# FILE muPowerConmpl.txt
dmudcSol:= simplify(solve(diff(mu(c)*tan(c*mu(c)),c)=0,diff(mu(c),c)));
removeSin:= u -> simplify(subs(sin(c*mu(c)) = cos(c*mu(c))/(beta*mu(c)),u));
removeCos:= u -> simplify(subs(cos(c*mu(c))^2 = 1/(1+ (1/(beta*mu(c)))^2),u));
dmudc:= removeCos(removeSin(dmudcSol));
lprint(dmudc);
# -mu(c)*(beta^2*mu(c)^2+1)/(beta+c*beta^2*mu(c)^2+c)
dmu2dc:= mu2(c)*(1+beta^2*mu2(c))/(beta+c*(1+beta^2*mu2(c))); # from earlier maple runs
d2mudc2:=factor(simplify(subs(diff(mu(c),c)=dmudc,diff(dmudc,c))));
d2logmudc2:=factor(simplify((mu(c)*d2mudc2-dmudc^2)/mu(c)^2));
#-1/2 concavity for lambda1 for N>=2?, OK at N=1 as it is
# 1/ mu concave, i.e. diff(1/(mu(c),c$2)<=0;
#
minus1condc2:= -factor(simplify((mu(c)*d2mudc2-2*dmudc^2)/mu(c)^3));
minus1condc2Again:= factor(simplify(subs(diff(mu(c),c)=dmudc,diff(-dmudc/mu(c)^2,c))));
# -2*mu(c)*(beta^2*mu(c)^2+1)*beta^3/(beta+c*beta^2*mu(c)^2+c)^3
check:= simplify(minus1condc2-minus1condc2Again);
# 0
#minushalfdc1:= -factor(dmudc/mu(c)^2);
#check:= simplify(diff(minus1dc1,c)
dmu2dc:= -2mu2(c)*(beta^2*mu2(c)+1)/(beta+c*beta^2*mu2(c)+c);
d11osqrtmu2dc:= -1/(2*mu2(c)^(3/2))*dmu2dc;
d21osqrtmu2dc:= factor(subs(diff(mu2(c),c)=dmu2dc,diff(d11osqrtmu2dc,c)));