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1: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
§23.2(ii) Weierstrass Elliptic Functions
§23.2(iii) Periodicity
2: 22.15 Inverse Functions
§22.15 Inverse Functions
§22.15(i) Definitions
The principal values satisfy …
§22.15(ii) Representations as Elliptic Integrals
3: 22.2 Definitions
§22.2 Definitions
where K ( k ) , K ( k ) 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. … … s s ( z , k ) = 1 . …
4: 19.16 Definitions
§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
§22.16(i) Jacobi’s Amplitude ( am ) Function
Definition
Quasi-Periodicity
Relation to Elliptic Integrals
Relation to the Elliptic Integral E ( ϕ , k )
6: 19.2 Definitions
§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
§19.2(iii) Bulirsch’s Integrals
7: 36.2 Catastrophes and Canonical Integrals
Normal Forms for Umbilic Catastrophes with Codimension K = 3
36.2.2 Φ ( E ) ( s , t ; x ) = s 3 - 3 s t 2 + z ( s 2 + t 2 ) + y t + x s , x = { x , y , z } ,
(elliptic umbilic). …
Canonical Integrals
36.2.25 Ψ ( E ) ( x , - y , z ) = Ψ ( E ) ( x , y , z ) .
8: 19.35 Other Applications
§19.35(i) Mathematical
§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
22.8.14 sn ( u + v ) = sn u cn u dn v + sn v cn v dn u cn u cn v + sn u dn u sn v dn v ,
§22.8(iii) Special Relations Between Arguments
If sums/differences of the z j ’s are rational multiples of K ( k ) , then further relations follow. …
10: 22.4 Periods, Poles, and Zeros
§22.4(i) Distribution
§22.4(ii) Graphical Interpretation via Glaisher’s Notation
For example, sn ( z + K , k ) = cd ( z , k ) . …