Making use of relation (13.2.37), along with expressions (13.2.8) for the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) of the transfer matrix \(\boldsymbol{P}\), determine the correlation length \(\xi(B, T)\) of the Ising chain in the presence of a magnetic field. Evaluate the critical exponent \(v^{c}\), as defined in Problem 12.25, and check that \(v^{c}=v / \Delta\), where \(v\) and \(\Delta\) are standard exponents defined in Sections 12.10 and 12.12.