asymptotic expansions for large variable

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11: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

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12: 8.12 Uniform Asymptotic Expansions for Large Parameter

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Inverse Function

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13: 10.41 Asymptotic Expansions for Large Order

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§10.41(ii) Uniform Expansions for Real Variable

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10.41.4 $$K_{\nu }\left(\nu z\right)\sim {\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(p)}{{\nu }^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.8

Permalink:

http://dlmf.nist.gov/10.41.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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14: Bibliography W

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An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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External Links:

FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§13.19.

Encodings:

BibTeX

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15: 10.20 Uniform Asymptotic Expansions for Large Order

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10.20.5 $$Y_{\nu }\left(\nu z\right)\sim -{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.36

Permalink:

http://dlmf.nist.gov/10.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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10.20.9 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu z\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu z\right)\end{array}\}\sim \frac{4{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}}{z}{\left(\frac{1-z^2}{4\zeta }\right)}^{\frac{1}{4}}\left(\frac{{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{4}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{C_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{2}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{D_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $C_k(\zeta )$: coefficients and $D_k(\zeta )$: coefficients

A&S Ref:

9.3.45

Referenced by:

§10.20(ii)

Permalink:

http://dlmf.nist.gov/10.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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16: 10.72 Mathematical Applications

… ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). …

17: 8.18 Asymptotic Expansions of $I_x(a,b)$

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§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

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Symmetric Case

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General Case

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Inverse Function

►For asymptotic expansions for large values of $a$ and/or $b$ of the $x$-solution of the equation …

18: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

… ►All fractional powers take their principal values. ►All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). … ►

19: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$ $(=x)$ are real. …

20: 10.40 Asymptotic Expansions for Large Argument

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10.40.2 $$K_{\nu }\left(z\right)\sim {\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.2

Referenced by:

§10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.41(iv), §10.45

Permalink:

http://dlmf.nist.gov/10.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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