# Eliminating three variables from an expression

I have an expression which reads

f1[kx_, ky_, kz_] = \[Nu]^3*m1 // Simplify;
m1= (2 d (kx kz + I ky Sqrt[kx^2 + ky^2 + kz^2])^2 (d (-I kx^10 (ky^2 + kz^2) +
I ky^6 kz^2 (ky^2 + kz^2)^2 -
4 kx^7 ky kz^3 Sqrt[kx^2 + ky^2 + kz^2] +
kx ky^5 kz Sqrt[
kx^2 + ky^2 + kz^2] (ky^4 - 4 ky^2 kz^2 - 5 kz^4) +
kx^5 ky kz Sqrt[kx^2 + ky^2 + kz^2] (ky^4 - kz^4) -
I kx^8 (3 ky^4 + kz^4) +
2 kx^3 ky^3 kz Sqrt[kx^2 + ky^2 + kz^2] (ky^4 + 5 kz^4) -
I kx^6 (3 ky^6 - 4 ky^4 kz^2 - 7 ky^2 kz^4) -
I kx^4 (ky^8 - 5 ky^6 kz^2 + ky^4 kz^4 - 5 ky^2 kz^6) +
I kx^2 (3 ky^8 kz^2 - 7 ky^6 kz^4 - 10 ky^4 kz^6)) +
Sqrt[3] kx (2 kx^8 ky kz + 3 kx^6 ky kz (ky^2 + kz^2) +
ky^5 kz (ky^2 + kz^2)^2 -
I kx^7 (ky^2 - kz^2) Sqrt[kx^2 + ky^2 + kz^2] -
2 I kx^5 ky^2 (ky^2 + kz^2) Sqrt[kx^2 + ky^2 + kz^2] +
4 I kx ky^4 kz^2 (ky^2 + kz^2) Sqrt[kx^2 + ky^2 + kz^2] +
kx^2 ky^3 kz (ky^4 - 5 ky^2 kz^2 - 6 kz^4) +
kx^4 ky kz (ky^4 - 4 ky^2 kz^2 + kz^4) -
I kx^3 ky^2 Sqrt[
kx^2 + ky^2 +
kz^2] (ky^4 - ky^2 kz^2 + 4 kz^4)) \[Nu]))/((kx^2 +
ky^2)^2 (kx^2 + ky^2 + kz^2)^(3/2) (I kx kz +
ky Sqrt[kx^2 + ky^2 + kz^2])^2 \[Nu] (-2 Sqrt[3]
d ky kz (-2 kx^2 + ky^2 + kz^2) + 3 I kx (ky^2 + kz^2) \[Nu]))


Here $$\nu$$ and $$d$$ are constant variables. I would like to evaluate this expression at $$\nu \sqrt{k_x^2 + k_y^2 +k_z^2}=\omega$$, where $$\omega$$ is a constant. I have tried Eliminate and Reduce in this thread but they are very slow and I was not bale to evaluate my expression using them. Do you have a any suggestion?

Edit:

I have applied the suggestion in the comments to the denominator of my expression as

 smp = Simplify[Denominator[f1[kx, ky, kz]], \[Nu]*Sqrt[kx^2 + ky^2 + kz^2] == 1];
smp - Denominator[f1[kx, ky, kz]]
(*0*)


The simplified expression is exactly the same as the denominator of $$f_1$$ even though I set $$\omega=1$$.

• I am looking for a final expression in terms of $\omega$, $d$, and $\nu$. Your suggested simplification won't remove all of the k variables. Sep 17, 2020 at 9:46
• You only have one constraint , how can you expect to remove all k variables? Sep 17, 2020 at 9:50
• Correct. I have to simplify my expression as much as I can. Sep 17, 2020 at 9:56
• Thank you, Bill. This works a bit better but there are still some terms like $(k_x^2 + k_y^2 + k_z^2)^(3/2)$ which have not been replaced by $\omgea^3$. Do you know how I get them? Sep 17, 2020 at 11:38