# increasing

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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 $\rho$ 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. … 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 $\ell$ for the regular solutions and increasing $\ell$ for the irregular solutions of §§33.2(iii) and 33.14(iii). …
##### 3: 10.37 Inequalities; Monotonicity
If $\nu$ $(\geq 0)$ is fixed, then throughout the interval $0, $I_{\nu}\left(x\right)$ is positive and increasing, and $K_{\nu}\left(x\right)$ is positive and decreasing. If $x$ $(>0)$ is fixed, then throughout the interval $0<\nu<\infty$, $I_{\nu}\left(x\right)$ is decreasing, and $K_{\nu}\left(x\right)$ 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 $0\leq x\leq 1$. …It, too, is strictly increasing when $0\leq x\leq 1$, and …
##### 6: 32.14 Combinatorics
With $1\leq m_{1}<\cdots, $\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})$ is said to be an increasing subsequence of $\boldsymbol{\pi}$ of length $n$ when $\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\cdots<\boldsymbol{\pi}(m_{n})$. Let $\ell_{N}(\boldsymbol{\pi})$ be the length of the longest increasing subsequence of $\boldsymbol{\pi}$. …
##### 7: 5.10 Continued Fractions
5.10.1 $\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$
##### 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 and …
##### 10: 9.17 Methods of Computation
However, in the case of $\operatorname{Ai}\left(z\right)$ and $\operatorname{Bi}\left(z\right)$ this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). …