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Jacobian elliptic-function form

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11: 22.13 Derivatives and Differential Equations
§22.13(i) Derivatives
Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.
d d z ( sn z ) = cn z dn z d d z ( dc z )  = k 2 sc z nc z
§22.13(ii) First-Order Differential Equations
22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 - sn 2 ( z , k ) ) ( 1 - k 2 sn 2 ( z , k ) ) ,
§22.13(iii) Second-Order Differential Equations
12: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.2 2 K k sn ( 2 K t , k ) = n = - π sin ( π ( t - ( n + 1 2 ) τ ) ) = n = - ( m = - ( - 1 ) m t - m - ( n + 1 2 ) τ ) ,
22.12.8 2 K dc ( 2 K t , k ) = n = - π sin ( π ( t + 1 2 - n τ ) ) = n = - ( m = - ( - 1 ) m t + 1 2 - m - n τ ) ,
22.12.11 2 K ns ( 2 K t , k ) = n = - π sin ( π ( t - n τ ) ) = n = - ( m = - ( - 1 ) m t - m - n τ ) ,
22.12.13 2 K cs ( 2 K t , k ) = lim N n = - N N ( - 1 ) n π tan ( π ( t - n τ ) ) = lim N n = - N N ( - 1 ) n ( lim M m = - M M 1 t - m - n τ ) .
13: Bille C. Carlson
In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …
14: 22.20 Methods of Computation
§22.20 Methods of Computation
§22.20(iii) Landen Transformations
§22.20(iv) Lattice Calculations
§22.20(v) Inverse Functions
15: 22.9 Cyclic Identities
§22.9 Cyclic Identities
§22.9(i) Notation
16: 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.
17: 22.7 Landen Transformations
§22.7(i) Descending Landen Transformation
22.7.3 cn ( z , k ) = cn ( z / ( 1 + k 1 ) , k 1 ) dn ( z / ( 1 + k 1 ) , k 1 ) 1 + k 1 sn 2 ( z / ( 1 + k 1 ) , k 1 ) ,
§22.7(ii) Ascending Landen Transformation
22.7.6 sn ( z , k ) = ( 1 + k 2 ) sn ( z / ( 1 + k 2 ) , k 2 ) cn ( z / ( 1 + k 2 ) , k 2 ) dn ( z / ( 1 + k 2 ) , k 2 ) ,
§22.7(iii) Generalized Landen Transformations
18: 19.25 Relations to Other Functions
Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). …
§19.25(v) Jacobian Elliptic Functions
For the use of R -functions with Δ ( p , q ) in unifying other properties of Jacobian elliptic functions, see Carlson (2004, 2006a, 2006b, 2008). Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of R F ( x , y , z ) . … …
19: 29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
29.2.3 ξ = sn 2 ( z , k ) .
we have …
20: 31.2 Differential Equations
§31.2(ii) Normal Form of Heun’s Equation
§31.2(iii) Trigonometric Form
§31.2(iv) Doubly-Periodic Forms
Jacobi’s Elliptic Form
Weierstrass’s Form