Bessel-function expansion

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31: 18.24 Hahn Class: Asymptotic Approximations

… ►Dunster (2001b) provides various asymptotic expansions for $C_n(x;a)$ as $n\to \mathrm{\infty }$, in terms of elementary functions or in terms of Bessel functions. …

32: 10.24 Functions of Imaginary Order

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33: 11.10 Anger–Weber Functions

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§11.10(viii) Expansions in Series of Products of Bessel Functions

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34: 10.21 Zeros

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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

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10.21.34 $$J_{\nu }^{\prime }\left(j_{\nu ,m}\right)\sim {(-1)}^m\frac{{(2/\nu )}^{\frac{2}{3}}}{\pi M\left(a_m\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k}{{\nu }^{2k/3}},$$

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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, $M\left(z\right)$: Airy modulus function, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and $\beta$: coefficients

Permalink:

http://dlmf.nist.gov/10.21.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

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§10.21(viii) Uniform Asymptotic Approximations for Large Order

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§10.21(x) Cross-Products

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35: 10.45 Functions of Imaginary Order

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36: 10.18 Modulus and Phase Functions

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§10.18(iii) Asymptotic Expansions for Large Argument

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10.18.18 $${\theta }_{\nu }\left(x\right)\sim x-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi +\frac{\mu -1}{2(4x)}+\frac{(\mu -1)(\mu -25)}{6{(4x)}^3}+\frac{(\mu -1)({\mu }^2-114\mu +1073)}{5{(4x)}^5}+\frac{(\mu -1)(5{\mu }^3-1535{\mu }^2+54703\mu -\mathrm{3\hspace{0.33em}75733})}{14{(4x)}^7}+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

Proof sketch:

See Nemes (2021, Theorem 2.1)

A&S Ref:

9.2.29

Referenced by:

§10.18(iii), §10.18(iii), Erratum (V1.1.8) for Section 10.18(iii)

Permalink:

http://dlmf.nist.gov/10.18.E18

Encodings:

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See also:

Annotations for §10.18(iii), §10.18 and Ch.10

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10.18.19 $$N_{\nu }^2\left(x\right)\sim \frac{2}{\pi x}\left(1-\frac{1}{2}\frac{\mu -3}{{(2x)}^2}-\frac{1}{2\cdot 4}\frac{(\mu -1)(\mu -45)}{{(2x)}^4}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.30 (corrected)

Referenced by:

§10.18(iii), §10.40(i), §10.40(i)

Permalink:

http://dlmf.nist.gov/10.18.E19

Encodings:

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See also:

Annotations for §10.18(iii), §10.18 and Ch.10

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10.18.21 $${\varphi }_{\nu }\left(x\right)\sim x-\left(\frac{1}{2}\nu -\frac{1}{4}\right)\pi +\frac{\mu +3}{2(4x)}+\frac{{\mu }^2+46\mu -63}{6{(4x)}^3}+\frac{{\mu }^3+185{\mu }^2-2053\mu +1899}{5{(4x)}^5}+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.31

Referenced by:

§10.18(iii), Erratum (V1.1.8) for Section 10.18(iii)

Permalink:

http://dlmf.nist.gov/10.18.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).

37: 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:

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See also:

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

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

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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:

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See also:

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

… ►Lastly, corresponding asymptotic approximations and expansions for ${𝐉}_{\nu }\left(\lambda \nu \right)$ and ${𝐄}_{\nu }\left(\lambda \nu \right)$, with or $\lambda >1$, follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$; see §10.19(ii). …

38: Bibliography W

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The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

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39: Bibliography N

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Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

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External Links:

ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt

Referenced by:

(9.7.16), (9.7.17), §9.7(iii), §9.7(iv).

Encodings:

BibTeX

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40: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►In regions in which the function $f(z)$ has a simple pole at $z=z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z=z_0$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho }$, where $\rho$ is the limiting value of ${(z-z_0)}^{2}g(z)$ as $z\to z_0$. …