WeierstrassEta1

WeierstrassEta1[{g2,g3}]

gives the value η1 of the Weierstrass zeta function ζ at the half-period TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW1].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • WeierstrassEta1 can be evaluated to arbitrary numerical precision.

Examples

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Basic Examples  (3)

Represent the value of WeierstrassZeta at the half-period ω1:

Evaluate numerically:

Plot the real and imaginary parts of η1:

Scope  (8)

Evaluate for complex arguments:

Evaluate to arbitrary numerical precision:

The precision of the output tracks the precision of the input:

Evaluate symbolically for the equianharmonic case:

Evaluate symbolically for the lemniscatic case:

WeierstrassEta1 has both singularities and discontinuities:

WeierstrassEta1 is neither non-negative nor non-positive:

It is inherently complex:

WeierstrassEta1 is neither convex nor concave:

TraditionalForm formatting:

Properties & Relations  (2)

WeierstrassZeta is quasi‐periodic on the lattice of periods of WeierstrassP:

The values of WeierstrassZeta at the half-periods are not linearly independent:

This identity holds for all arguments:

Wolfram Research (2017), WeierstrassEta1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassEta1.html.

Text

Wolfram Research (2017), WeierstrassEta1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassEta1.html.

CMS

Wolfram Language. 2017. "WeierstrassEta1." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeierstrassEta1.html.

APA

Wolfram Language. (2017). WeierstrassEta1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassEta1.html

BibTeX

@misc{reference.wolfram_2025_weierstrasseta1, author="Wolfram Research", title="{WeierstrassEta1}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassEta1.html}", note=[Accessed: 24-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_weierstrasseta1, organization={Wolfram Research}, title={WeierstrassEta1}, year={2017}, url={https://reference.wolfram.com/language/ref/WeierstrassEta1.html}, note=[Accessed: 24-July-2025 ]}