DirichletEta

DirichletEta[s]

gives the Dirichlet eta function TemplateBox[{s}, DirichletEta].

Details

  • The Dirichlet eta function is also known as the alternating zeta function.
  • DirichletEta is a mathematical function, suitable for both symbolic and numeric manipulation.
  • For , the Dirichlet eta function is defined as TemplateBox[{s}, DirichletEta]=sum_(n=0)^infty((-1)^n)/((n+1)^s).
  • For certain special arguments, DirichletEta automatically evaluates to exact values.
  • DirichletEta is an entire function with branch cut discontinuities.
  • DirichletEta can be evaluated to arbitrary numerical precision.
  • DirichletEta automatically threads over lists.
  • DirichletEta can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (3)

Plot over the reals:

Visualize in the complex plane:

Compute a special value:

Scope  (6)

DirichletEta is neither non-decreasing nor non-increasing:

DirichletEta is not injective:

DirichletEta is neither non-negative nor non-positive:

DirichletEta is neither convex nor concave:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix DirichletEta function using MatrixFunction:

Properties & Relations  (1)

Verify the interrelationship among the DirichletEta, DirichletLambda and Zeta functions:

Wolfram Research (2014), DirichletEta, Wolfram Language function, https://reference.wolfram.com/language/ref/DirichletEta.html (updated 2022).

Text

Wolfram Research (2014), DirichletEta, Wolfram Language function, https://reference.wolfram.com/language/ref/DirichletEta.html (updated 2022).

CMS

Wolfram Language. 2014. "DirichletEta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/DirichletEta.html.

APA

Wolfram Language. (2014). DirichletEta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DirichletEta.html

BibTeX

@misc{reference.wolfram_2025_dirichleteta, author="Wolfram Research", title="{DirichletEta}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/DirichletEta.html}", note=[Accessed: 26-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_dirichleteta, organization={Wolfram Research}, title={DirichletEta}, year={2022}, url={https://reference.wolfram.com/language/ref/DirichletEta.html}, note=[Accessed: 26-April-2025 ]}