WeierstrassInvariantG3[{ω,ω′}]
gives the invariant 
for the Weierstrass elliptic functions corresponding to the half‐periods {ω,ω′}.
WeierstrassInvariantG3
WeierstrassInvariantG3[{ω,ω′}]
gives the invariant 
for the Weierstrass elliptic functions corresponding to the half‐periods {ω,ω′}.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- WeierstrassInvariantG3 gives the second invariant from the pair given by WeierstrassInvariants.
- For certain special arguments, WeierstrassInvariantG3 automatically evaluates to exact values.
- WeierstrassInvariantG3 can be evaluated to arbitrary numerical precision.
- WeierstrassInvariantG3 can be used with CenteredInterval objects. »
Examples
open all close allBasic Examples (2)
Scope (6)
N[WeierstrassInvariantG3[{1, 2I}], 50]The precision of the output tracks the precision of the input:
WeierstrassInvariantG3[{1.00000000005000000000500000005, 2I}]Evaluate symbolically for the equianharmonic case:
WeierstrassInvariantG3[{1, E^I π / 3}]Evaluate symbolically for the lemniscatic case:
WeierstrassInvariantG3[{1, I}]WeierstrassInvariantG3 has both singularities and discontinuities:
FunctionSingularities[WeierstrassInvariantG3[{x, y}], x]//QuietFunctionDiscontinuities[WeierstrassInvariantG3[{x, y}], x]//QuietWeierstrassInvariantG3 can be used with CenteredInterval objects:
WeierstrassInvariantG3[{CenteredInterval[1 + I, (1 + I) / 10 ^ 9], CenteredInterval[3 / 4, 1 / 10 ^ 9]}]TraditionalForm formatting:
WeierstrassInvariantG3[{Subscript[ω, 1], Subscript[ω, 3]}]//TraditionalFormApplications (1)
Define the discriminant of the Weierstrass elliptic curve:
disc[g2_, g3_] = Block[{x}, Discriminant[4x ^ 3 - g2 x - g3, x] / 16]KleinInvariantJ can be computed as the ratio of a power of 
invariant and the discriminant:
j[τ_] := WeierstrassInvariantG2[{1, τ}] ^ 3 / disc[WeierstrassInvariantG2[{1, τ}], WeierstrassInvariantG3[{1, τ}]]j[0.5 I]Compare with the built‐in function value:
KleinInvariantJ[0.5 I]Text
Wolfram Research (2017), WeierstrassInvariantG3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html (updated 2023).
CMS
Wolfram Language. 2017. "WeierstrassInvariantG3." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html.
APA
Wolfram Language. (2017). WeierstrassInvariantG3. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html