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One of the most well-known identities involving $\wp$ is

$$\wp'^2=4 \wp^3-g_2 \wp-g_3. $$

However, when I run the command

WeierstrassPPrime[z, {g2, g3}]^2 - (4 WeierstrassP[z, {g2, g3}]^3 -
g2 WeierstrassP[z, {g2, g3}] - g3)

I get the same thing back, rather than $0$. I've also tried using FullSimplify as well as FunctionExpand, to no avail. Is there a way to make Mathematica see that this expression is zero, without manually removing it?

Thank you!

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    $\begingroup$ If you calculate the z derivative of the expression Mathematica simplifies it to zero. So the identity seems to be coded in somehow... $\endgroup$
    – ulvi
    Commented Jul 4, 2017 at 0:30
  • $\begingroup$ @ulvi Only the second-order ODE satisfied by WeierstrassP is required to show that the derivative of the identity in the question vanishes. The identity itself is not required. $\endgroup$
    – bbgodfrey
    Commented Jul 4, 2017 at 18:33
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    $\begingroup$ Strangely, FullSimplify[WeierstrassPPrime[z, {g2, g3}]^2 - (4 WeierstrassP[z, {g2, g3}]^3 - g2 WeierstrassP[z, {g2, g3}] - g3), z == 2 && g2 == 1 && g3 == 2] yields 0, but FullSimplify[(WeierstrassPPrime[z, {g2, g3}]^2 - (4 WeierstrassP[z, {g2, g3}]^3 - g2 WeierstrassP[z, {g2, g3}] - g3)) /. {z -> 2, g2 -> 1, g3 -> 2}] does not. $\endgroup$
    – bbgodfrey
    Commented Jul 4, 2017 at 18:41
  • $\begingroup$ Checking numerically, Table[ WeierstrassPPrime[z, {g2, g3}]^2 - (4 WeierstrassP[z, {g2, g3}]^3 - g2 WeierstrassP[z, {g2, g3}] - g3) /. Thread[{z, g2, g3} -> RandomComplex[{-200 (1 + I), 200 (1 + I)}, 3, WorkingPrecision -> 50]] // Chop[#, 10^-20] &, 1000] // Union evaluates to {0} $\endgroup$
    – Bob Hanlon
    Commented Jul 5, 2017 at 0:12

1 Answer 1

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The following is highly unsatisfying, but it works and may be of use in simplifying more complicated expressions involving WeierstrassPPrime. Define, 

tf[e_] := e /. WeierstrassPPrime[z_, {g2_, g3_}]^2 -> 
    (4 WeierstrassP[z, {g2, g3}]^3 - g2 WeierstrassP[z, {g2, g3}] - g3)

Then, not surprisingly, 

Simplify[(WeierstrassPPrime[z, {g2, g3}]^2 - (4 WeierstrassP[z, {g2, g3}]^3 - 
    g2 WeierstrassP[z, {g2, g3}] - g3)), TransformationFunctions -> {Automatic, tf}]

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