# modular transformations

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##### 1: 23.18 Modular Transformations
###### §23.18 ModularTransformations
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23.18.5 $\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$
##### 2: 21.5 Modular Transformations
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###### §21.5(i) Riemann Theta Functions
βΊEquation (21.5.4) is the modular transformation property for Riemann theta functions. … βΊ
###### §21.5(ii) Riemann Theta Functions with Characteristics
βΊFor explicit results in the case $g=1$, see §20.7(viii).
##### 3: 17.18 Methods of Computation
βΊThe 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
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20.7.33 $(-i\tau)^{1/2}\theta_{4}\left(z\middle|\tau\right)=\exp\left(i\tau^{\prime}z^{% 2}/\pi\right)\theta_{2}\left(z\tau^{\prime}\middle|\tau^{\prime}\right).$
βΊThese 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
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23.15.5 $f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),$ $\Im\tau>0$,
##### 6: 20.11 Generalizations and Analogs
βΊIf 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
βΊDedekind sums occur in the transformation theory of the Dedekind modular function $\eta\left(\tau\right)$, defined by …
##### 8: Bibliography V
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• G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
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• 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.
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• 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.
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• A. van Wijngaarden (1953) On the coefficients of the modular invariant $J(\tau)$ . Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 56, pp. 389–400.
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• R. VidΕ«nas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
• ##### 9: Bibliography C
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• R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
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• S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.
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• N. B. Christensen (1990) Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.
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• G. Cornell, J. H. Silverman, and G. Stevens (Eds.) (1997) Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.
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• J. E. Cremona (1997) Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, Cambridge.
• ##### 10: 24.19 Methods of Computation
βΊFor number-theoretic applications it is important to compute $B_{2n}\pmod{p}$ for $2n\leq p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0\pmod{p}$. … βΊ
• 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 $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).