orders ±12

SearchAdvancedHelp

(0.001 seconds)

21—30 of 67 matching pages

21: Bibliography C

… ►

A recurrence formula for $\zeta (2n)$ . Proc. Amer. Math. Soc. 12 (6), pp. 991–992.

ⓘ

External Links:

FullText, MathReview (R. D. James)

Referenced by:

§24.14(i).

Encodings:

BibTeX

… ►

Hadronic matter at high density. Phys. Rev. D 5 (12), pp. 3231–3242.

ⓘ

External Links:

Document

Referenced by:

§5.20.

Encodings:

BibTeX

… ►

Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

ⓘ

Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

Encodings:

BibTeX

… ►

Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

ⓘ

External Links:

ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

… ►

Algorithm 352: Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM 12 (7), pp. 399–407.

ⓘ

Notes:

Includes double-precision Fortran program, maximum accuracy 13D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§28.36(ii), §28.36(iii).

Encodings:

BibTeX

…

22: 2.8 Differential Equations with a Parameter

… ►For example, $u$ can be the order of a Bessel function or degree of an orthogonal polynomial. … ►

§2.8(iv) Case III: Simple Pole

… ►For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). … ►For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. … ►Lastly, for an example of a fourth-order differential equation, see Wong and Zhang (2007). …

23: Bibliography K

… ►

The Rayleigh function: Theory and computational methods. Zh. Vychisl. Mat. Mat. Fiz. 39 (12), pp. 1962–2006.

ⓘ

Notes:

Translation in Comput. Math. Math. Phys. 39 (1999), No. 12, pp. 1883–1925.

External Links:

ISSN 0044-4669, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

… ►

Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2153), pp. 20130008, 12.

ⓘ

External Links:

ISSN 1364-5021, Document, FullText, MathReview (Ulrich D. Jentschura), ZentralBlatt

Referenced by:

§2.4(vi), §2.4(vi).

Encodings:

BibTeX

… ►

Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.

ⓘ

External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Electromagnetic eigenfrequencies in a spheroidal cavity (calculation by spheroidal eigenvectors). J. Electromagn. Waves Appl. 12 (12), pp. 1601–1624.

ⓘ

External Links:

ISSN 0920-5071, Document, MathReview, ZentralBlatt

Referenced by:

§30.14(v).

Encodings:

BibTeX

… ►

Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1), pp. 177–232 (French).

ⓘ

External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§22.19(iv).

Encodings:

BibTeX

…

24: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
10.77(iii) Bessel Functions–Real Order and Argument	✓	✓	✓	✓	✓	✓		✓	✓		✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	NMS
10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions	✓	✓	✓									a	✓	✓			✓	✓	✓	✓	✓	✓	✓
…
10.77(vi) Bessel Functions–Imaginary Order and Real Argument	✓	✓										✓	✓								✓	✓	✓
10.77(viii) Bessel Functions–Complex Order and Argument	✓	✓										✓	✓								✓	✓	✓
…
12 Parabolic Cylinder Functions
…

…

25: 3.6 Linear Difference Equations

… ►Many special functions satisfy second-order recurrence relations, or difference equations, of the form … ►

§3.6(vii) Linear Difference Equations of Other Orders

►Similar considerations apply to the first-order equation … ►For a difference equation of order $k$ ($\ge 3$), …or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …

26: Bibliography T

… ►

Error bounds for the Liouville-Green approximation to initial-value problems. Z. Angew. Math. Mech. 58 (12), pp. 529–537.

ⓘ

External Links:

ISSN 0044-2267, Document, MathReview, ZentralBlatt

Referenced by:

§2.7(iii).

Encodings:

BibTeX

… ►

Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.

ⓘ

External Links:

ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

BibTeX

… ►

Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

MathReview (H. Pollard)

Referenced by:

§1.18(x).

Encodings:

BibTeX

… ►

Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.

ⓘ

Notes:

Papers from the conference held in honor of Ray A. Kunze at the AMS Special Session, Cincinnati, Ohio, January 12–14, 1994

External Links:

ISBN 0-8218-0310-7, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Donald St. P. Richards.

Encodings:

BibTeX

… ►

Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.

ⓘ

External Links:

ISSN 0002-9947, Document, FullText, MathReview (John M. Voight), ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

…

27: 27.13 Functions

… ►

27.13.3 $$n=x_1^2+x_2^2+\mathrm{\cdots }+x_k^2,$$

ⓘ

Symbols:

$k$: positive integer, $n$: positive integer and $x$: real number

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

►where the $x_j$ are integers, positive, negative, or zero, and the order of the summands is taken into account. … ►In fact, there are four representations, given by $5=2^2+1^2=2^2+{(-1)}^2={(-2)}^2+1^2={(-2)}^2+{(-1)}^2$, and four more with the order of summands reversed. … ►Explicit formulas for $r_k\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. …

28: Bibliography I

… ►

The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

ⓘ

External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

►

Computing zeros and orders of Bessel functions. J. Comput. Appl. Math. 38 (1-3), pp. 169–184.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vi), §3.8(iii).

Encodings:

BibTeX

… ►

Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

ⓘ

External Links:

ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

…

29: 3.5 Quadrature

… ►

3.5.1 $${\int }_a^{b}f(x)dx=\frac{1}{2}h(f(a)+f(b))-\frac{1}{12}{h}^3{f}^{\prime \prime }(\xi ),$$

ⓘ

Symbols:

$dx$: differential of $x$ and $\int$: integral

A&S Ref:

25.4.1 (second relation only)

Permalink:

http://dlmf.nist.gov/3.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(i), §3.5 and Ch.3

… ►

3.5.3 $$E_n(f)=-\frac{b-a}{12}{h}^2{f}^{\prime \prime }(\xi ),$$ .

ⓘ

Defines:

$E_n(f)$: error term (locally)

A&S Ref:

25.4.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/3.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(i), §3.5 and Ch.3

… ►If $k$ in (3.5.4) is not arbitrarily large, and if odd-order derivatives of $f$ are known at the end points $a$ and $b$, then the composite trapezoidal rule can be improved by means of the Euler–Maclaurin formula (§2.10(i)). … ►The Gauss nodes $x_k$ (the zeros of $p_n$) are the eigenvalues of the (symmetric tridiagonal) Jacobi matrix of order $n\times n$: … ►With $N$ function values, the Monte Carlo method aims at an error of order $1/\sqrt{N}$, independently of the dimension of the domain of integration. …

30: Bibliography R

… ►

A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

►

Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.

ⓘ

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii), 7th item, §10.77(ix).

Encodings:

BibTeX

… ►

On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

… ►

Mathieu functions of integral orders and real arguments. IEEE Trans. Microwave Theory Tech. 28 (3), pp. 276–277.

ⓘ

Notes:

Includes Fortran program

External Links:

FullText

Referenced by:

§28.36(ii), §28.36(iii).

Encodings:

BibTeX

… ►

Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

Referenced by:

§34.6.

Encodings:

BibTeX

…