Neumann addition theorem

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11: Tom M. Apostol

… ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …In addition, he was the co-author of New Horizons in Geometry, published by the MAA, which received the CHOICE “Outstanding Academic Title” award in 2013. …He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …

12: 23.20 Mathematical Applications

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§23.20(ii) Elliptic Curves

… ►It follows from the addition formula (23.10.1) that the points $P_j=P(z_j)$, $j=1,2,3$, have zero sum iff $z_1+z_2+z_3\in 𝕃$, so that addition of points on the curve $C$ corresponds to addition of parameters $z_j$ on the torus $ℂ/𝕃$; see McKean and Moll (1999, §§2.11, 2.14). … ►The addition law states that to find the sum of two points, take the third intersection with $C$ of the chord joining them (or the tangent if they coincide); then its reflection in the $x$-axis gives the required sum. 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). … ► $K$ always has the form $T\times {\mathbb{Z}}^r$ (Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of $r$, the rank of $K$, raises questions of great difficulty, many of which are still open. …

13: Errata

… ►We now include Markov’s Theorem. … ►

Equation (5.2.9).

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Equations (10.22.78), (10.22.79).

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Equation (16.16.5_5).

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Equation (4.13.5_3) (suggested by Warren Smith on 2023-08-10).

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14: 14.18 Sums

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§14.18(i) Expansion Theorem

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§14.18(ii) Addition Theorems

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15: 10.60 Sums

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§10.60(i) Addition Theorems

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16: 19.26 Addition Theorems

§19.26 Addition Theorems

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19.26.27 $$R_C(x^2,x^2-\theta )=2R_C(s^2,s^2-\theta ),$$ $s=x+\sqrt{x^2-\theta }$, $\theta \ne x^2$ or $s^2$.

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $x$: positive

Permalink:

http://dlmf.nist.gov/19.26.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

17: 23.10 Addition Theorems and Other Identities

§23.10 Addition Theorems and Other Identities

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§23.10(i) Addition Theorems

… ►For further addition-type identities for the $\sigma$-function see Lawden (1989, §6.4). …

18: 34.7 Basic Properties: $\mathit{9}j$ Symbol

… ►It constitutes an addition theorem for the $\mathit{9}j$ symbol. …

19: 22.8 Addition Theorems

§22.8 Addition Theorems

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20: 18.18 Sums

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§18.18(ii) Addition Theorems

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Ultraspherical

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Legendre

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Laguerre

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Hermite

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