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1: 27.17 Other Applications
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. …
2: 33.23 Methods of Computation
Cancellation errors increase with increases in ρ and | r | , 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. … 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. … This implies decreasing for the regular solutions and increasing for the irregular solutions of §§33.2(iii) and 33.14(iii). …
3: 10.37 Inequalities; Monotonicity
If ν ( 0 ) is fixed, then throughout the interval 0 < x < , I ν ( x ) is positive and increasing, and K ν ( x ) is positive and decreasing. If x ( > 0 ) is fixed, then throughout the interval 0 < ν < , I ν ( x ) is decreasing, and K ν ( x ) is increasing. …
4: 27.15 Chinese Remainder Theorem
This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …
5: 4.12 Generalized Logarithms and Exponentials
and is strictly increasing when 0 x 1 . …It, too, is strictly increasing when 0 x 1 , and …
6: 32.14 Combinatorics
With 1 m 1 < < m n N , π ( m 1 ) , π ( m 2 ) , , π ( m n ) is said to be an increasing subsequence of π of length n when π ( m 1 ) < π ( m 2 ) < < π ( m n ) . Let N ( π ) be the length of the longest increasing subsequence of π . …
7: 5.10 Continued Fractions
5.10.1 Ln Γ ( z ) + z - ( z - 1 2 ) ln z - 1 2 ln ( 2 π ) = a 0 z + a 1 z + a 2 z + a 3 z + a 4 z + a 5 z + ,
8: 29.20 Methods of Computation
A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …
9: 2.2 Transcendental Equations
Let f ( x ) be continuous and strictly increasing when a < x < and …
10: 9.17 Methods of Computation
However, in the case of Ai ( z ) and Bi ( z ) this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). …