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1: 4.2 Definitions

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§4.2(iv) Powers

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Powers with General Bases

►The general $a^{\mathrm{th}}$ power of $z$ is defined by …The principal value is … …

2: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

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Stieltjes Fractions

… ►We say that it corresponds to the formal power series …if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$ agrees with (3.10.7) up to and including the term in $z^{n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. …

3: 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 $\vartheta \left(x\right)$. …

4: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

5: 1.9 Calculus of a Complex Variable

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Powers

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§1.9(v) Infinite Sequences and Series

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§1.9(vi) Power Series

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Operations

… ►Lastly, a power series can be differentiated any number of times within its circle of convergence: …

6: 6.6 Power Series

§6.6 Power Series

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7: Guide to Searching the DLMF

… ►Wildcards allow matching patterns and marking parts of an expression that don’t matter (as for example, which variable name the author uses for a function): … ►You can use in math queries all the symbols and commands defined in LaTeX (you can omit the $\backslash$), and some additional convenient ones, as well as the special functions’ names: ►

Table 3: A sample of recognized symbols

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Symbols	Comments
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_, ^	For subscripting and superscripting (power)
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… ►DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. … ►Sometimes there are distinctions between various special function names based on font style, such as the use of bold or calligraphic letters. …

8: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …

9: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). …

10: 24.20 Tables

§24.20 Tables

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