Show that for the spherical model in one dimension the free energy at constant (lambda) is given by [ frac{beta A_{lambda}}{N}=frac{1}{2} ln left[frac{betaleft{lambda+sqrt{ }left(lambda^{2}-J^{2} ight)

Show that for the spherical model in one dimension the free energy at constant \(\lambda\) is given by

\[
\frac{\beta A_{\lambda}}{N}=\frac{1}{2} \ln \left[\frac{\beta\left\{\lambda+\sqrt{ }\left(\lambda^{2}-J^{2}\right)\right\}}{2 \pi}\right]-\frac{\beta \mu^{2} B^{2}}{4(\lambda-J)}
\]

while \(\lambda\) is determined by the constraint equation

\[
\frac{1}{2 \beta \sqrt{ }\left(\lambda^{2}-J^{2}\right)}+\frac{\mu^{2} B^{2}}{4(\lambda-J)^{2}}=1
\]

In the absence of the field \((B=0), \lambda=\sqrt{1+4 \beta^{2} J^{2}} / 2 \beta\); the free energy at constant \(\mathscr{S}\) is then given by

\[
\frac{\beta A_{\mathscr{S}}}{N}=\frac{1}{2} \ln \left[\frac{\sqrt{ }\left(1+4 \beta^{2} J^{2}\right)+1}{4 \pi}\right]-\frac{1}{2} \sqrt{ }\left(1+4 \beta^{2} J^{2}\right)
\]