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##### 1: 4.2 Definitions

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###### §4.2(iv) Powers

►###### 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

… ►###### 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

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►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.
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►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 $c$.
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►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 $\vartheta \left(x\right)$.
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##### 4: 12.18 Methods of Computation

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►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions.
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##### 5: 1.9 Calculus of a Complex Variable

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###### Powers

… ►###### §1.9(v) Infinite Sequences and Series

… ►###### §1.9(vi) Power Series

… ►###### 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

…##### 7: Guide to Searching the DLMF

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►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):
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►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:
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Table 3: A sample of recognized symbols
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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.
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►Sometimes there are distinctions between various special function names based on font style, such as the use of bold or calligraphic letters.
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Symbols | Comments |
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`_, ^` |
For subscripting and superscripting (power) |

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##### 8: 16.25 Methods of Computation

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►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations.
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##### 9: 14.32 Methods of Computation

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►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).
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