate efficiently at high precision:

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Compute average-case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix HurwitzZeta function using MatrixFunction:

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Specific Values  (5)

Simple exact values are generated automatically:

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HurwitzZeta[s,a] for symbolic a:

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HurwitzZeta[s,a] for symbolic s:

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Values at zero:

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Find a value of s for which HurwitzZeta[s,1]=1.05:

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Visualization  (3)

Plot the HurwitzZeta as a function of its parameter s:

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Plot the HurwitzZeta function for various orders:

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Plot the real part of HurwitzZeta function:

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Plot the imaginary part of HurwitzZeta function:

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Function Properties  (11)

Real domain of TemplateBox[{x, a}, HurwitzZeta]:

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For positive , this is simply :

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For negative integer , the domain is just the negative integers:

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Complex domain:

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For positive , this is again :

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Approximate function range of TemplateBox[{x, 3}, HurwitzZeta]:

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HurwitzZeta threads elementwise over lists:

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HurwitzZeta is not an analytic function:

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Nor is it meromorphic:

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TemplateBox[{x, 3}, HurwitzZeta] is neither non-decreasing nor non-increasing:

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TemplateBox[{x, 3}, HurwitzZeta] is not injective:

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TemplateBox[{3, a}, HurwitzZeta] is surjective:

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TemplateBox[{4, a}, HurwitzZeta] is not surjective:

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TemplateBox[{x, 3}, HurwitzZeta] is neither non-negative nor non-positive:

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TemplateBox[{2, a}, HurwitzZeta] has both singularity and discontinuity for negative integers:

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TemplateBox[{x, 3}, HurwitzZeta] is neither convex nor concave:

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TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to a:

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Higher derivatives with respect to a:

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Plot the higher derivatives with respect to a when s=3:

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Formula for the ^(th) derivative with respect to a:

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Integration  (3)

Compute the indefinite integral using Integrate:

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Verify the anti-derivative:

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Definite integral:

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More integrals:

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Series Expansions  (2)

Find the Taylor expansion using Series:

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Plots of the first three approximations around :

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Taylor expansion at a generic point:

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Function Identities and Simplifications  (2)

HurwitzZeta is defined through the identity:

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Recurrence identity:

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Applications  (1)Sample problems that can be solved with this function

The word count in a text follows a Zipf distribution:

Fit a ZipfDistribution to the word frequency data:

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Fit a truncated ZipfDistribution to counts at most 50 using rhohat as a starting value:

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Visualize the CDFs up to the truncation value:

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Estimate the proportion of the original data not included in the truncated model:

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Properties & Relations  (2)Properties of the function, and connections to other functions

HurwitzZeta can be generated by symbolic solvers:

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For , two-argument Zeta coincides with HurwitzZeta:

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Possible Issues  (2)Common pitfalls and unexpected behavior

HurwitzZeta differs from the two-argument form of Zeta by a different choice of branch cut:

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HurwitzZeta includes singular terms, unlike Zeta:

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Neat Examples  (1)Surprising or curious use cases

ComplexPlot of HurwitzZeta function, as a function of with :

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Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.
Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.

Text

Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.

Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.

CMS

Wolfram Language. 2008. "HurwitzZeta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HurwitzZeta.html.

Wolfram Language. 2008. "HurwitzZeta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HurwitzZeta.html.

APA

Wolfram Language. (2008). HurwitzZeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HurwitzZeta.html

Wolfram Language. (2008). HurwitzZeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HurwitzZeta.html

BibTeX

@misc{reference.wolfram_2025_hurwitzzeta, author="Wolfram Research", title="{HurwitzZeta}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/HurwitzZeta.html}", note=[Accessed: 08-June-2025 ]}

@misc{reference.wolfram_2025_hurwitzzeta, author="Wolfram Research", title="{HurwitzZeta}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/HurwitzZeta.html}", note=[Accessed: 08-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_hurwitzzeta, organization={Wolfram Research}, title={HurwitzZeta}, year={2008}, url={https://reference.wolfram.com/language/ref/HurwitzZeta.html}, note=[Accessed: 08-June-2025 ]}

@online{reference.wolfram_2025_hurwitzzeta, organization={Wolfram Research}, title={HurwitzZeta}, year={2008}, url={https://reference.wolfram.com/language/ref/HurwitzZeta.html}, note=[Accessed: 08-June-2025 ]}