modular functions
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31—40 of 44 matching pages
31: 26.21 Tables
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►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients for up to 50 and up to 25; extends Table 26.4.1 to ; tabulates Stirling numbers of the first and second kinds, and , for up to 25 and up to ; tabulates partitions and partitions into distinct parts for up to 500.
►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts , partitions into parts , and unrestricted plane partitions up to 100.
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32: 27.9 Quadratic Characters
§27.9 Quadratic Characters
… ►If does not divide , then has the value when the quadratic congruence has a solution, and the value when this congruence has no solution. The Legendre symbol , as a function of , is a Dirichlet character (mod ). …33: Frank Garvan
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►His research is in the areas of -series and modular forms, and he enjoys using MAPLE in his research.
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34: Bibliography K
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On the evaluation of the Gauss hypergeometric function.
C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.
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Generalized functions.
Mathematics in Science and Engineering, Vol. 171, Academic Press, Inc., Orlando, FL.
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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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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function.
Math. Comp. 24 (111), pp. 679–696.
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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument.
Comput. Phys. Comm. 4, pp. 221–226.
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35: Bibliography V
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Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions.
CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en
Informatica, Amsterdam.
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Modular hypergeometric residue sums of elliptic Selberg integrals.
Lett. Math. Phys. 58 (3), pp. 223–238.
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On the coefficients of the modular invariant
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Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes
Math. 15 56, pp. 389–400.
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An infinite series of Weber’s parabolic cylinder functions.
Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
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Transformations of some Gauss hypergeometric functions.
J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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36: Bibliography C
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The hypergeometric function and the -function near their branch points.
Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.
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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions.
Math. Comp. 75 (255), pp. 1309–1318.
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A compact mathematical function package.
Australian Computer Journal 16 (3), pp. 107–114.
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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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37: 24.19 Methods of Computation
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►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs
for which .
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38: 27.2 Functions
§27.2 Functions
… ►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. … ►This is Mangoldt’s function. …39: 27.12 Asymptotic Formulas: Primes
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27.12.1
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27.12.2
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►where depends only on , and is the Euler totient function (§27.2).
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►For example, if , then is composite.
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►A Carmichael number is a composite number for which for all .
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40: Mathematical Introduction
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