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21: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

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13.19.2 $$M_{\kappa ,\mu }\left(z\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{z}^{-s}+\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\mathrm{e}}^{-\frac{1}{2}z\pm (\frac{1}{2}+\mu -\kappa )\pi \mathrm{i}}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

… ►Again, $\delta$ denotes an arbitrary small positive constant. … ►

13.19.3 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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22: 10.75 Tables

… ►Also, for additional listings of tables pertaining to complex arguments see Babushkina et al. (1997). … ►

British Association for the Advancement of Science (1937) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $x=0(.001)16(.01)25$, 10D; $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0.01(.01)25$, 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $Y_0\left(x\right)$, $Y_1\left(x\right)$ for small values of $x$, as well as auxiliary functions to compute all four functions for large values of $x$.

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British Association for the Advancement of Science (1937) tabulates $I_0\left(x\right)$, $I_1\left(x\right)$, $x=0(.001)5$, 7–8D; $K_0\left(x\right)$, $K_1\left(x\right)$, $x=0.01(.01)5$, 7–10D; ${\mathrm{e}}^{-x}{I}_0\left(x\right)$, ${\mathrm{e}}^{-x}{I}_1\left(x\right)$, ${\mathrm{e}}^x{K}_0\left(x\right)$, ${\mathrm{e}}^x{K}_1\left(x\right)$, $x=5(.01)10(.1)20$, 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $K_0\left(x\right)$, $K_1\left(x\right)$ for small values of $x$.

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Young and Kirk (1964) tabulates ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, $n=0,1$, $x=0(.1)10$, 15D; ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, modulus and phase functions $M_n\left(x\right)$, ${\theta }_n\left(x\right)$, $N_n\left(x\right)$, ${\varphi }_n\left(x\right)$, $n=0,1,2$, $x=0(.01)2.5$, 8S, and $n=0(1)10$, $x=0(.1)10$, 7S. Also included are auxiliary functions to facilitate interpolation of the tables for $n=0(1)10$ for small values of $x$. (Concerning the phase functions see §10.68(iv).)

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23: Bibliography P

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On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).

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Notes:

English translation in Math. Notes 65 (1999) pp. 571-581.

External Links:

ISSN 0025-567X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

Encodings:

BibTeX

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Exponentially small expansions of the confluent hypergeometric functions. Appl. Math. Sci. (Ruse) 7 (133-136), pp. 6601–6609.

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External Links:

ISSN 1312-885X, Document, MathReview Entry

Referenced by:

§13.7(iii), §13.7(iii).

Encodings:

BibTeX

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The computation of Bessel functions on a small computer. Comput. Math. Appl. 10 (2), pp. 161–166.

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Notes:

Fortran code.

External Links:

ISSN 0898-1221, Document, MathReview (G. P. Bhattacharjee), ZentralBlatt

Referenced by:

§10.77(iii).

Encodings:

BibTeX

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24: DLMF Project News

error generating summary

25: 30.1 Special Notation

… ► ►►►

$x$	real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, .
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$\delta$	arbitrary small positive constant.

… ►Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. …

26: 1.10 Functions of a Complex Variable

… ►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in $ℂ$ with at most one exception. … ►

Phase (or Argument) Principle

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27: 9.7 Asymptotic Expansions

… ►Here $\delta$ denotes an arbitrary small positive constant and … ►

9.7.3 $$\chi (x)\equiv {\pi }^{1/2}\mathrm{\Gamma }\left(\frac{1}{2}x+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}x+\frac{1}{2}\right).$$

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Defines:

$\chi (x)$: function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $\equiv$: equals by definition

Source:

Olver (1997b, (13.02), p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the function $\chi$ was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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9.7.4 $$\chi (x)\sim {(\frac{1}{2}\pi x)}^{1/2}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $\chi (x)$: function

Source:

Olver (1997b, p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E4

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the formula was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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9.7.5 $$\mathrm{Ai}\left(z\right)\sim \frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

(9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.6 $${\mathrm{Ai}}^{\prime }\left(z\right)\sim -\frac{z^{1/4}{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

§9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

…

28: 2.11 Remainder Terms; Stokes Phenomenon

… ►However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►Owing to the factor ${\mathrm{e}}^{-\rho }$, that is, ${\mathrm{e}}^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_p\left(z\right)$. … ►Hence from §7.12(i) $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when $\rho$ is large. … ►In particular, on the ray $\theta =\pi$ greatest accuracy is achieved by (a) taking the average of the expansions (2.11.6) and (2.11.7), followed by (b) taking account of the exponentially-small contributions arising from the terms involving $h_{2s}(\theta ,\alpha )$ in (2.11.15). … ►As these lines are crossed exponentially-small contributions, such as that in (2.11.7), are “switched on” smoothly, in the manner of the graph in Figure 2.11.1. …

29: 4.45 Methods of Computation

… ►Then we take square roots repeatedly until $|y|$ is sufficiently small, where … ►

4.45.9 $$x_n=\frac{x_{n-1}}{1+{(1+x_{n-1}^2)}^{1/2}},$$ $n=1,2,3,\mathrm{\dots }$,

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Symbols:

$\arctan z$: arctangent function, $n$: integer and $x$: real variable

Referenced by:

(4.45.8), Erratum (V1.0.7) for Equations (4.45.8), (4.45.9)

Permalink:

http://dlmf.nist.gov/4.45.E9

Encodings:

pMML, png, TeX

Substitution (effective with 1.0.7):

The original recurrence given in this equation, $x_n=\frac{{(1+x_{n-1}^2)}^{1/2}-1}{x_{n-1}}$ is susceptible to cancellation error when $x_n$ is small. It has been replaced with another recurrence that is better for numerically computing $\arctan x$.

Suggested 2014-02-05 by Masataka Urago

See also:

Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

►until $x_n$ is sufficiently small. … ►The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. …

30: 2.10 Sums and Sequences

… ►Hence …the last step following from $|x^t|\le 1$ when $t$ is on the interval $[-\frac{1}{2},0]$, the imaginary axis, or the small semicircle. …