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1961
Padé approximants
Exact series expansions were most useful for high
temperatures, where the coefficients were uniformly
positive and the convergence behavior could be demonstrated
fairly directly. For low temperature expansions it was
more difficult to identify convergence patterns, since the
sign of the coefficients was inconsistent. Baker revived
Padé's method of approximating the ratio [L,M] of
two polynomials of degree L and M so that a given expansion
F(x) which contained many singularities (poles) could be
represented as F(x) = [L,M] + terms of order (L+M+1) and
higher. Baker showed that the Padé approximants
could be used in many instances to continue a series
expansion analytically to the critical point, thus yielding
more accurate critical exponents. In particular, Baker
analyzed spontaneous magnetization and found a value for
ß roughly equal to 0.30.
Primary:
George A. Baker, Jr., "Application of the Padé
approximant method to the investigation of some magnetic
properties of the Ising model," Phys. Rev. 124 (1961): 768-
774.
Secondary:
Brush 1983, 253.
Domb 1996, 165.
Fisher 1967, 687.
--Karl Hall
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