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Jacobi epsilon function

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1: 22.16 Related Functions
§22.16(ii) Jacobi’s Epsilon Function
Integral Representations
Relation to Theta Functions
See accompanying text
Figure 22.16.2: Jacobi’s epsilon function ( x , k ) for 0 x 10 π and k = 0.4 , 0.7 , 0.99 , 0.999999 . … Magnify
2: 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 ) . …
3: 22.21 Tables
§22.21 Tables
4: 22.18 Mathematical Applications
Ellipse
22.18.3 l ( u ) = a ( u , k ) ,
where ( u , k ) is Jacobi’s epsilon function22.16(ii)). …
5: 22.20 Methods of Computation
Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. … …
6: 31.16 Mathematical Applications
7: 31.7 Relations to Other Functions
§31.7 Relations to Other Functions
§31.7(i) Reductions to the Gauss Hypergeometric Function
§31.7(ii) Relations to Lamé Functions
With z = sn 2 ( ζ , k ) and …
γ = δ = ϵ = 1 2 ,
8: 31.2 Differential Equations
31.2.8 d 2 w d ζ 2 + ( ( 2 γ - 1 ) cn ζ dn ζ sn ζ - ( 2 δ - 1 ) sn ζ dn ζ cn ζ - ( 2 ϵ - 1 ) k 2 sn ζ cn ζ dn ζ ) d w d ζ + 4 k 2 ( α β sn 2 ζ - q ) w = 0 .
9: 18.15 Asymptotic Approximations
18.15.6 ( sin 1 2 θ ) α + 1 2 ( cos 1 2 θ ) β + 1 2 P n ( α , β ) ( cos θ ) = Γ ( n + α + 1 ) 2 1 2 ρ α n ! ( θ 1 2 J α ( ρ θ ) m = 0 M A m ( θ ) ρ 2 m + θ 3 2 J α + 1 ( ρ θ ) m = 0 M - 1 B m ( θ ) ρ 2 m + 1 + ε M ( ρ , θ ) ) ,
10: 20.15 Tables
Tables of Neville’s theta functions θ s ( x , q ) , θ c ( x , q ) , θ d ( x , q ) , θ n ( x , q ) (see §20.1) and their logarithmic x -derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for ε , α = 0 ( 5 ) 90 , where (in radian measure) ε = x / θ 3 2 ( 0 , q ) = π x / ( 2 K ( k ) ) , and α is defined by (20.15.1). …