gives the Hurwitz zeta function .
- 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.
Examplesopen allclose all
Basic 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:
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:
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 (11)
For positive , this is simply :
For negative integer , the domain is just the negative integers:
For positive , this is again :
Approximate function range of :
HurwitzZeta threads elementwise over lists:
HurwitzZeta is not an analytic function:
is neither non-decreasing nor non-increasing:
is neither non-negative nor non-positive:
has both singularity and discontinuity for negative integers:
is neither convex nor concave:
Compute the indefinite integral using Integrate:
Series Expansions (2)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
Function Identities and Simplifications (2)
HurwitzZeta is defined through the identity:
The word count in a text follows a Zipf distribution:
Fit a ZipfDistribution to the word frequency data:
Fit a truncated ZipfDistribution to counts at most 50 using rhohat as a starting value:
Visualize the CDFs up to the truncation value:
Estimate the proportion of the original data not included in the truncated model:
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:
Neat Examples (1)
ComplexPlot of HurwitzZeta function, as a function of with :
Wolfram Research (2008), HurwitzZeta, Wolfram Language function, 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.
Wolfram Language. (2008). HurwitzZeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HurwitzZeta.html