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2: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

… ►

§17.9(iv) Bibasic Series

►

Mixed-Base Heine-Type Transformations

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3: 22 Jacobian Elliptic Functions

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4: 23 Weierstrass Elliptic and Modular
Functions

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6: 20.12 Mathematical Applications

… ►For applications of

\theta_{3}\left(0,q\right)

to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s

\tau\left(n\right)

function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). … ►For the terminology and notation see McKean and Moll (1999, pp. 48–53). …

7: Bibliography M

… ►

Maxima (free interactive system)

ⓘ

Notes:: System for the manipulation of symbolic and numerical expressions. Includes a collection of special functions. Utilizes exact fractions, arbitrary precision integers, and arbitrary precision floating point numbers.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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H. McKean and V. Moll (1999) Elliptic Curves. Cambridge University Press, Cambridge.

ⓘ

Notes:: Reprint with corrections of original edition, published in 1997 by Cambridge University Press
External Links:: ISBN 0-521-58228-8; 0-521-65817-9, MathReview (Amílcar Pacheco), ZentralBlatt
Referenced by:: §20.1, §20.11(i), §20.11(ii), §20.12(i), §20.12(i), §20.12(ii), §20.7(iii), §20.7(v), §20.9(i), §20.9(ii), Ch.20, §22.18(ii), §22.18(iv), Ch.22, §23.1, §23.15(i), §23.20(i), §23.20(ii), §23.20(ii), §23.21(ii), §23.22(ii), §23.6(iv), Ch.23, Notations T.
Encodings:: BibTeX

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S. L. B. Moshier (1989) Methods and Programs for Mathematical Functions. Ellis Horwood Ltd., Chichester.

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Notes:: Includes diskette
External Links:: ISBN 0-13-578998-2, MathReview, ZentralBlatt
Referenced by:: 2nd item, CEPHES (free C library).
Encodings:: BibTeX

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MPFR (free C library)

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Notes:: MPFR is a C library for multiple-precision floating-point computations with correct rounding. The main goal of MPFR is to provide a library for multiple-precision floating-point computation which is both efficient and has well-defined semantics. MPFR includes most of the elementary functions. It also implements the gamma function, and more special functions are planned in the future.
External Links:: MPFR
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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mpmath (free python library)

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Notes:: Mpmath is a pure-Python library for multiprecision floating-point arithmetic. It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers, interval arithmetic, numerical integration and differentiation, root-finding, linear algebra, and much more.
External Links:: mpmath
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

…

8: 20.1 Special Notation

… ►McKean and Moll’s notation:

\vartheta_{j}(z|\tau)=\theta_{j}\left(\pi z\middle|\tau\right)

j=1,2,3,4

. See McKean and Moll (1999, p. 125). …

9: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

… ►

Gosper’s Bibasic Sum

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Gasper’s Extensions of Gosper’s Bibasic Sum

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10: 22.18 Mathematical Applications

… ►See Akhiezer (1990, Chapter 8) and McKean and Moll (1999, Chapter 2) for discussions of the inverse mapping. … ►Algebraic curves of the form

y^{2}=P(x)

, where

P

is a nonsingular polynomial of degree 3 or 4 (see McKean and Moll (1999, §1.10)), are elliptic curves, which are also considered in §23.20(ii). …This provides an abelian group structure, and leads to important results in number theory, discussed in an elementary manner by Silverman and Tate (1992), and more fully by Koblitz (1993, Chapter 1, especially §1.7) and McKean and Moll (1999, Chapter 3). …With the identification

x=\operatorname{sn}\left(z,k\right)

y=\ifrac{\mathrm{d}(\operatorname{sn}\left(z,k\right))}{\mathrm{d}z}

, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …