WeierstrassHalfPeriodW2
WeierstrassHalfPeriodW2[{g2,g3}]
gives the half-period ω2 for the Weierstrass elliptic functions corresponding to the invariants {g2,g3}.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The half-periods define the fundamental period parallelogram for the Weierstrass elliptic functions.
- WeierstrassHalfPeriodW2 can be evaluated to arbitrary precision.
- WeierstrassHalfPeriodW2 can be used with CenteredInterval objects. »
Examples
open all close allBasic Examples (3)
WeierstrassHalfPeriodW2[{11.2, 7}]Plot the real and imaginary parts of the second half-period:
Plot3D[{Re[WeierstrassHalfPeriodW2[{g2, g3}]], Im[WeierstrassHalfPeriodW2[{g2, g3}]]}, {g2, -4, 4}, {g3, 2, 10}]Compute the value of the Weierstrass 
function at the second half-period:
WeierstrassP[WeierstrassHalfPeriodW2[{11, 7}], {11, 7}]N[%]Scope (8)
Evaluate to arbitrary precision:
N[WeierstrassHalfPeriodW2[{11, 7}], 20]The precision of the output tracks the precision of the input:
WeierstrassHalfPeriodW2[{11, 7.0000000000000000000000000000}]Evaluate symbolically for the equianharmonic case:
WeierstrassHalfPeriodW2[{0, 1}]Evaluate symbolically for the lemniscatic case:
WeierstrassHalfPeriodW2[{1, 0}]WeierstrassHalfPeriodW2 has both singularities and discontinuities:
FunctionSingularities[WeierstrassHalfPeriodW2[{x, y}], x]//QuietFunctionDiscontinuities[WeierstrassHalfPeriodW2[{x, y}], x]//QuietWeierstrassHalfPeriodW2 is neither non-negative nor non-positive:
FunctionSign[WeierstrassHalfPeriodW2[{x, y}], {x, y}]WeierstrassHalfPeriodW2 is neither convex nor concave:
FunctionConvexity[WeierstrassHalfPeriodW2[{x, y}], {x, y}]WeierstrassHalfPeriodW2 can be used with CenteredInterval objects:
WeierstrassHalfPeriodW2[{CenteredInterval[1 + I, (1 + I) / 10 ^ 9], CenteredInterval[3 / 4, 1 / 10 ^ 9]}]TraditionalForm formatting:
WeierstrassHalfPeriodW2[{Subscript[g, 2], Subscript[g, 3]}]//TraditionalFormProperties & Relations (4)
Up to a change in sign, the half-period 
is equal to the sum of the half-periods 
and 
:
With[{g2 = 5, g3 = 1`20},
{WeierstrassHalfPeriodW2[{g2, g3}], WeierstrassHalfPeriodW1[{g2, g3}] + WeierstrassHalfPeriodW3[{g2, g3}]}]WeierstrassP is periodic with periods equal to twice the half-periods:
WeierstrassP[z + 2WeierstrassHalfPeriodW2[{g2, g3}], {g2, g3}]WeierstrassP[z + 2WeierstrassHalfPeriodW3[{g2, g3}], {g2, g3}]WeierstrassP[z + 2WeierstrassHalfPeriodW1[{g2, g3}], {g2, g3}]The half-periods 
, 
and 
of Weierstrass elliptic functions are not linearly independent:
Block[{g2 = 5, g3 = 1`20},
FindIntegerNullVector[{WeierstrassHalfPeriodW1[{g2, g3}], WeierstrassHalfPeriodW2[{g2, g3}], WeierstrassHalfPeriodW3[{g2, g3}]}]]This identity holds for all arguments:
0 == %.{WeierstrassHalfPeriodW1[{Subscript[g, 2], Subscript[g, 3]}], WeierstrassHalfPeriodW2[{Subscript[g, 2], Subscript[g, 3]}], WeierstrassHalfPeriodW3[{Subscript[g, 2], Subscript[g, 3]}]}//TraditionalFormWeierstrassP[z + WeierstrassHalfPeriodW1[{Subscript[g, 2], Subscript[g, 3]}] + WeierstrassHalfPeriodW2[{Subscript[g, 2], Subscript[g, 3]}] + WeierstrassHalfPeriodW3[{Subscript[g, 2], Subscript[g, 3]}], {Subscript[g, 2], Subscript[g, 3]}]WeierstrassHalfPeriodW2 gives a zero of WeierstrassPPrime in the lattice cell:
WeierstrassPPrime[WeierstrassHalfPeriodW2[{g2, g3}], {g2, g3}]Text
Wolfram Research (2017), WeierstrassHalfPeriodW2, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW2.html (updated 2023).
CMS
Wolfram Language. 2017. "WeierstrassHalfPeriodW2." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW2.html.
APA
Wolfram Language. (2017). WeierstrassHalfPeriodW2. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW2.html