power series

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31: 11.2 Definitions

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§11.2(i) Power-Series Expansions

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32: 11.9 Lommel Functions

… ►where $A$, $B$ are arbitrary constants, $s_{\mu ,\nu }\left(z\right)$ is the Lommel function defined by …

33: 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. … ►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)). …

34: 4.45 Methods of Computation

… ►The function $\ln x$ can always be computed from its ascending power series after preliminary scaling. … ►The function $\arctan x$ can always be computed from its ascending power series after preliminary transformations to reduce the size of $x$. … ►Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

35: 8.7 Series Expansions

§8.7 Series Expansions

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36: 4.38 Inverse Hyperbolic Functions: Further Properties

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§4.38(i) Power Series

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37: 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: …

38: 11.13 Methods of Computation

… ►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …

39: 30.4 Functions of the First Kind

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§30.4(iii) Power-Series Expansion

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40: 36.8 Convergent Series Expansions

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