Taylor's series expansion of a function of two variables: Consider a function having singularity at a point (a,b). The function can be expanded around (a,b) in Taylor's series as

Following the above expansion, the Landau free energy can be written as

Since, is an even function of m, no odd terms appear in the expansion. The coefficients are function of the reduced temperature only. The Helmholtz free energy is a convex function of magnetization m. Therefore, must be convex of m and thus the coefficients must be positive. The coefficients can be expanded in terms of t as:

.............

Therefore, the Landau free energy can be taken as

Since higher order terms can not alter the critical behavior of the system, one may terminate the series at the fourth order in m.