symmetric case

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1: 28.29 Definitions and Basic Properties

… ►In the symmetric case $Q(z)=Q(-z)$, $w_{\text{I}}(z,\lambda )$ is an even solution and $w_{\text{II}}(z,\lambda )$ is an odd solution; compare §28.2(ii). …

2: 8.18 Asymptotic Expansions of $I_x(a,b)$

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Symmetric Case

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3: 19.20 Special Cases

§19.20 Special Cases

… ►The general lemniscatic case is … ►where $x,y,z$ may be permuted. … ►The general lemniscatic case is … ►

4: 19.21 Connection Formulas

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19.21.15 $$p{R}_J(0,y,z,p)+q{R}_J(0,y,z,q)=3R_F(0,y,z),$$ $pq=yz$.

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Permalink:

http://dlmf.nist.gov/19.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(iii), §19.21 and Ch.19

5: 19.16 Definitions

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§19.16(iii) Various Cases of $R_{-a}(𝐛;𝐳)$

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6: Bille C. Carlson

… ►Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation. …

7: 20.9 Relations to Other Functions

… ►In the case of the symmetric integrals, with the notation of §19.16(i) we have …

8: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

… ► ►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)). … ►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). …

9: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►Define the elementary symmetric function $E_s(𝐳)$ by … ►The number of terms in $T_N$ can be greatly reduced by using variables $𝐙=\mathrm{𝟏}-(𝐳/A)$ with $A$ chosen to make $E_1(𝐙)=0$. … ►Special cases are given in (19.36.1) and (19.36.2).

10: 19.24 Inequalities

… ►The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. …