A=(1/2)((E^(-m (x^2)))+(E^((m+s2) (x^2))));
v=A-Sqrt[(A^2)-1]
u=A+Sqrt[(A^2)-1]
p=((((E^(m (x^2)))-v))/(u-v))

Series[u,{x,0,3}]
Series[v,{x,0,3}]
Series[p,{x,0,3}]