Certain Rational Function Approximations to (1+x2) −½

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Abstract
With a view to numerical applications, particular rational function approximations to (1+x2) −½ are constructed by using continued fraction expansions. It is found that the zeros of the denominator polynomials, which occur in the convergents of the continued fractions, can be determined explicitly. This fact enables the rational function approximations to be expressed conveniently in partial fractions. The resulting expressions are then applied to examples in which x is regarded either as a real variable, or as the Laplace operator, the latter case giving rise to some interesting approximations to Bessel functions of integral order and complex argument. The character of the errors of the various approximations is discussed, and expressions for the error bounds developed. These are in good agreement with computed results. The method followed in this paper is capable of further development. In particular it can be extended to those functions which possess suitable Taylor and asymptotic expansions.

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