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DirichletBeta[s]

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

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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:

Wolfram Language code: Plot[DirichletBeta[s], {s, -4, 4}]

Visualize in the complex plane:

Wolfram Language code: ComplexPlot3D[DirichletBeta[z], {z, 2}]

The Dirichlet beta function expands in terms of zeta functions:

Wolfram Language code: DirichletBeta[s]

Compute some special values:

Wolfram Language code: DirichletBeta /@ {2, 3}

Scope  (7)

DirichletBeta is neither non-decreasing nor non-increasing:

Wolfram Language code: FunctionMonotonicity[DirichletBeta[x], x]

DirichletBeta is not injective:

Wolfram Language code: FunctionInjective[DirichletBeta[x], x]
Wolfram Language code: Plot[{DirichletBeta[x], -.3}, {x, -4, 4}]

DirichletBeta is neither non-negative nor non-positive:

Wolfram Language code: FunctionSign[DirichletBeta[x], x]

DirichletBeta is neither convex nor concave:

Wolfram Language code: FunctionConvexity[DirichletBeta[x], x]

Compute special values of derivatives:

Wolfram Language code: DirichletBeta'[-1]//FullSimplify
Wolfram Language code: DirichletBeta'[0]//FullSimplify

Compute average-case statistical intervals using Around:

Wolfram Language code: DirichletBeta[ Around[2, 0.01]]

Compute the elementwise values of an array:

Wolfram Language code: DirichletBeta[{{1, -1}, {-1, 1}}]

Or compute the matrix DirichletBeta function using MatrixFunction:

Wolfram Language code: MatrixFunction[DirichletBeta, {{1, -1}, {-1, 1}}]//FullSimplify
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_2026_dirichletbeta, author="Wolfram Research", title="{DirichletBeta}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/DirichletBeta.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_dirichletbeta, organization={Wolfram Research}, title={DirichletBeta}, year={2014}, url={https://reference.wolfram.com/language/ref/DirichletBeta.html}, note=[Accessed: 12-June-2026]}