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31: 9.7 Asymptotic Expansions

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9.7.9 $$\mathrm{Ai}\left(-z\right)\sim \frac{1}{\sqrt{\pi }{z}^{1/4}}\left(\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k}}{{\zeta }^{2k}}+\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

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

Source:

Olver (1997b, (1.08), pp. 392 and 413)

Referenced by:

§9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E9

Encodings:

pMML, png, TeX

See also:

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

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9.7.10 $${\mathrm{Ai}}^{\prime }\left(-z\right)\sim \frac{z^{1/4}}{\sqrt{\pi }}\left(\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k}}{{\zeta }^{2k}}-\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

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

Source:

Olver (1997b, (1.09), pp. 392 and 413)

Permalink:

http://dlmf.nist.gov/9.7.E10

Encodings:

pMML, png, TeX

See also:

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

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9.7.11 $$\mathrm{Bi}\left(-z\right)\sim \frac{1}{\sqrt{\pi }{z}^{1/4}}\left(-\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k}}{{\zeta }^{2k}}+\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

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

Source:

Olver (1997b, (1.17), pp. 393 and 414)

Permalink:

http://dlmf.nist.gov/9.7.E11

Encodings:

pMML, png, TeX

See also:

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

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9.7.12 $${\mathrm{Bi}}^{\prime }\left(-z\right)\sim \frac{z^{1/4}}{\sqrt{\pi }}\left(\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k}}{{\zeta }^{2k}}+\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$.

ⓘ

Symbols:

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

Source:

Olver (1997b, (1.18), pp. 393 and 414)

Referenced by:

§9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E12

Encodings:

pMML, png, TeX

See also:

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

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9.7.14 $${\mathrm{Bi}}^{\prime }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/3}\right)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\mp \pi \mathrm{i}/6}{z}^{1/4}\left(-\sin \left(\zeta -\frac{1}{4}\pi \mp \frac{1}{2}\mathrm{i}\ln 2\right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k}}{{\zeta }^{2k}}+\cos \left(\zeta -\frac{1}{4}\pi \mp \frac{1}{2}\mathrm{i}\ln 2\right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$.

ⓘ

Symbols:

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

Proof sketch:

See Olver (1997b, pp. 392–393 and 413–414).

Referenced by:

§9.7(iv), §9.7(v)

Permalink:

http://dlmf.nist.gov/9.7.E14

Encodings:

pMML, png, TeX

See also:

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

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32: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.2 $${𝐉}_{\nu }\left(z\right)\sim J_{\nu }\left(z\right)+\frac{\sin \left(\pi \nu \right)}{\pi z}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu }{z}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}}\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions

Proof sketch:

Derivable from (11.10.15).

Referenced by:

§11.11(i)

Permalink:

http://dlmf.nist.gov/11.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(i), §11.11 and Ch.11

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11.11.3 $${𝐄}_{\nu }\left(z\right)\sim -Y_{\nu }\left(z\right)-\frac{1+\cos \left(\pi \nu \right)}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu (1-\cos \left(\pi \nu \right))}{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}},$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions

Proof sketch:

Derivable from (11.10.16).

Permalink:

http://dlmf.nist.gov/11.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(i), §11.11 and Ch.11

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33: 10.17 Asymptotic Expansions for Large Argument

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10.17.3 $$J_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}-\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.5

Referenced by:

§10.17(i), §10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.49(i), §10.61(v), §10.7(ii)

Permalink:

http://dlmf.nist.gov/10.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

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10.17.4 $$Y_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}+\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.6

Referenced by:

§10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.7(ii), §11.6(i)

Permalink:

http://dlmf.nist.gov/10.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

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10.17.9 $$J_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_{2k}(\nu )}{z^{2k}}+\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.2.11

Permalink:

http://dlmf.nist.gov/10.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(ii), §10.17 and Ch.10

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10.17.10 $$Y_{\nu }^{\prime }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_{2k}(\nu )}{z^{2k}}-\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.2.12

Permalink:

http://dlmf.nist.gov/10.17.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(ii), §10.17 and Ch.10

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34: Bibliography F

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Error bounds for a uniform asymptotic expansion of the Legendre function $Q_n^{-m}(\cosh z)$ . SIAM J. Math. Anal. 21 (2), pp. 523–535.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26.

Encodings:

BibTeX

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35: 10.12 Generating Function and Associated Series

… ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, ►

$\cos \left(z\sin \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

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$\sin \left(z\sin \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{J}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right),$

►

$\cos \left(z\cos \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

►

$\sin \left(z\cos \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k+1}\left(z\right)\cos \left((2k+1)\theta \right).$

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36: 6.10 Other Series Expansions

§6.10 Other Series Expansions

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§6.10(i) Inverse Factorial Series

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§6.10(ii) Expansions in Series of Spherical Bessel Functions

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6.10.4 $$\mathrm{Si}\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{\left({𝗃}_n\left(\frac{1}{2}z\right)\right)}^2,$$

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Symbols:

$\mathrm{Si}\left(z\right)$: sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

… ►An expansion for $E_1\left(z\right)$ can be obtained by combining (6.2.4) and (6.10.8).

37: 24.19 Methods of Computation

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Buhler et al. (1992) uses the expansion

24.19.3 $$\frac{t^2}{\cosh t-1}=-2\sum _{n=0}^{\mathrm{\infty }}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},$$

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Symbols:

$B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $n$: integer and $t$: real or complex

Referenced by:

§24.19(ii)

Permalink:

http://dlmf.nist.gov/24.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

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38: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ► … ►Series expansions of $F(\varphi ,k)$ and $E(\varphi ,k)$ are surveyed and improved in Van de Vel (1969), and the case of $F(\varphi ,k)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when see Erdélyi et al. (1953b, §13.6(9)). …

39: 24.8 Series Expansions

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24.8.9 $$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $k$: integer and $n$: integer

Keywords:

Euler polynomials, infinite series expansions, other

Permalink:

http://dlmf.nist.gov/24.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

40: 8.12 Uniform Asymptotic Expansions for Large Parameter

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8.12.16 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin \left(\pi a\right)}Q(-a,a{\mathrm{e}}^{\pm \pi \mathrm{i}})\sim \pm \frac{1}{2}-\frac{\mathrm{i}}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(0){(-a)}^{-k},$$ $|\mathrm{ph}a|\le \pi -\delta$,

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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, $\mathrm{i}$: imaginary unit, $Q(a,z)$: normalized incomplete gamma function, $\mathrm{ph}$: phase, $\sin z$: sine function, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant and $c_k(\eta )$: coefficients

Referenced by:

(8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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