elementary symmetric functions
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1: 19.36 Methods of Computation
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►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated.
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19.36.1
►where the elementary symmetric functions
are defined by (19.19.4).
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19.36.2
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19.36.4
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2: 19.19 Taylor and Related Series
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►Define the elementary symmetric function
by
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19.19.4
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19.19.5
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►The number of terms in can be greatly reduced by using variables with chosen to make .
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,
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3: 19.15 Advantages of Symmetry
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►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)).
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4: 1.11 Zeros of Polynomials
5: 19.16 Definitions
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§19.16(i) Symmetric Integrals
… ►Just as the elementary function (§19.2(iv)) is the degenerate case … ►§19.16(ii)
… ► … ►§19.16(iii) Various Cases of
…6: Bibliography Z
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Numerical analysis of Struve functions with applications to other special functions.
Ann. Numer. Math. 2 (1-4), pp. 199–208.
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On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval.
J. Approx. Theory 94 (1), pp. 73–106.
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Symmetric elliptic integrals of the third kind.
Math. Comp. 24 (109), pp. 199–214.
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Fast evaluation of elementary mathematical functions with correctly rounded last bit.
ACM Trans. Math. Software 17 (3), pp. 410–423.
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The summation of series of hyperbolic functions.
SIAM J. Math. Anal. 10 (1), pp. 192–206.
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7: 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. …8: 22.15 Inverse Functions
§22.15 Inverse Functions
… ►Each of these inverse functions is multivalued. … ►can be transformed into normal form by elementary change of variables. … ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …9: 22.14 Integrals
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