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21: 13.14 Definitions and Basic Properties

… ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►

§13.14(iii) Limiting Forms as $z\to 0$

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§13.14(iv) Limiting Forms as $z\to \mathrm{\infty }$

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22: 26.12 Plane Partitions

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§26.12(iv) Limiting Form

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23: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

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24: 10.2 Definitions

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Bessel Functions of the Third Kind (Hankel Functions)

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25: 18.7 Interrelations and Limit Relations

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§18.7(iii) Limit Relations

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26: 10.5 Wronskians and Cross-Products

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27: 10.50 Wronskians and Cross-Products

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28: 10.45 Functions of Imaginary Order

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$${\tilde{K}}_{\nu }\left(x\right)={(\pi /(2x))}^{\frac{1}{2}}{\mathrm{e}}^{-x}\left(1+O\left(x^{-1}\right)\right).$$

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29: 10.25 Definitions

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30: 18.34 Bessel Polynomials

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18.34.8 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,a-\alpha -2)}\left(1+\alpha x\right)}{P_n^{(\alpha ,a-\alpha -2)}\left(1\right)}=y_n(x;a).$$

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Symbols:

$y_n(x;a)$: Bessel polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.34(iii)

Permalink:

http://dlmf.nist.gov/18.34.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

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