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general elliptic functions

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11: 22.16 Related Functions
22.16.1 am ( x , k ) = Arcsin ( sn ( x , k ) ) , x ,
12: 22.8 Addition Theorems
§22.8 Addition Theorems
§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. …Generalizations are given in §22.9.
13: 12.15 Generalized Parabolic Cylinder Functions
§12.15 Generalized Parabolic Cylinder Functions
can be viewed as a generalization of (12.2.4). This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …
14: 19.1 Special Notation
All derivatives are denoted by differentials, not by primes. … We use also the function D ( ϕ , k ) , introduced by Jahnke et al. (1966, p. 43). The functions (19.1.1) and (19.1.2) are used in Erdélyi et al. (1953b, Chapter 13), except that Π ( α 2 , k ) and Π ( ϕ , α 2 , k ) are denoted by Π 1 ( ν , k ) and Π ( ϕ , ν , k ) , respectively, where ν = - α 2 . In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by K ( α ) , E ( α ) , Π ( n \ α ) , F ( ϕ \ α ) , E ( ϕ \ α ) , and Π ( n ; ϕ \ α ) , where α = arcsin k and n is the α 2 (not related to k ) in (19.1.1) and (19.1.2). … R - a ( b 1 , b 2 , , b n ; z 1 , z 2 , , z n ) is a multivariate hypergeometric function that includes all the functions in (19.1.3). …
15: 36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes Φ K ( t ; x ) ; umbilic catastrophes with codimension three Φ ( E ) ( s , t ; x ) , Φ ( H ) ( s , t ; x ) ; canonical integrals Ψ K ( x ) , Ψ ( E ) ( x ) , Ψ ( H ) ( x ) ; diffraction catastrophes Ψ K ( x ; k ) , Ψ ( E ) ( x ; k ) , Ψ ( H ) ( x ; k ) generated by the catastrophes. …
16: 22.9 Cyclic Identities
§22.9 Cyclic Identities
§22.9(i) Notation
17: 22.4 Periods, Poles, and Zeros
The set of points z = m K + n i K , m , n , comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by m K + n i K , where again m , n . …
18: 22.15 Inverse Functions
§22.15 Inverse Functions
§22.15(i) Definitions
The principal values satisfy …
§22.15(ii) Representations as Elliptic Integrals
19: 19.17 Graphics
§19.17 Graphics
See Figures 19.17.119.17.8 for symmetric elliptic integrals with real arguments. Because the R -function is homogeneous, there is no loss of generality in giving one variable the value 1 or - 1 (as in Figure 19.3.2). …The case y = 1 corresponds to elementary functions. …
See accompanying text
Figure 19.17.8: R J ( 0 , y , 1 , p ) , 0 y 1 , - 1 p 2 . … Magnify 3D Help
20: 32.10 Special Function Solutions
§32.10 Special Function Solutions
More generally, if n = 1 , 2 , 3 , , then … Next, let Λ = Λ ( u , z ) be the elliptic function (§§22.15(ii), 23.2(iii)) defined by …Then P VI , with α = β = γ = 0 and δ = 1 2 , has the general solution …