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relation to amplitude (am) function

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1: 22.16 Related Functions
22.16.4 am ( x , 0 ) = x ,
Relation to Elliptic Integrals
2: 29.1 Special Notation
(For other notation see Notation for the Special Functions.) … The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2). … The relation to the Lamé functions L c ν ( m ) , L s ν ( m ) of Jansen (1977) is given by …where ψ = am ( z , k ) ; see §22.16(i). The relation to the Lamé functions Ec ν m , Es ν m of Ince (1940b) is given by …
3: 29.6 Fourier Series
§29.6 Fourier Series
§29.6(i) Function Ec ν 2 m ( z , k 2 )
With ϕ = 1 2 π - am ( z , k ) , as in (29.2.5), we have … In addition, if H satisfies (29.6.2), then (29.6.3) applies. … Consequently, Ec ν 2 m ( z , k 2 ) reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …
4: 22.20 Methods of Computation
A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument z and the modulus k is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. …
§22.20(vi) Related Functions
am ( x , k ) can be computed from its definition (22.16.1) or from its Fourier series (22.16.9). Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute am ( x , k ) . … For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947). …
5: 19.25 Relations to Other Functions
§19.25 Relations to Other Functions
§19.25(iv) Theta Functions
§19.25(vii) Hypergeometric Function
6: 29.2 Differential Equations
§29.2(ii) Other Forms
29.2.5 ϕ = 1 2 π - am ( z , k ) .
For am ( z , k ) see §22.16(i). … we have …For the Weierstrass function see §23.2(ii). …