power-series expansions

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31: 30.16 Methods of Computation

… ►For small $|{\gamma }^2|$ we can use the power-series expansion (30.3.8). …

32: 16.5 Integral Representations and Integrals

… ►In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta$ is an arbitrary small positive constant. …

33: 11.10 Anger–Weber Functions

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§11.10(iii) Maclaurin Series

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34: Bibliography L

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New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.

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External Links:

ISSN 0025-5718, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§28.24, §28.24.

Encodings:

BibTeX

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35: 1.10 Functions of a Complex Variable

… ►Let $F(x,z)$ have a converging power series expansion of the form …

36: 3.11 Approximation Techniques

… ►When $F$ has an explicit power-series expansion a possible choice of $R$ is a Padé approximation to $F$. …

37: 13.14 Definitions and Basic Properties

… ►The series … ►

13.14.9 $$W_{\kappa ,\pm \frac{1}{2}n}\left(z\right)={(-1)}^{\kappa -\frac{1}{2}n-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}n+\frac{1}{2}}\sum _{k=0}^{\kappa -\frac{1}{2}n-\frac{1}{2}}\left(\genfrac{}{}{0.0pt}{}{\kappa -\frac{1}{2}n-\frac{1}{2}}{k}\right){\left(n+1+k\right)}_{\kappa -k-\frac{1}{2}n-\frac{1}{2}}{(-z)}^k,$$ $\kappa -\frac{1}{2}n-\frac{1}{2}=0,1,2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.14(i)

Permalink:

http://dlmf.nist.gov/13.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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38: 23.17 Elementary Properties

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§23.17(ii) Power and Laurent Series

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39: 3.10 Continued Fractions

… ►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 $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. …

40: 28.34 Methods of Computation

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

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

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

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Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

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Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.