Carlson symmetric form
In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa. The Carlson elliptic integrals are: formula_1 formula_2 formula_3 formula_4 Since formula_5 and formula_6 are special cases of formula_7 and formula_8, all elliptic integrals can ultimately be evaluated in terms of just formula_7 and formula_8. The term "symmetric" refers to the fact that in contrast to the Legendre forms, these  functions are unchanged by the exchange of certain subsets of their arguments. The value of formula_11 is the same for any permutation of its arguments, and the value of formula_12 is the same for any permutation of its first three arguments. The Carlson elliptic integrals are named after Bille C. Carlson (1924-2013). Relation to the Legendre forms. Incomplete elliptic integrals. Incomplete elliptic integrals can be calculated easily using Carlson symmetric forms: Complete elliptic integrals. Complete elliptic integrals can be calculated by substituting φ = π: Special cases. When any two, or all three of the arguments of formula_7 are  the same, then a substitution of formula_22 renders the integrand rational. The integral can then be expressed in terms of elementary transcendental functions. Similarly, when at least two of the first three arguments of formula_8 are the same, Properties. Homogeneity. By substituting in the integral definitions formula_26 for any constant formula_27, it is found that Duplication theorem. where formula_31. where formula_33 and formula_34 Series Expansion. In obtaining a Taylor series expansion for formula_35 or formula_36 it proves convenient to expand about the mean value of the several arguments. So for formula_35, letting the mean value of the arguments be formula_38 , and using homogeneity, define formula_39, formula_40 and formula_41 by that is formula_43 etc. The differences formula_39, formula_40 and formula_41 are defined with this sign (such that they are "subtracted"), in order to be in agreement with Carlson's papers. Since formula_47 is symmetric under permutation of formula_48, formula_49 and formula_50, it is also symmetric in the quantities formula_39, formula_40 and formula_41. It follows that both the integrand of formula_35 and its integral can be expressed as functions of the elementary symmetric polynomials in formula_39, formula_40 and formula_41 which are Expressing the integrand in terms of these polynomials, performing a multidimensional  Taylor expansion and integrating term-by-term... The advantage of expanding about the mean value of the arguments is now apparent; it reduces formula_62 identically to zero, and so eliminates all terms involving formula_62 - which otherwise would be the most numerous. An ascending series for formula_36 may be found in a similar way. There is a slight difficulty because formula_36 is not fully symmetric; its dependence on its fourth argument, formula_66, is different from its dependence on formula_48, formula_49 and formula_50. This is overcome by treating formula_36 as a fully symmetric function of "five" arguments, two of which happen to  have the same value formula_66. The mean value of the arguments is therefore taken to be and the differences formula_39, formula_40 formula_41 and formula_76 defined by The elementary symmetric polynomials in formula_39, formula_40, formula_41, formula_76 and (again) formula_76 are in full However, it is possible to simplify the formulae for formula_88, formula_89 and formula_90 using the fact that formula_91. Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term as before... As with formula_36, by expanding about the mean value of the arguments, more than half the terms (those involving formula_62) are  eliminated. Negative arguments. In general, the arguments x, y, z of Carlson's integrals may not be real and negative, as this would place a branch point on the path of integration, making the integral ambiguous. However, if the second argument of formula_5, or the fourth argument, p, of formula_8 is negative, then this results in a simple pole on the path of integration. In these cases the Cauchy principal value (finite part) of the integrals may be of interest; these are and where which must be greater than zero for formula_100 to be evaluated. This may be arranged by  permuting x, y and z so that the value of y is between that of x and z. Numerical evaluation. The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate formula_11: first, define formula_102, formula_103 and formula_104. Then iterate the series until the desired precision is reached: if formula_48, formula_49 and formula_50 are non-negative, all of the series will converge quickly to a given value, say, formula_110. Therefore, Evaluating formula_112 is much the same  due to the relation