§10.69 Uniform Asymptotic Expansions for Large Order

ⓘ

Keywords:: Kelvin functions, asymptotic approximations and expansions, double asymptotic properties, double asymptotic property, exponentially-small contributions, uniform asymptotic expansions for large order
Notes:: Combine the results given in §§10.41(ii) and 10.41(iii) with the definitions (10.61.1) and (10.61.2).
Permalink:: http://dlmf.nist.gov/10.69
See also:: Annotations for Ch.10

Let $U_{k}(p)$ and $V_{k}(p)$ be the polynomials defined in §10.41(ii), and

10.69.1		$\xi=(1+ix^{2})^{\ifrac{1}{2}}.$
ⓘ Defines: $\xi$ (locally) Symbols: $\mathrm{i}$ : imaginary unit and $x$ : real variable A&S Ref: 9.10.41 Permalink: http://dlmf.nist.gov/10.69.E1 Encodings: TeX, pMML, png See also: Annotations for §10.69 and Ch.10

Then as $\nu\to+\infty$ ,

10.69.2	$\displaystyle\operatorname{ber}_{\nu}\left(\nu x\right)+i\operatorname{bei}_{% \nu}\left(\nu x\right)$	$\displaystyle\sim\frac{e^{\nu\xi}}{(2\pi\nu\xi)^{\ifrac{1}{2}}}\left(\frac{xe^% {3\pi i/4}}{1+\xi}\right)^{\nu}\sum_{k=0}^{\infty}\frac{U_{k}(\xi^{-1})}{\nu^{% k}},$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin 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, $x$ : real variable, $\nu$ : complex parameter, $U_{k}(p)$ : polynomial coefficient and $\xi$ A&S Ref: 9.10.37 Referenced by: §10.69 Permalink: http://dlmf.nist.gov/10.69.E2 Encodings: TeX, pMML, png See also: Annotations for §10.69 and Ch.10
10.69.3	$\displaystyle\operatorname{ker}_{\nu}\left(\nu x\right)+i\operatorname{kei}_{% \nu}\left(\nu x\right)$	$\displaystyle\sim e^{-\nu\xi}\left(\frac{\pi}{2\nu\xi}\right)^{\ifrac{1}{2}}% \left(\frac{xe^{3\pi i/4}}{1+\xi}\right)^{-\nu}\sum_{k=0}^{\infty}(-1)^{k}% \frac{U_{k}(\xi^{-1})}{\nu^{k}},$
ⓘ Symbols: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin 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, $x$ : real variable, $\nu$ : complex parameter, $U_{k}(p)$ : polynomial coefficient and $\xi$ A&S Ref: 9.10.38 Permalink: http://dlmf.nist.gov/10.69.E3 Encodings: TeX, pMML, png See also: Annotations for §10.69 and Ch.10
10.69.4	$\displaystyle\operatorname{ber}_{\nu}'\left(\nu x\right)+i\operatorname{bei}_{% \nu}'\left(\nu x\right)$	$\displaystyle\sim\frac{e^{\nu\xi}}{x}\left(\frac{\xi}{2\pi\nu}\right)^{\ifrac{% 1}{2}}\left(\frac{xe^{3\pi i/4}}{1+\xi}\right)^{\nu}\sum_{k=0}^{\infty}\frac{V% _{k}(\xi^{-1})}{\nu^{k}},$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin 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, $x$ : real variable, $\nu$ : complex parameter, $V_{k}(p)$ : polynomial coefficient and $\xi$ A&S Ref: 9.10.39 Referenced by: §10.69 Permalink: http://dlmf.nist.gov/10.69.E4 Encodings: TeX, pMML, png See also: Annotations for §10.69 and Ch.10
10.69.5	$\displaystyle\operatorname{ker}_{\nu}'\left(\nu x\right)+i\operatorname{kei}_{% \nu}'\left(\nu x\right)$	$\displaystyle\sim-\frac{e^{-\nu\xi}}{x}\left(\frac{\pi\xi}{2\nu}\right)^{% \ifrac{1}{2}}\left(\frac{xe^{3\pi i/4}}{1+\xi}\right)^{-\nu}\*\sum_{k=0}^{% \infty}(-1)^{k}\frac{V_{k}(\xi^{-1})}{\nu^{k}},$
ⓘ Symbols: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin 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, $x$ : real variable, $\nu$ : complex parameter, $V_{k}(p)$ : polynomial coefficient and $\xi$ A&S Ref: 9.10.40 Permalink: http://dlmf.nist.gov/10.69.E5 Encodings: TeX, pMML, png See also: Annotations for §10.69 and Ch.10

uniformly for $x$ $\in(0,\infty)$ . All fractional powers take their principal values.

All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv).

Accuracy in (10.69.2) and (10.69.4) can be increased by including exponentially-small contributions as in (10.67.3), (10.67.4), (10.67.7), and (10.67.8) with $x$ replaced by $\nu x$ .