About the Project
NIST

modular transformations

AdvancedHelp

(0.001 seconds)

1—10 of 13 matching pages

1: 23.18 Modular Transformations
§23.18 Modular Transformations
►
23.18.3 λ ⁡ ( 𝒜 ⁡ τ ) = λ ⁡ ( τ ) ,
►
23.18.4 J ⁡ ( 𝒜 ⁡ τ ) = J ⁡ ( τ ) .
►
23.18.5 η ⁡ ( 𝒜 ⁡ τ ) = ε ⁡ ( 𝒜 ) ⁢ ( - i ⁢ ( c ⁢ τ + d ) ) 1 / 2 ⁢ η ⁡ ( τ ) ,
2: 21.5 Modular Transformations
§21.5 Modular Transformations
►
§21.5(i) Riemann Theta Functions
►Equation (21.5.4) is the modular transformation property for Riemann theta functions. … ►
§21.5(ii) Riemann Theta Functions with Characteristics
►For explicit results in the case g = 1 , see §20.7(viii).
3: 17.18 Methods of Computation
►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …
4: 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 ) .
►These are specific examples of modular transformations as discussed in §23.15; the corresponding results for the general case are given by Rademacher (1973, pp. 181–183). …
5: 23.15 Definitions
►
23.15.5 f ⁡ ( 𝒜 ⁡ τ ) = c 𝒜 ⁢ ( c ⁢ τ + d ) ℓ ⁢ f ⁡ ( τ ) , ⁡ τ > 0 ,
6: 20.11 Generalizations and Analogs
►If both m , n are positive, then G ⁡ ( m , n ) allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …
7: 27.14 Unrestricted Partitions
►Dedekind sums occur in the transformation theory of the Dedekind modular function η ⁡ ( τ ) , defined by …
8: Bibliography V
►
  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
  • ►
  • J. F. Van Diejen and V. P. Spiridonov (2001) Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.
  • ►
  • C. Van Loan (1992) Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • ►
  • A. van Wijngaarden (1953) On the coefficients of the modular invariant J ⁢ ( τ ) . Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 56, pp. 389–400.
  • ►
  • R. VidÅ«nas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
  • 9: Bibliography C
    ►
  • R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
  • ►
  • S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.
  • ►
  • N. B. Christensen (1990) Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.
  • ►
  • G. Cornell, J. H. Silverman, and G. Stevens (Eds.) (1997) Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.
  • ►
  • J. E. Cremona (1997) Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, Cambridge.
  • 10: 24.19 Methods of Computation
    ►For number-theoretic applications it is important to compute B 2 ⁢ n ( mod p ) for 2 ⁢ n p - 3 ; in particular to find the irregular pairs ( 2 ⁢ n , p ) for which B 2 ⁢ n 0 ( mod p ) . … ►
  • A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs ( 2 ⁢ n , p ) . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).