I have a strange case here! The story begins with this equation
t22b=(r - rh)*(-2*rh*\[Kappa]g - (24*G^2*\[Eta]*\[Kappa]g^2)/rh^2 +
(Q^2*(3 - 8*b^2*G*Sqrt[Pi]*(-3*Sqrt[Pi] +
Sqrt[4*Pi + Q^2/(b^2*rh^4)])*rh^2 -
4*b*G*Sqrt[Pi]*Sqrt[Q^2 + 4*b^2*Pi*rh^4] - 3*rh^2*\[CapitalLambda]) +
12*b^2*Pi*rh^4*(1 + 8*b^2*G*Pi*rh^2 -
4*b*G*Sqrt[Pi]*Sqrt[Q^2 + 4*b^2*Pi*rh^4] - rh^2*\[CapitalLambda]))/
(3*(Q^2 + 4*b^2*Pi*rh^4)))
Firstly, one can expand t22b
as follows
(Series[t22b, {b, Infinity, 0}] // Normal // PowerExpand //
FullSimplify // Expand) // FullSimplify
The result will be
-(((r - rh) (G (Q^2 + 24 G \[Eta] \[Kappa]g^2) +
rh^2 (-1 + 2 rh \[Kappa]g + rh^2 \[CapitalLambda])))/rh^2)
Now, solving above term equal to zero for \[Kappa]g
we have (note that, this answer is true for my problem)
\[Kappa]g -> (-2 rh^3 + Sqrt[
4 rh^6 - 96 G^2 \[Eta] (G Q^2 - rh^2 + rh^4 \[CapitalLambda])])/(
48 G^2 \[Eta]).
In the second method, one can find \[Kappa]g
from the eq22b=0
and then expand the answer.
sol1 = Solve[(t22b) == 0, \[Kappa]g] // PowerExpand // FullSimplify
\[Kappa]a = \[Kappa]g /. sol1[[1]];
\[Kappa]b = \[Kappa]g /. sol1[[2]];
The expanded result is
Series[\[Kappa]a, {b, Infinity, 0}] // PowerExpand // FullSimplify
Series[\[Kappa]b, {b, Infinity, 0}] // PowerExpand // FullSimplify
Non of these expanded answers reproduce the \[Kappa]g
of previous procedure.
Why?! wHy?! WhY?!!!
It is driving me crazy :))
Can you help?
PowerExpand
andFullSimplify
.Assuming[b > 0 && rh > 0, Series[t22b, {b, Infinity, 0}]]
does the job. Notice extra term. $\endgroup$