![TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW1]](Files/WeierstrassEta1.en/3.png "TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW1]"){kind=small}
WeierstrassEta1
WeierstrassEta1[{g2,g3}]
gives the value η1 of the Weierstrass zeta function ζ at the half-period ![TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW1]](Files/WeierstrassEta1.en/1.png "TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW1]"){kind=small}
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Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- WeierstrassEta1 can be evaluated to arbitrary numerical precision.
Examples
open all close allBasic Examples (3)
Represent the value of WeierstrassZeta at the half-period ω1:
WeierstrassZeta[WeierstrassHalfPeriodW1[{g2, g3}], {g2, g3}]WeierstrassEta1[{12, 8}]WeierstrassEta1[{9., 3.}]Plot the real and imaginary parts of η1:
Plot3D[{Re[WeierstrassEta1[{g2, g3}]], Im[WeierstrassEta1[{g2, g3}]]}, {g2, -4, 4}, {g3, 1, 7}]Scope (8)
Evaluate for complex arguments:
WeierstrassEta1[{0.5 + I, 1.}]Evaluate to arbitrary numerical precision:
N[WeierstrassEta1[{23, 7}], 50]The precision of the output tracks the precision of the input:
WeierstrassEta1[{7, 3.12345678910111213141516}]WeierstrassEta1[{7, 3.1234567891011121314151617181920}]Evaluate symbolically for the equianharmonic case:
WeierstrassEta1[{0, 1}]Evaluate symbolically for the lemniscatic case:
WeierstrassEta1[{1, 0}]WeierstrassEta1 has both singularities and discontinuities:
FunctionSingularities[WeierstrassEta1[{x, y}], {x, y}]//QuietFunctionDiscontinuities[WeierstrassEta1[{x, y}], {x, y}]//QuietWeierstrassEta1 is neither non-negative nor non-positive:
FunctionSign[WeierstrassEta1[{x, y}], {x, y}]ReImPlot[WeierstrassEta1[{x, x}], {x, -5, 5}]WeierstrassEta1 is neither convex nor concave:
FunctionConvexity[WeierstrassEta1[{x, y}], {x, y}]TraditionalForm formatting:
WeierstrassEta1[{Subscript[g, 2], Subscript[g, 3]}]//TraditionalFormProperties & Relations (2)
WeierstrassZeta is quasi‐periodic on the lattice of periods of WeierstrassP:
WeierstrassZeta[z + 2WeierstrassHalfPeriodW1[{g2, g3}], {g2, g3}]WeierstrassZeta[z + 2WeierstrassHalfPeriodW2[{g2, g3}], {g2, g3}]WeierstrassZeta[z + 2WeierstrassHalfPeriodW3[{g2, g3}], {g2, g3}]The values of WeierstrassZeta at the half-periods are not linearly independent:
Block[{g2 = 5, g3 = 1`20},
FindIntegerNullVector[{WeierstrassEta1[{g2, g3}], WeierstrassEta2[{g2, g3}], WeierstrassEta3[{g2, g3}]}]]This identity holds for all arguments:
0 == %.{WeierstrassEta1[{Subscript[g, 2], Subscript[g, 3]}], WeierstrassEta2[{Subscript[g, 2], Subscript[g, 3]}], WeierstrassEta3[{Subscript[g, 2], Subscript[g, 3]}]}//TraditionalFormText
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