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1: 27.17 Other Applications
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►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series.
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2: 33.23 Methods of Computation
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►Cancellation errors increase with increases in and , and may be estimated by comparing the final sum of the series with the largest partial sum.
Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21.
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►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii.
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►This implies decreasing for the regular solutions and increasing
for the irregular solutions of §§33.2(iii) and 33.14(iii).
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3: 10.37 Inequalities; Monotonicity
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►If
is fixed, then throughout the interval , is positive and increasing, and is positive and decreasing.
►If
is fixed, then throughout the interval , is decreasing, and is increasing.
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4: 27.15 Chinese Remainder Theorem
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►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation.
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5: 4.12 Generalized Logarithms and Exponentials
6: 32.14 Combinatorics
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►With , is said to be an increasing
subsequence of of length
when .
Let be the length of the longest increasing subsequence of .
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7: 5.10 Continued Fractions
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►
5.10.1
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8: 29.20 Methods of Computation
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►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree.
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9: 19.30 Lengths of Plane Curves
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►As
increases, the eccentricity decreases and the rate of change of arclength for a fixed value of is given by
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►See Carlson (1977b, Ex. 9.4-1 and (9.4-4)) for arclengths of hyperbolas and ellipses in terms of that differ only in the sign of .
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10: 2.2 Transcendental Equations
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►Let be continuous and strictly increasing when and
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