WeierstrassHalfPeriods
WeierstrassHalfPeriods[{g2,g3}]
gives the half‐periods {ω1,ω3} for Weierstrass elliptic functions corresponding to the invariants {g2,g3}.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The half‐periods {ω1,ω3} define the fundamental period parallelogram for the Weierstrass elliptic functions.
- WeierstrassHalfPeriods is the inverse of WeierstrassInvariants.
- WeierstrassHalfPeriods can be evaluated to arbitrary numerical precision.
- WeierstrassHalfPeriods can be used with CenteredInterval objects. »
Examples
open all close allBasic Examples (2)
WeierstrassHalfPeriods[{1., 2.}]
WeierstrassHalfPeriodW1[{1., 2.}]WeierstrassHalfPeriodW3[{1., 2.}]Plot the half-periods over a subset of the reals:
Plot[Re[WeierstrassHalfPeriods[{g2, 1 - I}]], {g2, -4, 4}]Scope (4)
N[WeierstrassHalfPeriods[{1, 2}], 50]The precision of the output tracks the precision of the input:
WeierstrassHalfPeriods[{1, 2.00000000000000000000000000000}]Symbolic evaluation of the equianharmonic case of WeierstrassHalfPeriods:
WeierstrassHalfPeriods[{0, 1}]Symbolic evaluation of the lemniscatic case of WeierstrassHalfPeriods:
WeierstrassHalfPeriods[{1, 0}]WeierstrassHalfPeriods can be used with CenteredInterval objects:
WeierstrassHalfPeriods[{CenteredInterval[1 + I, (1 + I) / 10 ^ 9], CenteredInterval[3 / 4, 1 / 10 ^ 9]}]Applications (1)
Plot an elliptic function over a period parallelogram:
Module[{g2 = 2., g3 = 3, ω1, ω3},
{ω1, ω3} = WeierstrassHalfPeriods[{g2, g3}];Graphics[GeometricTransformation[ContourPlot[Evaluate[Im[Sign[WeierstrassPPrime[(ω1 + I ω3) x - I (x + I y) ω3, {g2, g3}]]]], {x, -1, 1}, {y, -1, 1}, MaxRecursion -> 1]//First, Transpose[{ReIm[ω1], ReIm[ω3]}]]]]Properties & Relations (2)
![WeierstrassHalfPeriods[{g_2,g_3}]={TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW1],TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]}](Files/WeierstrassHalfPeriods.en/3.png "WeierstrassHalfPeriods[{g_2,g_3}]={TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW1],TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]}"){kind=small}
WeierstrassHalfPeriods[{1., 1.}] == {
WeierstrassHalfPeriodW1[{1., 1.}], WeierstrassHalfPeriodW3[{1., 1.}]}WeierstrassHalfPeriods is effectively the inverse of WeierstrassInvariants:
WeierstrassInvariants[WeierstrassHalfPeriods[{0.5, 1 - I}]]// ChopPossible Issues (1)
Assignment of half‐periods corresponding to symbolic or exact invariants is impossible as the right‐hand side is not a list:
{ω1, ω3} = WeierstrassHalfPeriods[{g2, g3}]
Use WeierstrassHalfPeriodW1 and WeierstrassHalfPeriodW3 instead:
ω1 = WeierstrassHalfPeriodW1[{g2, g3}]ω3 = WeierstrassHalfPeriodW3[{g2, g3}]Neat Examples (1)
A doubly periodic function over the complex plane:
Module[{g2 = 2., g3 = 1 + I, ω1, ω3, f, ℘},
℘ = WeierstrassP[#1, {g2, g3}]&;f[z_] = (2 ℘[2 z + 1] - 2 ℘[2 z - 1]) / ℘[z];{ω1, ω3} = N[WeierstrassHalfPeriods[{g2, g3}]];ParametricPlot3D[{Re[x ω1 + y ω3], Im[x ω1 + y ω3], Re[f[x ω1 + y ω3]]}, {x, -2, 2}, {y, -2, 2}, PlotRange -> {-1, 1}, Mesh -> False]]Related Links
MathWorld
The Wolfram Functions SiteText
Wolfram Research (1996), WeierstrassHalfPeriods, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriods.html (updated 2023).
CMS
Wolfram Language. 1996. "WeierstrassHalfPeriods." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassHalfPeriods.html.
APA
Wolfram Language. (1996). WeierstrassHalfPeriods. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassHalfPeriods.html