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Jacobi function

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11: 22.21 Tables
§22.21 Tables
12: 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 ) . … The notation sn ( z , k ) , cn ( z , k ) , dn ( z , k ) is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). 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). …
13: 22.17 Moduli Outside the Interval [0,1]
22.17.3 cn ( z , 1 / k ) = dn ( z / k , k ) ,
22.17.4 dn ( z , 1 / k ) = cn ( z / k , k ) .
22.17.6 sn ( z , i k ) = k 1 sd ( z / k 1 , k 1 ) ,
22.17.7 cn ( z , i k ) = cd ( z / k 1 , k 1 ) ,
22.17.8 dn ( z , i k ) = nd ( z / k 1 , k 1 ) .
14: 22.7 Landen Transformations
22.7.2 sn ( z , k ) = ( 1 + k 1 ) sn ( z / ( 1 + k 1 ) , k 1 ) 1 + k 1 sn 2 ( z / ( 1 + k 1 ) , k 1 ) ,
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 ) ,
22.7.4 dn ( z , k ) = dn 2 ( z / ( 1 + k 1 ) , k 1 ) ( 1 k 1 ) 1 + k 1 dn 2 ( z / ( 1 + k 1 ) , k 1 ) .
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 ) ,
22.7.7 cn ( z , k ) = ( 1 + k 2 ) ( dn 2 ( z / ( 1 + k 2 ) , k 2 ) k 2 ) k 2 2 dn ( z / ( 1 + k 2 ) , k 2 ) ,
15: 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 ) ,
22.2.5 cn ( z , k ) = θ 4 ( 0 , q ) θ 2 ( 0 , q ) θ 2 ( ζ , q ) θ 4 ( ζ , q ) = 1 nc ( z , k ) ,
22.2.6 dn ( z , k ) = θ 4 ( 0 , q ) θ 3 ( 0 , q ) θ 3 ( ζ , q ) θ 4 ( ζ , q ) = 1 nd ( z , k ) ,
22.2.7 sd ( z , k ) = θ 3 2 ( 0 , q ) θ 2 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 3 ( ζ , q ) = 1 ds ( z , k ) ,
22.2.8 cd ( z , k ) = θ 3 ( 0 , q ) θ 2 ( 0 , q ) θ 2 ( ζ , q ) θ 3 ( ζ , q ) = 1 dc ( z , k ) ,
16: 22.13 Derivatives and Differential Equations
22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 sn 2 ( z , k ) ) ( 1 k 2 sn 2 ( z , k ) ) ,
22.13.2 ( d d z cn ( z , k ) ) 2 = ( 1 cn 2 ( z , k ) ) ( k 2 + k 2 cn 2 ( z , k ) ) ,
22.13.4 ( d d z cd ( z , k ) ) 2 = ( 1 cd 2 ( z , k ) ) ( 1 k 2 cd 2 ( z , k ) ) ,
22.13.7 ( d d z dc ( z , k ) ) 2 = ( dc 2 ( z , k ) 1 ) ( dc 2 ( z , k ) k 2 ) ,
22.13.10 ( d d z ns ( z , k ) ) 2 = ( ns 2 ( z , k ) k 2 ) ( ns 2 ( z , k ) 1 ) ,
17: 20.11 Generalizations and Analogs
For m = 1 , 2 , 3 , 4 , n = 1 , 2 , 3 , 4 , and m n , define twelve combined theta functions φ m , n ( z , q ) by
20.11.6 φ m , 1 ( z , q ) = θ 1 ( 0 , q ) θ m ( z , q ) θ m ( 0 , q ) θ 1 ( z , q ) , m = 2 , 3 , 4 ,
20.11.7 φ 1 , n ( z , q ) = θ n ( 0 , q ) θ 1 ( z , q ) θ 1 ( 0 , q ) θ n ( z , q ) , n = 2 , 3 , 4 ,
20.11.8 φ m , n ( z , q ) = θ n ( 0 , q ) θ m ( z , q ) θ m ( 0 , q ) θ n ( z , q ) , m , n = 2 , 3 , 4 .
18: 22.18 Mathematical Applications
Ellipse
where ( u , k ) is Jacobi’s epsilon function22.16(ii)). … By use of the functions sn and cn , parametrizations of algebraic equations, such as …
§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
19: 25.1 Special Notation
The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
20: 22.3 Graphics
Line graphs of the functions sn ( x , k ) , cn ( x , k ) , dn ( x , k ) , cd ( x , k ) , sd ( x , k ) , nd ( x , k ) , dc ( x , k ) , nc ( x , k ) , sc ( x , k ) , ns ( x , k ) , ds ( x , k ) , and cs ( x , k ) for representative values of real x and real k illustrating the near trigonometric ( k = 0 ), and near hyperbolic ( k = 1 ) limits. … sn ( x , k ) , cn ( x , k ) , and dn ( x , k ) as functions of real arguments x and k . …
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Figure 22.3.25: sn ( 5 , k ) as a function of complex k 2 , 1 ( k 2 ) 3.5 , 1 ( k 2 ) 1 . … Magnify 3D Help
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Figure 22.3.26: Density plot of | sn ( 5 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
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Figure 22.3.27: Density plot of | sn ( 10 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify