power-series expansions in ϵ

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1: 33.19 Power-Series Expansions in $r$

§33.19 Power-Series Expansions in $r$

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2: 33.6 Power-Series Expansions in $\rho$

§33.6 Power-Series Expansions in $\rho$

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3: 28.15 Expansions for Small $q$

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§28.15(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$

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4: 33.20 Expansions for Small $|ϵ|$

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§33.20(ii) Power-Series in $ϵ$ for the Regular Solution

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5: 28.6 Expansions for Small $q$

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§28.6(i) Eigenvalues

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28.6.19 $$a-{(2n+2)}^2-\frac{q^2}{a-{(2n)}^2-}\frac{q^2}{a-{(2n-2)}^2-}\mathrm{\cdots }\frac{q^2}{a-2^2}=-\frac{q^2}{{(2n+4)}^2-a-}\frac{q^2}{{(2n+6)}^2-a-}\mathrm{\cdots },$$ $a=b_{2n+2}\left(q\right)$.

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter, $n$: integer and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.6.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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6: 6.18 Methods of Computation

… ►For small or moderate values of $x$ and $|z|$, the expansion in power series (§6.6) or in series of spherical Bessel functions (§6.10(ii)) can be used. …

7: 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. …

8: 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 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)). …

9: 27.13 Functions

… ►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 $\vartheta \left(x\right)$. …

10: 8.7 Series Expansions

§8.7 Series Expansions

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