modular%20functions
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1: Peter L. Walker
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►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004.
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2: William P. Reinhardt
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►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals.
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►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
3: 23 Weierstrass Elliptic and Modular
Functions
Chapter 23 Weierstrass Elliptic and Modular Functions
…4: Bibliography B
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Bessel functions and modular relations of higher type and hyperbolic differential equations.
Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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5: Software Index
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support.
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►In the list below we identify four main sources of software for computing special functions.
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Commercial Software.
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►The following are web-based software repositories with significant holdings in the area of special functions.
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.
6: 20.7 Identities
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§20.7(v) Watson’s Identities
… ►§20.7(vi) Landen Transformations
… ►§20.7(vii) Derivatives of Ratios of Theta Functions
… ►See Lawden (1989, pp. 19–20). … ►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). …7: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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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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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Generalized elliptic integrals and modular equations.
Pacific J. Math. 192 (1), pp. 1–37.
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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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8: 26.10 Integer Partitions: Other Restrictions
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►The set is denoted by .
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§26.10(ii) Generating Functions
… ►where the last right-hand side is the sum over of the generating functions for partitions into distinct parts with largest part equal to . … ►§26.10(vi) Bessel-Function Expansion
… ►where is the modified Bessel function (§10.25(ii)), and …9: 27.2 Functions
§27.2 Functions
… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►and if is the smallest positive integer such that , then is a primitive root mod . …This is Jordan’s function. … ►This is Liouville’s function. …10: Bibliography C
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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Modular Forms and Fermat’s Last Theorem.
Springer-Verlag, New York.
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Algorithms for Modular Elliptic Curves.
2nd edition, Cambridge University Press, Cambridge.
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