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1: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 20.12 Mathematical Applications
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§20.12(i) Number Theory
►For applications of 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 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). ►The space of complex tori (that is, the set of complex numbers in which two of these numbers and are regarded as equivalent if there exist integers such that ) is mapped into the projective space via the identification . …3: Customize DLMF
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You can customize the appearance and functionality of the DLMF site using these selections. Note: Javascript and Cookies must be enabled to support customizations.4: 17.9 Further Transformations of Functions
5: 22 Jacobian Elliptic Functions
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6: 23 Weierstrass Elliptic and Modular
Functions
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7: 22.18 Mathematical Applications
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►See Akhiezer (1990, Chapter 8) and McKean and Moll (1999, Chapter 2) for discussions of the inverse mapping.
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►Algebraic curves of the form , where 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 , , 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).
The theory of elliptic functions brings together complex analysis, algebraic curves, number theory, and geometry: Lang (1987), Siegel (1988), and Serre (1973).
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8: 20 Theta Functions
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9: Bibliography M
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Elliptic Curves.
Cambridge University Press, Cambridge.
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Methods and Programs for Mathematical Functions.
Ellis Horwood Ltd., Chichester.
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10: 23.20 Mathematical Applications
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►It follows from the addition formula (23.10.1) that the points , , have zero sum iff , so that addition of points on the curve corresponds to addition of parameters on the torus ; see McKean and Moll (1999, §§2.11, 2.14).
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►The geometric nature of this construction is illustrated in McKean and Moll (1999, §2.14), Koblitz (1993, §§6, 7), and Silverman and Tate (1992, Chapter 1, §§3, 4): each of these references makes a connection with the addition theorem (23.10.1).
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►To determine , we make use of the fact that if then must be a divisor of ; hence there are only a finite number of possibilities for .
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