expansions in modified Bessel functions

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11: 10.41 Asymptotic Expansions for Large Order

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12: 10.35 Generating Function and Associated Series

§10.35 Generating Function and Associated Series

►For $z\in ℂ$ and $t\in ℂ\setminus \{0\}$, … ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, … ►

10.35.4 $$1=I_0\left(z\right)-2I_2\left(z\right)+2I_4\left(z\right)-2I_6\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.6.36

Permalink:

http://dlmf.nist.gov/10.35.E4

Encodings:

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See also:

Annotations for §10.35 and Ch.10

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10.35.5 $${\mathrm{e}}^{\pm z}=I_0\left(z\right)\pm 2I_1\left(z\right)+2I_2\left(z\right)\pm 2I_3\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.6.37,9.6.38

Permalink:

http://dlmf.nist.gov/10.35.E5

Encodings:

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See also:

Annotations for §10.35 and Ch.10

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13: 10.74 Methods of Computation

… ►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. …

14: 28.34 Methods of Computation

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

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§28.34(iv) Modified Mathieu Functions

►For the modified functions we have: ►

Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

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Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

15: 11.4 Basic Properties

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§11.4(iv) Expansions in Series of Bessel Functions

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16: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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§10.40(i) Hankel’s Expansions

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Products

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§10.40(iv) Exponentially-Improved Expansions

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17: Bibliography T

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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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External Links:

ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§13.8(iii), §13.8(iii).

Encodings:

BibTeX

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18: 18.15 Asymptotic Approximations

… ►These expansions are in terms of Bessel functions and modified Bessel functions, respectively. …

19: 10.66 Expansions in Series of Bessel Functions

§10.66 Expansions in Series of Bessel Functions

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10.66.1 $${\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x=\sum _{k=0}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{(3\nu +k)\pi \mathrm{i}/4}{x}^k{J}_{\nu +k}\left(x\right)}{2^{k/2}k!}=\sum _{k=0}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{(3\nu +3k)\pi \mathrm{i}/4}{x}^k{I}_{\nu +k}\left(x\right)}{2^{k/2}k!}.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.32

Referenced by:

§10.66

Permalink:

http://dlmf.nist.gov/10.66.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.66 and Ch.10

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${\mathrm{ber}}_n\left(x\sqrt{2}\right)={\displaystyle \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}}{(-1)}^{n+k}{J}_{n+2k}\left(x\right)I_{2k}\left(x\right),$

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${\mathrm{bei}}_n\left(x\sqrt{2}\right)={\displaystyle \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}}{(-1)}^{n+k}{J}_{n+2k+1}\left(x\right)I_{2k+1}\left(x\right).$

20: 2.8 Differential Equations with a Parameter

… ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). …