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elliptic cases

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1: 23.4 Graphics
§23.4(i) Real Variables
See accompanying text
Figure 23.4.6: σ ( x ; 0 , g 3 ) for 5 x 5 , g 3 = 0. … Magnify
See accompanying text
Figure 23.4.7: ( x ) with ω 1 = K ( k ) , ω 3 = i K ( k ) for 0 x 9 , k 2 = 0. … Magnify
2: 19.16 Definitions
All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …
§19.16(iii) Various Cases of R a ( 𝐛 ; 𝐳 )
All other elliptic cases are integrals of the second kind. …
3: 22.21 Tables
Spenceley and Spenceley (1947) tabulates sn ( K x , k ) , cn ( K x , k ) , dn ( K x , k ) , am ( K x , k ) , ( K x , k ) for arcsin k = 1 ( 1 ) 89 and x = 0 ( 1 90 ) 1 to 12D, or 12 decimals of a radian in the case of am ( K x , k ) . …
4: 23.5 Special Lattices
§23.5(iii) Lemniscatic Lattice
§23.5(iv) Rhombic Lattice
As a function of e 3 the root e 1 is increasing. For the case ω 3 = e π i / 3 ω 1 see §23.5(v).
§23.5(v) Equianharmonic Lattice
5: 22.5 Special Values
In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. … For values of K , K when k 2 = 1 2 (lemniscatic case) see §23.5(iii), and for k 2 = e i π / 3 (equianharmonic case) see §23.5(v).
6: 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 …
7: 19.21 Connection Formulas
19.21.15 p R J ( 0 , y , z , p ) + q R J ( 0 , y , z , q ) = 3 R F ( 0 , y , z ) , p q = y z .
8: 19.7 Connection Formulas
§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k )
The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of Π ( ϕ , α 2 , k ) when α 2 > csc 2 ϕ (see (19.6.5) for the complete case). …
9: 19.6 Special Cases
§19.6 Special Cases
10: 19.15 Advantages of Symmetry
Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s F D (Carlson (1961b)). …