quadratic reciprocity law

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11: 13.12 Products

… ►For generalizations of this quadratic relation see Majima et al. (2000). …

12: 27.22 Software

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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below ${10}^{16}$. Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$, and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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13: 16.6 Transformations of Variable

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Quadratic

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14: 27.19 Methods of Computation: Factorization

… ►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …

15: 5.3 Graphics

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Figure 5.3.1: $\mathrm{\Gamma }\left(x\right)$ and $1/\mathrm{\Gamma }\left(x\right)$. … Magnify

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Figure 5.3.5: $1/|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|$. Magnify 3D Help

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16: Bibliography F

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On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

Encodings:

BibTeX

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17: 8.23 Statistical Applications

… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

18: 18.7 Interrelations and Limit Relations

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§18.7(ii) Quadratic Transformations

… ►Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). …

19: 19.36 Methods of Computation

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§19.36(ii) Quadratic Transformations

►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … ►As $n\to \mathrm{\infty }$, $c_n$, $a_n$, and $t_n$ converge quadratically to limits $0$, $M$, and $T$, respectively; hence … ►Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …

20: 15.8 Transformations of Variable

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§15.8(iii) Quadratic Transformations

►A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. … ►

§15.8(iv) Quadratic Transformations (Continued)

… ►This is a quadratic transformation between two cases in Group 1. … ►which is a quadratic transformation between two cases in Group 3. …