= Solution
A further field derivative of the <magnetization> scaling form gives
$$
\chi=\partial_hm=s^{2-\alpha-2\Delta}g'_{m,\pm}(x).
$$
At zero field, on a branch with a finite nonzero susceptibility <amplitude>, comparison with $\chi\sim s^{-\gamma}$ yields
$$
\boxed{\gamma=2\Delta-2+\alpha=\Delta-\beta.}
$$
Consequently $\alpha+2\beta+\gamma=2$ and $\gamma=\beta(\delta-1)$, the Rushbrooke and <Widom scaling relations>. Regular analytic terms do not change the leading divergent susceptibility.
Back to article page
