# reduction to symmetric elliptic integrals

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##### 1: 19.29 Reduction of General Elliptic Integrals
###### §19.29(i) Reduction Theorems
19.29.33 $(x-y)^{2}U=\left(\sqrt{x^{4}+a^{4}}+\sqrt{y^{4}+a^{4}}\right)^{2}-(x^{2}-y^{2}% )^{2}.$
##### 2: Bibliography C
• B. C. Carlson and J. FitzSimons (2000) Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.
• ##### 3: 19.15 Advantages of Symmetry
###### §19.15 Advantages of Symmetry
Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. … For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …
##### 4: 19.36 Methods of Computation
###### §19.36 Methods of Computation
Because of cancellations in (19.26.21) it is advisable to compute $R_{G}$ from $R_{F}$ and $R_{D}$ by (19.21.10) or else to use §19.36(ii). Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. …
##### 5: Bibliography N
• W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
• E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.
• E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.
• E. Neuman (2003) Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.
• E. H. Neville (1951) Jacobian Elliptic Functions. 2nd edition, Clarendon Press, Oxford.