today I tried to simplify the following expression:
expr = Sqrt[(Cosh[ν] - Cos[u])/(a Sinh[ν])] /. Thread[{ν, u, ϕ} ->
CoordinateTransformData[{"Cartesian" -> {"Toroidal", a}}, "Mapping", {x, y, z}]]
Even FullSimplify
doesn't give it a try:
FullSimplify[expr, Assumptions -> {a > 0, Element[{x, y, z}, Reals]}]
after a few seconds returns the original expression. However, this is simply equal to (x^2+y^2)^(-1/4)
, which can be seen by either manipulating toroidal coordinates, or trying
(expr/(x^2 + y^2)^(-1/4)) /. {a -> 1, x -> 1, y -> 2, z -> 3} // N
1.
(expr/(x^2 + y^2)^(-1/4)) /. {a -> -5, x -> 10, y -> -2, z -> 0.4} // N
1.
What am I doing wrong? How come MMA thinks that the long clumsy expression is simpler than (x^2+y^2)^(-1/4)
? Notice that it doesn't even depend on z
, however, if you don't specify z
, MMA acts like it's not a definite number
(expr/(x^2 + y^2)^(-1/4)) /. {a -> -5, x -> 10, y -> -2} // N
1.4221 Sqrt[-1. (-((1. (77.25 + z^2))/Sqrt[
100. z^2 + (77.25 + z^2)^2]) +
Cosh[0.5 Log[(26.1313 + z^2)/(228.369 + z^2)]]) Csch[
0.5 Log[(26.1313 + z^2)/(228.369 + z^2)]]]
What's wrong? Thank you.