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1: 22.15 Inverse Functions
§22.15 Inverse Functions
§22.15(i) Definitions
The principal values satisfy …
§22.15(ii) Representations as Elliptic Integrals
2: 22.2 Definitions
§22.2 Definitions
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. … … The Jacobian functions are related in the following way. …
3: 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).
4: 22.18 Mathematical Applications
§22.18 Mathematical Applications
§22.18(i) Lengths and Parametrization of Plane Curves
Lemniscate
5: 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.
6: 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
7: 22.4 Periods, Poles, and Zeros
§22.4(i) Distribution
For the distribution of the k -zeros of the Jacobian elliptic functions see Walker (2009).
§22.4(ii) Graphical Interpretation via Glaisher’s Notation
8: 22.14 Integrals
§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions
§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions
The indefinite integral of the 3rd power of a Jacobian function can be expressed as an elementary function of Jacobian functions and a product of Jacobian functions. …
§22.14(iv) Definite Integrals
9: 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
10: 22.1 Special Notation
The functions treated in this chapter are the three principal Jacobian elliptic functions sn ( z , k ) , cn ( z , k ) , dn ( z , k ) ; the nine subsidiary Jacobian elliptic functions cd ( z , k ) , sd ( z , k ) , nd ( z , k ) , dc ( z , k ) , nc ( z , k ) , sc ( z , k ) , ns ( z , k ) , ds ( z , k ) , cs ( z , k ) ; the amplitude function am ( x , k ) ; Jacobi’s epsilon and zeta functions ( x , k ) and Z ( x | k ) . … Other notations for sn ( z , k ) are sn ( z | m ) and sn ( z , m ) with m = k 2 ; see Abramowitz and Stegun (1964) and Walker (1996). …