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1: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►A fourth integral that is symmetric in only two variables is defined by … ►which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\mathrm{\dots },n$. … … ►

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

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2: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

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§23.2(iii) Periodicity

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3: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

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§22.15(ii) Representations as Elliptic Integrals

… ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …

4: 22.2 Definitions

§22.2 Definitions

… ►where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). … ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ► $\mathrm{s}\mathrm{s}(z,k)=1$. …

5: 22.16 Related Functions

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§22.16(i) Jacobi’s Amplitude ($\mathrm{am}$) Function

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Definition

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Quasi-Periodicity

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Relation to Elliptic Integrals

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Relation to the Elliptic Integral $E(\varphi ,k)$

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6: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

… ►is called an elliptic integral. …Thus the elliptic part of (19.2.1) is … ►

§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

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7: 36.2 Catastrophes and Canonical Integrals

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Normal Forms for Umbilic Catastrophes with Codimension $K=3$

… ►(elliptic umbilic). … ►

Canonical Integrals

… ►with the contour passing to the lower right of $u=0$. …with the contour passing to the upper right of $u=0$. …

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: 19.35 Other Applications

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§19.35(i) Mathematical

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§19.35(ii) Physical

►Elliptic integrals appear in lattice models of critical phenomena (Guttmann and Prellberg (1993)); theories of layered materials (Parkinson (1969)); fluid dynamics (Kida (1981)); string theory (Arutyunov and Staudacher (2004)); astrophysics (Dexter and Agol (2009)).

10: 19.36 Methods of Computation

§19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). 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. …where the elementary symmetric functions $E_s$ are defined by (19.19.4). … ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). …