inhomogeneous equations

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21: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.9_5 $$a_{k+1}(\lambda )=\frac{\lambda }{1-{\lambda }^2}\frac{\lambda a_k^{\prime \prime }(\lambda )+a_k^{\prime }(\lambda )}{(2k+1)(2k+2)},$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

$k$: nonnegative integer, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Referenced by:

Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/11.11.E9_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.13_5 $${\left(\frac{1}{2}\right)}_k{b}_k(\lambda )=\frac{{(-1)}^k}{{\left(1-{\lambda }^2\right)}^{1/4}}{U}_k\left(\frac{1}{\sqrt{1-{\lambda }^2}}\right),$$ $k=0,1,2,\mathrm{\dots }$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $k$: nonnegative integer, $\lambda$: parameter and $b_k(\lambda )$: expansion function

Referenced by:

Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/11.11.E13_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.17 $${𝐀}_{-\nu }\left(\nu +a{\nu }^{1/3}\right)=2^{1/3}{\nu }^{-1/3}\mathrm{Hi}\left(-2^{1/3}a\right)+O\left({\nu }^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), ${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\nu$: real or complex order and $a_k(\lambda )$: expansion function

Sources:

Olver (1997b, pp. 103 and 352); Nemes (2020)

Referenced by:

§11.11(iii), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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