modular%20transformations

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1: 20.7 Identities

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§20.7(iv) Reduction Formulas for Products

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§20.7(vi) Landen Transformations

… ►See Lawden (1989, pp. 19–20). … ►

§20.7(viii) Transformations of Lattice Parameter

… ►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). …

2: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

External Links:

Code, Document

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1st item.

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Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

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External Links:

ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

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Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97127-0, MathReview, ZentralBlatt

Referenced by:

§17.2(i), §23.10(iv), §23.15(i), §23.17(ii), §23.18, §23.18, §23.19, §23.20(i), §23.20(v), Ch.23, §27.14(iii), §27.14(iv), §27.14(vi), §27.7, Ch.27.

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3: Bibliography C

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

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External Links:

Document

Referenced by:

§10.74(vii).

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

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Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

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Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-59820-6, MathReview (Joseph H. Silverman), ZentralBlatt

Referenced by:

§23.20(ii).

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4: 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)): …

5: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

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Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

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ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt

Referenced by:

§19.35(i).

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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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External Links:

MathReview (J. Lehner), ZentralBlatt

Referenced by:

§23.17(ii).

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Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

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ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§15.8(v).

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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6: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

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Introduction to Elliptic Curves and Modular Forms. 2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97966-2, MathReview, ZentralBlatt

Referenced by:

§20.10(i), §20.12(i), §20.9(iii), §22.18(iv), §23.1, §23.20(ii), §23.20(v), Ch.23.

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A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:

Translated from the Russian by George Yankovsky. Reprint of the 1977 translation

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.5(viii).

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