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WeierstrassEta3
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WeierstrassEta3

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WeierstrassEta3[{g2,g3}]

gives the value η3 of the Weierstrass zeta function ζ at the half-period TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • WeierstrassEta3 can be evaluated to arbitrary numerical precision.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Represent the value of WeierstrassZeta at the half-period ω3:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-dzadk4
Out[1]=1

Evaluate numerically:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-dgzfnh
Out[1]=1

Plot the real and imaginary parts of η3:

In[1]:=1
✖
https://wolfram.com/xid/0btotzhnsfnp-fcuh4
Out[1]=1

Scope  (8)Survey of the scope of standard use cases

Evaluate for complex arguments:

In[1]:=1
✖
https://wolfram.com/xid/0btotzhnsfnp-ep8m36
Out[1]=1

Evaluate to arbitrary numerical precision:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-edgtwd
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-bapeoy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btotzhnsfnp-czortl
Out[2]=2

Evaluate symbolically for the equianharmonic case:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-dmeqrb
Out[1]=1

Evaluate symbolically for the lemniscatic case:

In[2]:=2
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https://wolfram.com/xid/0btotzhnsfnp-bs95ea
Out[2]=2

WeierstrassEta3 has both singularities and discontinuities:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-mdtl3h
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btotzhnsfnp-mn5jws
Out[2]=2

WeierstrassEta3 is neither non-negative nor non-positive:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-84dui
Out[1]=1

It is inherently complex:

In[2]:=2
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https://wolfram.com/xid/0btotzhnsfnp-t326s
Out[2]=2

WeierstrassEta3 is neither convex nor concave:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-8kku21
Out[1]=1

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-bnbf44

Properties & Relations  (2)Properties of the function, and connections to other functions

WeierstrassZeta is quasiperiodic on the lattice of periods of WeierstrassP:

In[1]:=1
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https://wolfram.com/xid/0btotzhnsfnp-xeg4u
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btotzhnsfnp-f62sqy
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0btotzhnsfnp-bp047i
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0btotzhnsfnp-bwnl1o
Out[1]=1

This identity holds for all arguments:

In[2]:=2
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https://wolfram.com/xid/0btotzhnsfnp-b0y60x
Wolfram Research (2017), WeierstrassEta3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassEta3.html.
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Wolfram Research (2017), WeierstrassEta3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassEta3.html.

Text

Wolfram Research (2017), WeierstrassEta3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassEta3.html.

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Wolfram Research (2017), WeierstrassEta3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassEta3.html.

CMS

Wolfram Language. 2017. "WeierstrassEta3." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeierstrassEta3.html.

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Wolfram Language. 2017. "WeierstrassEta3." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeierstrassEta3.html.

APA

Wolfram Language. (2017). WeierstrassEta3. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassEta3.html

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Wolfram Language. (2017). WeierstrassEta3. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassEta3.html

BibTeX

@misc{reference.wolfram_2025_weierstrasseta3, author="Wolfram Research", title="{WeierstrassEta3}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassEta3.html}", note=[Accessed: 08-June-2025 ]}

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@misc{reference.wolfram_2025_weierstrasseta3, author="Wolfram Research", title="{WeierstrassEta3}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassEta3.html}", note=[Accessed: 08-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_weierstrasseta3, organization={Wolfram Research}, title={WeierstrassEta3}, year={2017}, url={https://reference.wolfram.com/language/ref/WeierstrassEta3.html}, note=[Accessed: 08-June-2025 ]}

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@online{reference.wolfram_2025_weierstrasseta3, organization={Wolfram Research}, title={WeierstrassEta3}, year={2017}, url={https://reference.wolfram.com/language/ref/WeierstrassEta3.html}, note=[Accessed: 08-June-2025 ]}