![TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]](Files/WeierstrassEta3.en/3.png "TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]"){kind=small}
WeierstrassEta3
WeierstrassEta3[{g2,g3}]
gives the value η3 of the Weierstrass zeta function ζ at the half-period ![TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]](Files/WeierstrassEta3.en/1.png "TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]"){kind=small}
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Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- WeierstrassEta3 can be evaluated to arbitrary numerical precision.
Examples
open all close allBasic Examples (3)
Represent the value of WeierstrassZeta at the half-period ω3:
WeierstrassZeta[WeierstrassHalfPeriodW3[{g2, g3}], {g2, g3}]WeierstrassEta3[{9., 3.}]Plot the real and imaginary parts of η3:
Plot3D[{Re[WeierstrassEta3[{g2, g3}]], Im[WeierstrassEta3[{g2, g3}]]}, {g2, -4, 4}, {g3, 1, 7}]Scope (8)
Evaluate for complex arguments:
WeierstrassEta3[{0.3 - 0.4I, 0.5}]Evaluate to arbitrary numerical precision:
N[WeierstrassEta3[{25, 7}], 50]The precision of the output tracks the precision of the input:
WeierstrassEta3[{7, 3.12345678910111213141516}]WeierstrassEta3[{7, 3.1234567891011121314151617181920}]Evaluate symbolically for the equianharmonic case:
WeierstrassEta3[{0, 1}]Evaluate symbolically for the lemniscatic case:
WeierstrassEta3[{1, 0}]WeierstrassEta3 has both singularities and discontinuities:
FunctionSingularities[WeierstrassEta3[{x, y}], {x, y}]//QuietFunctionDiscontinuities[WeierstrassEta3[{x, y}], {x, y}]//QuietWeierstrassEta3 is neither non-negative nor non-positive:
FunctionSign[WeierstrassEta3[{x, y}], {x, y}]ReImPlot[WeierstrassEta3[{x, x}], {x, -5, 5}]WeierstrassEta3 is neither convex nor concave:
FunctionConvexity[WeierstrassEta3[{x, y}], {x, y}]TraditionalForm formatting:
WeierstrassEta3[{g2, g3}]//TraditionalFormProperties & Relations (2)
WeierstrassZeta is quasiperiodic 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), 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.
APA
Wolfram Language. (2017). WeierstrassEta3. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassEta3.html