HurwitzZeta
HurwitzZeta[s,a]
gives the Hurwitz zeta function .
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Hurwitz zeta function is defined as an analytic continuation of
.
- 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
open allclose allBasic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Scope (26)
Numerical Evaluation (4)
Specific Values (5)
Simple exact values are generated automatically:
HurwitzZeta[s,a] for symbolic a:
HurwitzZeta[s,a] for symbolic s:
Find a value of s for which HurwitzZeta[s,1]=1.05:
Visualization (3)
Plot the HurwitzZeta as a function of its parameter s:
Plot the HurwitzZeta function for various orders:
Plot the real part of HurwitzZeta function:
Plot the imaginary part of HurwitzZeta function:
Function Properties (4)
Real domain of HurwitzZeta for non-negative integers:
Real domain of HurwitzZeta for negative integers:
Approximate function range of HurwitzZeta:
HurwitzZeta threads elementwise over lists:
TraditionalForm formatting:
Differentiation (3)
Integration (3)
Compute the indefinite integral using Integrate:
Series Expansions (2)
Find the Taylor expansion using Series:
Function Identities and Simplifications (2)
Properties & Relations (2)
HurwitzZeta can be generated by symbolic solvers:
For , two-argument Zeta coincides with HurwitzZeta:
Possible Issues (2)
HurwitzZeta differs from the two-argument form of Zeta by a different choice of branch cut:
HurwitzZeta includes singular terms, unlike Zeta:
Text
Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.
BibTeX
BibLaTeX
CMS
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