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Jacobi function

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1: 22.16 Related Functions
§22.16(i) Jacobi’s Amplitude ( am ) Function
§22.16(ii) Jacobi’s Epsilon Function
Relation to Theta Functions
§22.16(iii) Jacobi’s Zeta Function
Definition
2: 22.6 Elementary Identities
22.6.1 sn 2 ( z , k ) + cn 2 ( z , k ) = k 2 sn 2 ( z , k ) + dn 2 ( z , k ) = 1 ,
22.6.2 1 + cs 2 ( z , k ) = k 2 + ds 2 ( z , k ) = ns 2 ( z , k ) ,
22.6.5 sn ( 2 z , k ) = 2 sn ( z , k ) cn ( z , k ) dn ( z , k ) 1 - k 2 sn 4 ( z , k ) ,
22.6.14 ns ( 2 z , k ) = ns 4 ( z , k ) - k 2 2 cs ( z , k ) ds ( z , k ) ns ( z , k ) ,
§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
3: 35.7 Gaussian Hypergeometric Function of Matrix Argument
Jacobi Form
35.7.2 P ν ( γ , δ ) ( T ) = Γ m ( γ + ν + 1 2 ( m + 1 ) ) Γ m ( γ + 1 2 ( m + 1 ) ) F 1 2 ( - ν , γ + δ + ν + 1 2 ( m + 1 ) γ + 1 2 ( m + 1 ) ; T ) , 0 < T < I ; γ , δ , ν ; ( γ ) > - 1 .
4: 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 ,
22.8.3 dn ( u + v ) = dn u dn v - k 2 sn u cn u sn v cn v 1 - k 2 sn 2 u sn 2 v .
22.8.5 sd ( u + v ) = sd u cd v nd v + sd v cd u nd u 1 + k 2 k 2 sd 2 u sd 2 v ,
22.8.6 nd ( u + v ) = nd u nd v + k 2 sd u cd u sd v cd v 1 + k 2 k 2 sd 2 u sd 2 v ,
5: 14.31 Other Applications
§14.31(ii) Conical Functions
The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …
6: 15.17 Mathematical Applications
§15.17(iii) Group Representations
For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …
7: 22.14 Integrals
22.14.1 sn ( x , k ) d x = k - 1 ln ( dn ( x , k ) - k cn ( x , k ) ) ,
22.14.4 cd ( x , k ) d x = k - 1 ln ( nd ( x , k ) + k sd ( x , k ) ) ,
8: 22.4 Periods, Poles, and Zeros
For example, the poles of sn ( z , k ) , abbreviated as sn in the following tables, are at z = 2 m K + ( 2 n + 1 ) i K . … Then: (a) In any lattice unit cell p q ( z , k ) has a simple zero at z = p and a simple pole at z = q . (b) The difference between p and the nearest q is a half-period of p q ( z , k ) . … For example, sn ( z + K , k ) = cd ( z , k ) . …
9: 22.5 Special Values
For example, at z = K + i K , sn ( z , k ) = 1 / k , d sn ( z , k ) / d z = 0 . … Table 22.5.2 gives sn ( z , k ) , cn ( z , k ) , dn ( z , k ) for other special values of z . For example, sn ( 1 2 K , k ) = ( 1 + k ) - 1 / 2 . … In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …
10: 22.15 Inverse Functions
22.15.1 sn ( ξ , k ) = x , - 1 x 1 ,
22.15.2 cn ( η , k ) = x , - 1 x 1 ,
22.15.3 dn ( ζ , k ) = x , k x 1 ,
are denoted respectively by …
22.15.11 x = 0 sn ( x , k ) d t ( 1 - t 2 ) ( 1 - k 2 t 2 ) , - 1 x 1 , 0 k 1 .