Rigorous Scaling-Law Equation of State near the Critical Point

Making use of the scaling properties of $H(\epsilon , M)=M^{\delta }h\left(\frac{\epsilon }{M^{\frac{1}{\beta }}}\right)$, it is shown that the "scaling function" $h\left(x\right)\mathrm{}$ can be expressed rigorously as $\frac{h\left(x\right)\mathrm{}}{h\left(0\right)\mathrm{}}={\left[\frac{(x_0+x)}{x_0}\right]}^{\beta (\delta -1)}$ in the neighborhood of the critical point. This result is in agreement with the known mean-field-model prediction, directly obtainable from power-series expansion, and with accurate numerical results for the Ising and Heisenberg models.