Verify that expression (32) of Section 14.2 indeed satisfies the functional equation (31) for the field-free spherical model in one dimension. Next show (or at least verify) that, with the field present, the functional equation (27), with \(\boldsymbol{K}^{\prime}\) given by (25), is satisfied by the more general expression

\[
f\left(K_{1}, K_{2}, \Lambda\right)=\frac{1}{2} \ln \left[\frac{\Lambda+\sqrt{\Lambda^{2}-K_{1}^{2}}}{2 \pi}\right]-\frac{K_{2}^{2}}{4\left(\Lambda-K_{1}\right)},
\]

where \(\Lambda\) is determined by the constraint equation

\[
\frac{\partial f}{\partial \Lambda}=\frac{1}{2 \sqrt{\Lambda^{2}-K_{1}^{2}}}+\frac{K_{2}^{2}}{4\left(\Lambda-K_{1}\right)^{2}}=1
\]