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1: 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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2: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

… ►The principal values satisfy … ►

§22.15(ii) Representations as Elliptic Integrals

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3: 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. … … ► $\mathrm{s}\mathrm{s}(z,k)=1$. …

4: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … … ►

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

… ►All other elliptic cases are integrals of the second kind. …

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 $K=3$

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36.2.2 $${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)=s^3-3s{t}^2+z(s^2+t^2)+yt+xs,$$ $𝐱=\{x,y,z\}$,

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Defines:

${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$: elliptic umbilic catastrophe

Symbols:

$y$: real parameter, $z$: real parameter, $t$: variable, $s$: variable and $x$: real parameter

Referenced by:

§36.10(iii), §36.2(i), §36.7(iii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

►(elliptic umbilic). … ►

Canonical Integrals

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36.2.25 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(x,-y,z\right)={\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right).$$

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §36.2 and Ch.36

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8: 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)).

9: 22.8 Addition Theorems

§22.8 Addition Theorems

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22.8.14 $$\mathrm{sn}(u+v)=\frac{\mathrm{sn}u\mathrm{cn}u\mathrm{dn}v+\mathrm{sn}v\mathrm{cn}v\mathrm{dn}u}{\mathrm{cn}u\mathrm{cn}v+\mathrm{sn}u\mathrm{dn}u\mathrm{sn}v\mathrm{dn}v},$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(ii), §22.8 and Ch.22

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§22.8(iii) Special Relations Between Arguments

… ►If sums/differences of the $z_j$’s are rational multiples of $K\left(k\right)$, then further relations follow. …

10: 22.4 Periods, Poles, and Zeros

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§22.4(i) Distribution

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§22.4(ii) Graphical Interpretation via Glaisher’s Notation

… ►For example, $\mathrm{sn}(z+K,k)=\mathrm{cd}(z,k)$. …