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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
are denoted respectively by …The principal values satisfy …
3: 22.2 Definitions
§22.2 Definitions
22.2.4 sn ( z , k ) = θ 3 ( 0 , q ) θ 2 ( 0 , q ) θ 1 ( ζ , q ) θ 4 ( ζ , q ) = 1 ns ( z , k ) ,
22.2.9 sc ( z , k ) = θ 3 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 2 ( ζ , q ) = 1 cs ( z , k ) .
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. … …
4: 22.16 Related Functions
§22.16(i) Jacobi’s Amplitude ( am ) Function
Definition
Quasi-Periodicity
Integral Representation
Relation to Elliptic Integrals
5: 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 …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 )
6: 19.2 Definitions
19.2.4 F ( ϕ , k ) = 0 ϕ d θ 1 - k 2 sin 2 θ = 0 sin ϕ d t 1 - t 2 1 - k 2 t 2 ,
19.2.5 E ( ϕ , k ) = 0 ϕ 1 - k 2 sin 2 θ d θ = 0 sin ϕ 1 - k 2 t 2 1 - t 2 d t .
19.2.6 D ( ϕ , k ) = 0 ϕ sin 2 θ d θ 1 - k 2 sin 2 θ = 0 sin ϕ t 2 d t 1 - t 2 1 - k 2 t 2 = ( F ( ϕ , k ) - E ( ϕ , k ) ) / k 2 .
19.2.11_5 el1 ( x , k c ) = 0 arctan x 1 cos 2 θ + k c 2 sin 2 θ d θ ,
19.2.16 el3 ( x , k c , p ) = 0 arctan x d θ ( cos 2 θ + p sin 2 θ ) cos 2 θ + k c 2 sin 2 θ = Π ( arctan x , 1 - p , k ) , x 2 - 1 / p .
7: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
§22.17(i) Real or Purely Imaginary Moduli
§22.17(ii) Complex Moduli
When z is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of k 2 . …For proofs of these results and further information see Walker (2003).
8: 23.13 Zeros
§23.13 Zeros
9: 22.6 Elementary Identities
§22.6 Elementary Identities
§22.6(ii) Double Argument
§22.6(iii) Half Argument
§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
See §22.17.
10: 22.8 Addition Theorems
§22.8 Addition Theorems
22.8.1 sn ( u + v ) = sn u cn v dn v + sn v cn u dn u 1 - k 2 sn 2 u sn 2 v ,
22.8.2 cn ( u + v ) = cn u cn v - sn u dn u sn v dn v 1 - k 2 sn 2 u sn 2 v ,
§22.8(iii) Special Relations Between Arguments