About the Project

Jacobi transform

AdvancedHelp

(0.002 seconds)

1—10 of 47 matching pages

1: 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. …
2: 15.17 Mathematical Applications
Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform1.14(ii)) or as a specialization of a group Fourier transform. …
3: 15.9 Relations to Other Functions
The Jacobi transform is defined as …with inverse … …
4: 22.6 Elementary Identities
§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.
sn ( i z , k ) = i sc ( z , k ) dc ( i z , k ) = dn ( z , k )
5: Bibliography K
  • T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.
  • 6: 1.14 Integral Transforms
    §1.14 Integral Transforms
    7: 18.17 Integrals
    Jacobi
    Jacobi
    Jacobi
    18.17.36 1 1 ( 1 x ) z 1 ( 1 + x ) β P n ( α , β ) ( x ) d x = 2 β + z Γ ( z ) Γ ( 1 + β + n ) ( 1 + α z ) n n ! Γ ( 1 + β + z + n ) , z > 0 .
    8: 37.4 Disk with Weight Function ( 1 x 2 y 2 ) α
    ( f ) ( ρ cos θ , ρ sin θ ) = i m n e m n ( θ ) 0 1 r m n + 1 ( 1 r 2 ) α R n ( α , m n ) ( 2 r 2 1 ) J m n ( ρ r ) d r = 2 α Γ ( α + 1 ) i m + n e m n ( θ ) ρ ( α + 1 ) J m + n + α + 1 ( ρ ) , ρ > 0 ,
    First, the Jacobi polynomials (37.3.3) on and the ultraspherical polynomials (37.4.4) on 𝔻 are related by the quadratic transformations …Second, the Jacobi polynomials (37.3.9) on are related to the real disk polynomials (37.4.15) by the quadratic transformations
    9: 37.12 Orthogonal Polynomials on Quadratic Surfaces
    S , m n ( z 2 , | z | 2 ; 0 ) = { z 1 S n + m + 1 , n m ( z , z ¯ ) z ¯ 1 S n m , n + m + 1 ( z , z ¯ ) } , 0 m n , z .
    10: 29.10 Lamé Functions with Imaginary Periods
    transform (29.2.1) into
    29.10.3 d 2 w d z 2 + ( h ν ( ν + 1 ) k 2 sn 2 ( z , k ) ) w = 0 .