Methods of series analysis. III. Integral approximant methods

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Abstract
We discuss the approximation of functions by the solution of certain differential equations derived from their power-series coefficients. We call these approximations integral-curve approximants, or more simply integral approximants, and find that they include as special cases many of the currently used methods. We investigate their invariance and singularity properties, and test the power of the first-order ones on a series of known functions. These integral approximants do as well, and often better, than any of the now current (nonexact) methods of approximation. We have applied them to the low-temperature Ising-model susceptibility and find, for the first time by series methods, reasonably good evidence that the low-temperature critical index is equal to the high-temperature one, as expected from the scaling and renormalization-group approaches.