Gram%E2%80%93Schmidt%20procedure
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1: 3.11 Approximation Techniques
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►A more efficient procedure is as follows.
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►A general procedure is to approximate by a rational function (vanishing at infinity) and then approximate by .
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►A set of functions that is linearly independent on the set (compare (3.11.36)) can always be orthogonalized in the sense given in the preceding paragraph by the Gram–Schmidt procedure; see Gautschi (1997a).
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2: 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.
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3: Bibliography H
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The Laplace transform for expressions that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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Integrals that contain a probability function of complicated arguments.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).
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Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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Some properties and applications of the repeated integrals of the error function.
Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
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4: Bibliography K
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A proof of the -Macdonald-Morris conjecture for
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Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
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Linear convergence and the bisection algorithm.
Amer. Math. Monthly 93 (1), pp. 48–51.
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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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Tom Koornwinder’s Personal Collection of Maple Procedures
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5: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions and , ,
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ACM Trans. Math. Software 3 (1), pp. 93–95.
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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument.
ACM Trans. Math. Software 16 (2), pp. 178–182.
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Special Functions.
Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.
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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.
J. Math. Anal. Appl. 416 (1), pp. 52–80.
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6: Bibliography D
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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Uniform asymptotic expansions for Charlier polynomials.
J. Approx. Theory 112 (1), pp. 93–133.
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7: 8.23 Statistical Applications
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►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).
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8: 32.8 Rational Solutions
9: 26.13 Permutations: Cycle Notation
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