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21: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

BibTeX

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Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

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Integral transforms with generalized Legendre functions as kernels. Compositio Math. 18, pp. 235–287.

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External Links:

MathReview (R. K. Saxena), ZentralBlatt

Referenced by:

§14.20(vi), §14.29, §14.31(ii).

Encodings:

BibTeX

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22: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:

Reprinted by Dover Publications, Inc., New York, 1979.

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.

Encodings:

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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Singular integrals whose kernels involve certain Sturm-Liouville functions. I. J. Math. Mech. 19 (10), pp. 855–873.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.10(ii).

Encodings:

BibTeX

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23: 1.15 Summability Methods

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Poisson Kernel

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1.15.14 $$P(r,\theta )\to 0,$$

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Symbols:

$P(r,\theta )$: Poisson kernel

Permalink:

http://dlmf.nist.gov/1.15.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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Fejér Kernel

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Poisson Kernel

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Fejér Kernel

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24: 10.68 Modulus and Phase Functions

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${\ker }_{\nu }x=N_{\nu }\left(x\right)\cos {\varphi }_{\nu }\left(x\right),$

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$N_{\nu }\left(x\right)={({\ker }_{\nu }^{2}x+{\mathrm{kei}}_{\nu }^{2}x)}^{1/2},$

… ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. … ►20) and (Eqs. …However, numerical tabulations show that if the second of these equations applies and ${\varphi }_1\left(x\right)$ is continuous, then ${\varphi }_1\left(0\right)=-\frac{3}{4}\pi$; compare Abramowitz and Stegun (1964, p. 433).

25: Errata

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18.2.12 $$K_n(x,y)\equiv \sum _{\mathrm{\ell }=0}^n\frac{p_{\mathrm{\ell }}(x)p_{\mathrm{\ell }}(y)}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\frac{p_{n+1}(x)p_n(y)-p_n(x)p_{n+1}(y)}{x-y},$$ $x\ne y$

18.2.13 $$K_n(x,x)=\sum _{\mathrm{\ell }=0}^n\frac{{(p_{\mathrm{\ell }}(x))}^2}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\left(p_{n+1}^{\prime }(x)p_n(x)-p_n^{\prime }(x)p_{n+1}(x)\right)$$

The left-hand sides were updated to include the definition of the Christoffel–Darboux kernel $K_n(x,y)$.

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Equation (16.16.5_5).

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Four of the terms were rewritten for improved clarity.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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26: 10.71 Integrals

… ►In the following equations $f_{\nu },g_{\nu }$ is any one of the four ordered pairs given in (10.63.1), and ${\widehat{f}}_{\nu },{\widehat{g}}_{\nu }$ is either the same ordered pair or any other ordered pair in (10.63.1). … ►

${\displaystyle \int x{N}_{\nu }^2\left(x\right)dx}=x({\ker }_{\nu }x{\mathrm{kei}}_{\nu }^{\prime }x-{\ker }_{\nu }^{\prime }x{\mathrm{kei}}_{\nu }x),$

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27: 18.2 General Orthogonal Polynomials

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Kernel property

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Kernel Polynomials

… ►Then the kernel polynomials … ►

Poisson kernel

… ►Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12). …

28: 10.67 Asymptotic Expansions for Large Argument

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§10.67(i) ${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x,{\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x$, and Derivatives

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10.67.1 $${\ker }_{\nu }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right),$$

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Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin 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, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.3, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(ii), §10.68(iii)

Permalink:

http://dlmf.nist.gov/10.67.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

… ►The contributions of the terms in ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$, ${\ker }_{\nu }^{\prime }x$, and ${\mathrm{kei}}_{\nu }^{\prime }x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). … ►

10.67.14 $$\ker x{\mathrm{kei}}^{\prime }x-{\ker }^{\prime }x\mathrm{kei}x\sim -\frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}-\frac{1}{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^2}-\frac{39}{512}\frac{1}{x^3}+\frac{75}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.32

Permalink:

http://dlmf.nist.gov/10.67.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

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10.67.15 $$\ker x{\ker }^{\prime }x+\mathrm{kei}x{\mathrm{kei}}^{\prime }x\sim -\frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}+\frac{3}{8}\frac{1}{x}-\frac{15}{64\sqrt{2}}\frac{1}{x^2}+\frac{45}{512}\frac{1}{x^3}+\frac{315}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.33

Permalink:

http://dlmf.nist.gov/10.67.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

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29: 28.32 Mathematical Applications

… ►The two-dimensional wave equation … ► … ►Kernels $K$ can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. … ►When the Helmholtz equation …

30: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …with $\kappa =-\mathrm{i}\pi /4$. … ►Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.