§10.69 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).
Keywords:: Kelvin functions, asymptotic approximations and expansions
Permalink:: http://dlmf.nist.gov/10.69

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

10.69.1 $\xi=(1+ix^{2})^{{\ifrac{1}{2}}}.$

Symbols:: $x$ : real variable and $\xi$
A&S Ref:: 9.10.41
Permalink:: http://dlmf.nist.gov/10.69.E1
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Then as $\nu\to+\infty$ ,

10.69.2 $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(\nu x\right)+i\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(\nu x\right)\sim\frac{e^{{\nu\xi}}}{(2\pi\nu\xi)^{{\ifrac{1}{2}}}}\left(\frac{xe^{{3\pi i/4}}}{1+\xi}\right)^{\nu}\sum _{{k=0}}^{\infty}\frac{U_{k}(\xi^{{-1}})}{\nu^{k}},$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $e$ : base of exponential function, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $x$ : real variable, $\nu$ : complex parameter, $U_{k}(p)$ : polynomials and $\xi$
A&S Ref:: 9.10.37
Referenced by:: §10.69
Permalink:: http://dlmf.nist.gov/10.69.E2
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10.69.3 $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(\nu x\right)+i\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(\nu x\right)\sim e^{{-\nu\xi}}\left(\frac{\pi}{2\nu\xi}\right)^{{\ifrac{1}{2}}}\left(\frac{xe^{{3\pi i/4}}}{1+\xi}\right)^{{-\nu}}\sum _{{k=0}}^{\infty}(-1)^{k}\frac{U_{k}(\xi^{{-1}})}{\nu^{k}},$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $e$ : base of exponential function, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $x$ : real variable, $\nu$ : complex parameter, $U_{k}(p)$ : polynomials and $\xi$
A&S Ref:: 9.10.38
Permalink:: http://dlmf.nist.gov/10.69.E3
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10.69.4 ${\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu x\right)+i{\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu x\right)\sim\frac{e^{{\nu\xi}}}{x}\left(\frac{\xi}{2\pi\nu}\right)^{{\ifrac{1}{2}}}\left(\frac{xe^{{3\pi i/4}}}{1+\xi}\right)^{\nu}\sum _{{k=0}}^{\infty}\frac{V_{k}(\xi^{{-1}})}{\nu^{k}},$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $e$ : base of exponential function, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $x$ : real variable, $\nu$ : complex parameter, $V_{k}(p)$ : polynomials and $\xi$
A&S Ref:: 9.10.39
Referenced by:: §10.69
Permalink:: http://dlmf.nist.gov/10.69.E4
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10.69.5 ${\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu x\right)+i{\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu x\right)\sim-\frac{e^{{-\nu\xi}}}{x}\left(\frac{\pi\xi}{2\nu}\right)^{{\ifrac{1}{2}}}\left(\frac{xe^{{3\pi i/4}}}{1+\xi}\right)^{{-\nu}}\*\sum _{{k=0}}^{\infty}(-1)^{k}\frac{V_{k}(\xi^{{-1}})}{\nu^{k}},$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $e$ : base of exponential function, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $x$ : real variable, $\nu$ : complex parameter, $V_{k}(p)$ : polynomials and $\xi$
A&S Ref:: 9.10.40
Permalink:: http://dlmf.nist.gov/10.69.E5
Encodings:: TeX, pMML, png

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 $\nu x$ .