# increasing

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## 1—10 of 46 matching pages

##### 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. …##### 2: 33.23 Methods of Computation

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►Cancellation errors increase with increases in $\rho $ 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.
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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 $\mathrm{\ell}$ for the regular solutions and increasing
$\mathrm{\ell}$ 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 $\nu $
$(\ge 0)$ is fixed, then throughout the interval $$, ${I}_{\nu}\left(x\right)$ is positive and increasing, and ${K}_{\nu}\left(x\right)$ is positive and decreasing.
►If $x$
$(>0)$ is fixed, then throughout the interval $$, ${I}_{\nu}\left(x\right)$ is decreasing, and ${K}_{\nu}\left(x\right)$ 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

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►and is strictly increasing when $0\le x\le 1$.
…It, too, is strictly increasing when $0\le x\le 1$, and
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##### 6: 32.14 Combinatorics

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►With $$, $\mathit{\pi}({m}_{1}),\mathit{\pi}({m}_{2}),\mathrm{\dots},\mathit{\pi}({m}_{n})$ is said to be an

*increasing subsequence*of $\mathit{\pi}$ of*length*$n$ when $$. Let ${\mathrm{\ell}}_{N}(\mathit{\pi})$ be the length of the longest increasing subsequence of $\mathit{\pi}$. …##### 7: 5.10 Continued Fractions

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►

5.10.1
$$\mathrm{Ln}\mathrm{\Gamma}\left(z\right)+z-\left(z-\frac{1}{2}\right)\mathrm{ln}z-\frac{1}{2}\mathrm{ln}\left(2\pi \right)=\frac{{a}_{0}}{z+}\frac{{a}_{1}}{z+}\frac{{a}_{2}}{z+}\frac{{a}_{3}}{z+}\frac{{a}_{4}}{z+}\frac{{a}_{5}}{z+}\mathrm{\cdots},$$

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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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