Gauss%20sum
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9 matching pages
1: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
…It can be expressed as a sum over all primes :
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►Gauss and Legendre conjectured that is asymptotic to as :
…(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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►the sum of the th powers of the positive integers that are relatively prime to .
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2: 20.11 Generalizations and Analogs
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§20.11(i) Gauss Sum
►For relatively prime integers with and even, the Gauss sum is defined by ►
20.11.1
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►Similar identities can be constructed for , , and .
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3: 18.5 Explicit Representations
4: 19.36 Methods of Computation
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►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated.
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►The step from to is an ascending Landen transformation if (leading ultimately to a hyperbolic case of ) or a descending Gauss transformation if (leading to a circular case of ).
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►Descending Gauss transformations of (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)).
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►The function is computed by descending Landen transformations if is real, or by descending Gauss transformations if is complex (Bulirsch (1965b)).
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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5: Bibliography V
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Modular hypergeometric residue sums of elliptic Selberg integrals.
Lett. Math. Phys. 58 (3), pp. 223–238.
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Transformations of some Gauss hypergeometric functions.
J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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Representation of an odd number as a sum of three primes (Russian).
Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).
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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
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6: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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A -analog of the Gauss summation theorem for hypergeometric series in
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Adv. in Math. 72 (1), pp. 59–131.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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7: Bibliography G
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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer.
J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.
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Werke. Band II.
pp. 436–447 (German).
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Construction of Gauss-Christoffel quadrature formulas.
Math. Comp. 22, pp. 251–270.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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8: 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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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively -binomial sums and basic hypergeometric series.
Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
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9: Errata
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Equation (18.38.3)
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Subsection 17.7(iii)
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Subsection 15.19(v)
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Chapters 8, 20, 36
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References
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18.38.3
, ,
This equation was updated to include the value of the sum in terms of the function. Also the constraint was previously , .
The title of the paragraph which was previously “Andrews’ Terminating -Analog of (17.7.8)” has been changed to “Andrews’ -Analog of the Terminating Version of Watson’s Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating -Analog” has been changed to “Andrews’ -Analog of the Terminating Version of Whipple’s Sum (16.4.7)”.
A new Subsection Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.