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Bounds for symmetric elliptic integrals. (English) Zbl 1041.33015
In this paper the author derives lower and upper bounds for the four standard incomplete symmetric elliptic integrals. The bounding functions are expressed in terms of the elementary transcendental functions. All elliptic integrals delt with in this paper can be represented by the $R$-hypergeometric functions introduced in [B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York (1977; Zbl 0394.33001)]. Sharp bounds for the ratio of the complete elliptic integrals of the first and second kind are also obtained. These results can be used to obtain bounds for the product of these integrals. It is shown that an iterative numerical algorithm for computing the ratios and products of complete integrals has the second order of convergence.
##### MSC:
 3.3e+06 Elliptic functions and integrals