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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: 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 .
    7: 1.14 Integral Transforms
    §1.14 Integral Transforms
    8: 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 .
    9: 20.7 Identities
    20.7.33 ( - i τ ) 1 / 2 θ 4 ( z | τ ) = exp ( i τ z 2 / π ) θ 2 ( z τ | τ ) .
    10: 20.10 Integrals
    20.10.1 0 x s - 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 - 2 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) ,
    20.10.2 0 x s - 1 ( θ 3 ( 0 | i x 2 ) - 1 ) d x = π - s / 2 Γ ( 1 2 s ) ζ ( s ) ,
    20.10.3 0 x s - 1 ( 1 - θ 4 ( 0 | i x 2 ) ) d x = ( 1 - 2 1 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) .