I am trying to solve this inequality for $\eta$,
where B, J, $\eta$, and $\beta$ are reals and $\beta$>0. I used
f[\[Eta]_]:=(B^2+2 J^2+B^2 Cosh[2 \[Beta] \[Eta]])/(2 \[Eta]^2 Cosh[J \[Beta]] Cosh[\[Beta] \[Eta]]+2 J \[Eta] Sinh[J \[Beta]] Sinh[\[Beta] \[Eta]])
Reduce[f[\[Eta]]>=1,\[Eta]]
But I got nothing. Is this possible using Mathematica?
How do you conclude that this equation has an analytical solution?
– No name Jun 16 '20 at 22:11\[Beta] , J, Bone can solve this inequality e.g.With[{\[Beta] = 1, J = 1, B = 1/2}, Reduce[(B^2 + 2 J^2 + B^2 Cosh[2 \[Beta] \[Eta]])/(2 \[Eta]^2 Cosh[ J \[Beta]] Cosh[\[Beta] \[Eta]] + 2 J \[Eta] Sinh[J \[Beta]] Sinh[\[Beta] \[Eta]]) >= 1, \[Eta], Reals]]. Otherwise one cannot solve it symbolically, see e.g. Solve symbolically a transcendental trigonometric equation and plot its solutions and references therein. – Artes Jun 16 '20 at 22:43