10.66 Expansions in Series of Bessel Functions10.68 Modulus and Phase Functions

§10.67 Asymptotic Expansions for Large Argument

Contents

§10.67(i) \mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x, and Derivatives

Define a_{k}(\nu) and b_{k}(\nu) as in §§10.17(i) and 10.17(ii). Then as x\to\infty with \nu fixed,

10.67.3 \mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x\sim\frac{e^{{x/\sqrt{2}}}}{(2\pi x)^{{\frac{1}{2}}}}\*\sum _{{k=0}}^{\infty}\frac{a_{k}(\nu)}{x^{k}}\mathop{\cos\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3k}{4}-\frac{1}{8}\right)\pi\right)-\frac{1}{\pi}(\mathop{\sin\/}\nolimits\!\left(2\nu\pi\right)\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x+\mathop{\cos\/}\nolimits\!\left(2\nu\pi\right)\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x),
10.67.4 \mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x\sim\frac{e^{{x/\sqrt{2}}}}{(2\pi x)^{{\frac{1}{2}}}}\*\sum _{{k=0}}^{\infty}\frac{a_{k}(\nu)}{x^{k}}\mathop{\sin\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3k}{4}-\frac{1}{8}\right)\pi\right)+\frac{1}{\pi}(\mathop{\cos\/}\nolimits\!\left(2\nu\pi\right)\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x-\mathop{\sin\/}\nolimits\!\left(2\nu\pi\right)\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x).
10.67.7 {\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x\sim\frac{e^{{x/\sqrt{2}}}}{(2\pi x)^{{\frac{1}{2}}}}\*\sum _{{k=0}}^{\infty}\frac{b_{k}(\nu)}{x^{k}}\mathop{\cos\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3k}{4}+\frac{1}{8}\right)\pi\right)-\frac{1}{\pi}(\mathop{\sin\/}\nolimits\!\left(2\nu\pi\right){\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}x+\mathop{\cos\/}\nolimits\!\left(2\nu\pi\right){\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits^{{\prime}}}x),
10.67.8 {\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}x\sim\frac{e^{{x/\sqrt{2}}}}{(2\pi x)^{{\frac{1}{2}}}}\*\sum _{{k=0}}^{\infty}\frac{b_{k}(\nu)}{x^{k}}\mathop{\sin\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3k}{4}+\frac{1}{8}\right)\pi\right)+\frac{1}{\pi}(\mathop{\cos\/}\nolimits\!\left(2\nu\pi\right){\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}x-\mathop{\sin\/}\nolimits\!\left(2\nu\pi\right){\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits^{{\prime}}}x).

The contributions of the terms in \mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x, \mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x, {\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}x, and {\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits^{{\prime}}}x on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). However, their inclusion improves numerical accuracy.

§10.67(ii) Cross-Products and Sums of Squares in the Case \nu=0

As x\to\infty

10.67.9 {\mathop{\mathrm{ber}\/}\nolimits^{{2}}}x+{\mathop{\mathrm{bei}\/}\nolimits^{{2}}}x\sim\frac{e^{{x\sqrt{2}}}}{2\pi x}\left(1+\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^{2}}-\frac{33}{256\sqrt{2}}\frac{1}{x^{3}}-\frac{1797}{8192}\frac{1}{x^{4}}+\cdots\right),
10.67.10 \mathop{\mathrm{ber}\/}\nolimits x{\mathop{\mathrm{bei}\/}\nolimits^{{\prime}}}x-{\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x\mathop{\mathrm{bei}\/}\nolimits x\sim\frac{e^{{x\sqrt{2}}}}{2\pi x}\left(\frac{1}{\sqrt{2}}+\frac{1}{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{39}{512}\frac{1}{x^{3}}+\frac{75}{8192\sqrt{2}}\frac{1}{x^{4}}+\cdots\right),
10.67.11 \mathop{\mathrm{ber}\/}\nolimits x{\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x+\mathop{\mathrm{bei}\/}\nolimits x{\mathop{\mathrm{bei}\/}\nolimits^{{\prime}}}x\sim\frac{e^{{x\sqrt{2}}}}{2\pi x}\left(\frac{1}{\sqrt{2}}-\frac{3}{8}\frac{1}{x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{45}{512}\frac{1}{x^{3}}+\frac{315}{8192\sqrt{2}}\frac{1}{x^{4}}+\cdots\right),
10.67.12 \left({\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x\right)^{2}+\left({\mathop{\mathrm{bei}\/}\nolimits^{{\prime}}}x\right)^{2}\sim\frac{e^{{x\sqrt{2}}}}{2\pi x}\left(1-\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9}{64}\frac{1}{x^{2}}+\frac{75}{256\sqrt{2}}\frac{1}{x^{3}}+\frac{2475}{8192}\frac{1}{x^{4}}+\cdots\right).
10.67.13 {\mathop{\mathrm{ker}\/}\nolimits^{{2}}}x+{\mathop{\mathrm{kei}\/}\nolimits^{{2}}}x\sim\frac{\pi}{2x}e^{{-x\sqrt{2}}}\left(1-\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^{2}}+\frac{33}{256\sqrt{2}}\frac{1}{x^{3}}-\frac{1797}{8192}\frac{1}{x^{4}}+\cdots\right),
10.67.14 \mathop{\mathrm{ker}\/}\nolimits x{\mathop{\mathrm{kei}\/}\nolimits^{{\prime}}}x-{\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}}x\mathop{\mathrm{kei}\/}\nolimits x\sim-\frac{\pi}{2x}e^{{-x\sqrt{2}}}\left(\frac{1}{\sqrt{2}}-\frac{1}{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{39}{512}\frac{1}{x^{3}}+\frac{75}{8192\sqrt{2}}\frac{1}{x^{4}}+\cdots\right),
10.67.15 \mathop{\mathrm{ker}\/}\nolimits x{\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}}x+\mathop{\mathrm{kei}\/}\nolimits x{\mathop{\mathrm{kei}\/}\nolimits^{{\prime}}}x\sim-\frac{\pi}{2x}e^{{-x\sqrt{2}}}\left(\frac{1}{\sqrt{2}}+\frac{3}{8}\frac{1}{x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{45}{512}\frac{1}{x^{3}}+\frac{315}{8192\sqrt{2}}\frac{1}{x^{4}}+\cdots\right),
10.67.16 \left({\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}}x\right)^{2}+\left({\mathop{\mathrm{kei}\/}\nolimits^{{\prime}}}x\right)^{2}\sim\frac{\pi}{2x}e^{{-x\sqrt{2}}}\left(1+\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9}{64}\frac{1}{x^{2}}-\frac{75}{256\sqrt{2}}\frac{1}{x^{3}}+\frac{2475}{8192}\frac{1}{x^{4}}+\cdots\right).