power

SearchAdvancedHelp

(0.001 seconds)

31—40 of 142 matching pages

31: 27.19 Methods of Computation: Factorization

… ►The snfs can be applied only to numbers that are very close to a power of a very small base. …

32: 33.6 Power-Series Expansions in $\rho$

§33.6 Power-Series Expansions in $\rho$

…

33: 33.19 Power-Series Expansions in $r$

§33.19 Power-Series Expansions in $r$

…

34: 35.9 Applications

… ►For other statistical applications of ${}_p{}^{}F_q^{}$ functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). …

35: 27.7 Lambert Series as Generating Functions

… ►If , then the quotient $x^n/(1-x^n)$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: … ►

27.7.5 $$\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha }\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)x^n,$$

ⓘ

Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

…

36: 28.34 Methods of Computation

… ►

Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

… ►

Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

…

37: 10.74 Methods of Computation

… ►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. … ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). … ►If values of the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, or the other functions treated in this chapter, are needed for integer-spaced ranges of values of the order $\nu$, then a simple and powerful procedure is provided by recurrence relations typified by the first of (10.6.1). …

38: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►

39: 27.13 Functions

… ►This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer $n$ is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. … ►Hardy and Littlewood (1925) conjectures that when $k$ is not a power of 2, and that $G\left(k\right)\le 4k$ when $k$ is a power of 2, but the most that is known (in 2009) is for some constant $c$. … ►Mordell (1917) notes that $r_k\left(n\right)$ is the coefficient of $x^n$ in the power-series expansion of the $k$th power of the series for $\vartheta \left(x\right)$. …

40: 36.8 Convergent Series Expansions

… ►For multinomial power series for ${\mathrm{\Psi }}_K\left(𝐱\right)$, see Connor and Curtis (1982). …