Simplify and Jacobi Elliptic functions

Questions
1

Simplify knows about some special functions.

But, am I correct that it doesn’t handle the Jacobi Elliptic functions?

shouldBeZero = (JacobiDN[s, m] - Sqrt[1 - m JacobiSN[s, m]^2])

FullSimplify[shouldBeZero , Assumptions :> 0 < m < 1 && s > 0]  (*doesn't simplify*)
2

As this doesn’t work

FullSimplify[ JacobiDN[s, m] == Sqrt[1 - m JacobiSN[s, m]^2], {0 < m < 1, s > 0}]

(* Out[8] = Sqrt[1 - m JacobiSN[s, m]^2] == JacobiDN[s, m] *)

“with a little help from a friend” … taking squares

FullSimplify[JacobiDN[s, m]^2 == (Sqrt[1 - m JacobiSN[s, m]^2])^2]

(* Out[9] = True *)

It’s alright, even without any specifications of the parameters m and s.

Regards,
Wolfgang

3

Thanks Wolfgang,
That works, but the expression that I am trying to Simplify has many instances of

Sqrt[1 - m JacobiSN[s, m]^2]

and other relations between Jacobi ellliptic functions.

I can force the simplification by using a rule-replace (which is what I ended up doing). But, still, isn’t this something that Simplify should be able to deal with?

Thanks, Craig

4

Craig,

you are right, it should.

But many years of (good!) experience showed me that a prudent cooperation between Mathematica and the user is the best way to reach the goal.

Sorry for “teaching” philosophy :wink:

Regards,
Wolfgang