symmetric
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11: Bille C. Carlson
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►If some of the parameters are equal, then the -function is symmetric in the corresponding variables.
This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory.
Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation.
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►This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions.
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12: 19.38 Approximations
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13: 19.24 Inequalities
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§19.24(i) Complete Integrals
►The condition for (19.24.1) and (19.24.2) serves only to identify as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity. … ► ►§19.24(ii) Incomplete Integrals
… ►The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. …14: 19.27 Asymptotic Approximations and Expansions
15: 35.5 Bessel Functions of Matrix Argument
16: 19.22 Quadratic Transformations
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19.22.2
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Bartky’s Transformation
… ►§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
… ► … ►17: 19.25 Relations to Other Functions
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§19.25(i) Legendre’s Integrals as Symmetric Integrals
… ►§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
… ► … ►§19.25(vii) Hypergeometric Function
… ►18: 19.1 Special Notation
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►However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF.
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, , and are the symmetric (in , , and ) integrals of the first, second, and third kinds; they are complete if exactly one of , , and is identically 0.
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nonnegative integers. | |
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