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Adding new rules to Simplify


I'm trying to teach Mathematica the obvious simplification rules for Jacobi elliptic functions, which i had hoped would be built in! Grumble

Having defined


sn[x_ /; !FreeQ[x, JacobiSN] := 
x /. JacobiSN[u_, m_]-> Sqrt[(1 - JacobiDN[u, m]^2)/m];

sn[x_] := x;

mySimplify = 
Simplify[#, TransformationFunctions -> {sn, Automatic}] &;

I find that

mySimplify[1 - (1 - k^2) JacobiSN[p, 1 - k^2]]

gives me

JacobiDN[p, 1 - k^2]^2

as I had hoped, but that the equivalent expression

mySimplify[1 + (-1 + k^2) JacobiSN[p, 1 - k^2]

leaves the formula unsimplified.

What Mathematica subtlety am I missing here??

Thanks,

John


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