DirichletBeta

DirichletBeta[s]

gives the Dirichlet beta function TemplateBox[{s}, DirichletBeta].

Details

  • The Dirichlet beta function is also known as the Catalan beta function.
  • DirichletBeta is a mathematical function, suitable for both symbolic and numeric manipulation.
  • For , the Dirichlet beta function is defined as TemplateBox[{s}, DirichletBeta]=sum_(n=0)^infty((-1)^n)/((2 n+1)^s).
  • For certain special arguments, DirichletBeta automatically evaluates to exact values.
  • DirichletBeta is an entire function with branch cut discontinuities.
  • DirichletBeta can be evaluated to arbitrary numerical precision.
  • DirichletBeta automatically threads over lists.

Examples

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Basic Examples  (4)

Plot on the real axis:

Visualize in the complex plane:

The Dirichlet beta function expands in terms of zeta functions:

Compute some special values:

Scope  (5)

DirichletBeta is neither non-decreasing nor non-increasing:

DirichletBeta is not injective:

DirichletBeta is neither non-negative nor non-positive:

DirichletBeta is neither convex nor concave:

Compute special values of derivatives:

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

Text

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

CMS

Wolfram Language. 2014. "DirichletBeta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DirichletBeta.html.

APA

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

BibTeX

@misc{reference.wolfram_2023_dirichletbeta, author="Wolfram Research", title="{DirichletBeta}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/DirichletBeta.html}", note=[Accessed: 01-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_dirichletbeta, organization={Wolfram Research}, title={DirichletBeta}, year={2014}, url={https://reference.wolfram.com/language/ref/DirichletBeta.html}, note=[Accessed: 01-March-2024 ]}