relation to modified Bessel functions

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21: 28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►

Barrett’s Expansions

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions

\operatorname{me}_{\nu}\left(z,q\right)

(§28.12(ii)) and modified Mathieu functions

{\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)

(§28.20(iii)). ►For related results see Langer (1934) and Sharples (1967, 1971). …

22: 28.20 Definitions and Basic Properties

… ►

28.20.1

w^{\prime\prime}-\left(a-2q\cosh\left(2z\right)\right)w=0,

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Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function, $q=h^{2}$ : parameter, $z$ : complex variable, $a$ : parameter and $w(z)$ : Mathieu’s equation solution
A&S Ref:: 20.1.2 (in slightly different form) 20.8.6
Referenced by:: §28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)
Permalink:: http://dlmf.nist.gov/28.20.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.20(i), §28.20 and Ch.28

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§28.20(ii) Solutions $\operatorname{Ce}_{\nu}$ , $\operatorname{Se}_{\nu}$ , $\operatorname{Me}_{\nu}$ , $\operatorname{Fe}_{n}$ , $\operatorname{Ge}_{n}$

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§28.20(iv) Radial Mathieu Functions ${\operatorname{Mc}^{(j)}_{n}}$ , ${\operatorname{Ms}^{(j)}_{n}}$

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§28.20(vi) Wronskians

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§28.20(vii) Shift of Variable

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23: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality … ► … ► … ► … ►SUSY leads to algebraic simplifications in generating excited states, and partner potentials with closely related energy spectra, from knowledge of a single ground state wave function. …

24: Errata

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Equation (18.34.2)

18.34.2

y_{n}(x)=y_{n}\left(x;2\right)=2{\pi}^{-1}x^{-1}{\mathrm{e}}^{1/x}\mathsf{k}_{% n}\left(x^{-1}\right),

\theta_{n}(x)=x^{n}y_{n}(x^{-1})=2{\pi}^{-1}x^{n+1}{\mathrm{e}}^{x}\mathsf{k}_% {n}\left(x\right)

This equation was updated to include definitions in terms of the modified spherical Bessel function of the second kind.

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Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, $\mathsf{P}_{n},\mathsf{Q}_{n},P_{n},Q_{n},\boldsymbol{Q}_{n}$ and the Laguerre polynomial, $L_{n}$ , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

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Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function

The Gegenbauer function $C^{(\lambda)}_{\alpha}\left(z\right)$ , was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C^{(\lambda)}_{n}\left(z\right)$ . In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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Chapter 25 Zeta and Related Functions

A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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25: 10.6 Recurrence Relations and Derivatives

§10.6 Recurrence Relations and Derivatives

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§10.6(i) Recurrence Relations

… ►For results on modified quotients of the form

\ifrac{z\mathscr{C}_{\nu\pm 1}\left(z\right)}{\mathscr{C}_{\nu}\left(z\right)}

see Onoe (1955) and Onoe (1956). ►

§10.6(ii) Derivatives

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§10.6(iii) Cross-Products

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26: Bibliography Z

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F. A. Zafiropoulos, T. N. Grapsa, O. Ragos, and M. N. Vrahatis (1996) On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §10.74(vi).
Encodings:: BibTeX

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D. Zagier (1989) The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:: MathReview (J. Browkin), ZentralBlatt
Referenced by:: §25.12(i).
Encodings:: BibTeX

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J. Zhang (1996) A note on the

\tau

-method approximations for the Bessel functions

Y_{0}(z)

and

Y_{1}(z)

. Comput. Math. Appl. 31 (9), pp. 63–70.

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External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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J. Zhang and J. A. Belward (1997) Chebyshev series approximations for the Bessel function

Y_{n}(z)

of complex argument. Appl. Math. Comput. 88 (2-3), pp. 275–286.

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External Links:: ISSN 0096-3003, Document, MathReview (Sven Ehrich), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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M. I. Žurina and L. N. Karmazina (1967) Tablitsy modifitsirovannykh funktsii Besselya s mnimym indeksom

K_{i\tau}(x)

. Vyčisl. Centr Akad. Nauk SSSR, Moscow.

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Notes:: Translated title: Tables of the Modified Bessel Functions with Imaginary Index $K_{i\tau}(x)$
External Links:: MathReview (John Todd), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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27: Bibliography R

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Ju. M. Rappoport (1979) Tablitsy modifitsirovannykh funktsii Besselya

K_{\frac{1}{2}+i\beta}(x)

. “Nauka”, Moscow (Russian).

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Notes:: In Russian. Translated title: Tables of Modified Bessel Functions $K_{\frac{1}{2}+i\beta}(x)$
External Links:: MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:: Double-precision Fortran, maximum accuracy 18D.
External Links:: ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt
Referenced by:: 6th item.
Encodings:: BibTeX

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F. E. Relton (1965) Applied Bessel Functions. Dover Publications Inc., New York.

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Notes:: Reprint of original Blackie & Son Limited, London, 1946.
External Links:: MathReview, ZentralBlatt
Referenced by:: §10.73(iii).
Encodings:: BibTeX

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M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:: ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt
Referenced by:: §10.23(ii).
Encodings:: BibTeX

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J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

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28: 28.34 Methods of Computation

… ►

(b)

Representations for $w_{\mbox{\tiny I}}(\pi;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$ ; see Schäfke (1961a).

… ►

(f)

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

… ►

§28.34(iv) Modified Mathieu Functions

►For the modified functions we have: ►

(a)

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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29: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

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Stieltjes Fractions

… ►For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … ►This forward algorithm achieves efficiency and stability in the computation of the convergents

C_{n}=A_{n}/B_{n}

, and is related to the forward series recurrence algorithm. … ►Alternatives to Steed’s algorithm are the Lentz algorithm Lentz (1976) and the modified Lentz algorithm Thompson and Barnett (1986). …

30: Bibliography L

… ►

A. Laforgia (1991) Bounds for modified Bessel functions. J. Comput. Appl. Math. 34 (3), pp. 263–267.

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External Links:: ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

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K. V. Leung and S. S. Ghaderpanah (1979) An application of the finite element approximation method to find the complex zeros of the modified Bessel function

K_{n}(z)

. Math. Comp. 33 (148), pp. 1299–1306.

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External Links:: ISSN 0025-5718, FullText, MathReview (R. P. Boas, Jr.), ZentralBlatt
Referenced by:: §10.74(vi), 2nd item.
Encodings:: BibTeX

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S. Lewanowicz (1985) Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.

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External Links:: ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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S. Lewanowicz (1987) Corrigenda: “Recurrence relations for hypergeometric functions of unit argument” [Math. Comp. 45 (1985), no. 172, 521–535; MR 86m:33004]. Math. Comp. 48 (178), pp. 853.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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L. Lorch and M. E. Muldoon (2008) Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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External Links:: ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §10.21(iv), §10.21(iv).
Encodings:: BibTeX

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