weight functions

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31—40 of 44 matching pages

31: 18.18 Sums

… ►In all three cases of Jacobi, Laguerre and Hermite, if

f(x)

L^{2}

on the corresponding interval with respect to the corresponding weight function and if

a_{n},b_{n},d_{n}

are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in

L^{2}

sense. …

32: 18.27 $q$ -Hahn Class

… ►Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. …

33: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

\,\mathrm{d}\alpha(x)=w(x)\,\mathrm{d}x

, see §1.4(v), where the nonnegative weight function

w(x)

is Lebesgue measurable on

X

. …

34: Errata

… ►

Chapter 18 Additions

The following additions were made in Chapter 18:

Section 18.2

In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)–18.2(xii), with Equations (18.2.21)–(18.2.46),
Section 18.3

A new introduction, minor changes in the presentation, and three new paragraphs.
Section 18.5

Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).
Section 18.8

Line numbers and two extra rows were added to Table 18.8.1.
Section 18.9

Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.
Section 18.12

Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).
Section 18.14

Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).
Section 18.15

Equation (18.15.4_5).
Section 18.16

The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).
Section 18.17

Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).
Section 18.18

Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.
Section 18.19

A new introduction.
Section 18.21

Equation (18.21.13).
Section 18.25

Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.
Section 18.26

In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).
Section 18.27

Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).
Section 18.28

A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)–18.28(xi), with Equations (18.28.23)–(18.28.34).
Section 18.30

Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)–18.30(viii), with Equations (18.30.8)–(18.30.31).
Section 18.32

This short section has been expanded, with Equation (18.32.2).
Section 18.33

Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).
Section 18.34

This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).
Section 18.35

This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).
Section 18.36

This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.
Section 18.38

The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).
Section 18.39

This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

The original text of 18.39(i) Quantum Mechanics was:

“Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$ :

errata.1 $\left(\frac{-\hbar^{2}}{2m}\frac{{\partial}^{2}}{{\partial x}^{2}}+V(x)\right)% \psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t),$

where $\hbar$ is the reduced Planck’s constant. On substituting $\psi(x,t)=\eta(x)\zeta(t)$ , we obtain two ordinary differential equations, each of which involve the same constant $E$ . The equation for $\eta(x)$ is

errata.2 $\frac{{\mathrm{d}}^{2}\eta}{{\mathrm{d}x}^{2}}+\frac{2m}{\hbar^{2}}\left(E-V(x% )\right)\eta=0.$

For a harmonic oscillator, the potential energy is given by

errata.3 $V(x)=\tfrac{1}{2}m\omega^{2}x^{2},$

where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval $-\infty<x<\infty$ , the constant $E$ (the total energy of the particle) must satisfy

errata.4 $E=E_{n}=\left(n+\tfrac{1}{2}\right)\hbar\omega,$ $n=0,1,2,\dots$ .

The corresponding eigenfunctions are

errata.5 $\eta_{n}(x)=\pi^{-\frac{1}{4}}2^{-\frac{1}{2}n}(n!\,b)^{-\frac{1}{2}}H_{n}% \left(x/b\right){\mathrm{e}}^{-x^{2}/2b^{2}},$

where $b=(\hbar/m\omega)^{1/2}$ , and $H_{n}$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

A second example is provided by the three-dimensional time-independent Schrödinger equation

errata.6 $\nabla^{2}\psi+\frac{2m}{\hbar^{2}}\left(E-V(\mathbf{x})\right)\psi=0,$

when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”
Section 18.40

The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

…

35: Bibliography G

… ►

G. Gasper (1981) Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.

ⓘ

External Links:: ISSN 0008-414X, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

…

36: Bibliography Z

… ►

M. R. Zaghloul and A. N. Ali (2011) Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: 2nd item, 5th item.
Encodings:: BibTeX

… ►

M. R. Zaghloul (2017) Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.

ⓘ

External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 4th item, §7.25(iii).
Encodings:: BibTeX

… ►

R. Zanovello (1995) Numerical analysis of Struve functions with applications to other special functions. Ann. Numer. Math. 2 (1-4), pp. 199–208.

ⓘ

External Links:: ISSN 1021-2655, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §11.7(ii).
Encodings:: BibTeX

… ►

J. Zeng (1992) Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:: ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

… ►

I. J. Zucker (1979) The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §4.41.
Encodings:: BibTeX

…

37: 3.3 Interpolation

… ►and the weights

\widetilde{\omega}_{k}

are … ►

Example

… ►For interpolation of a bounded function

f

\mathbb{R}

the cardinal function of

f

is defined by …where …is called the Sinc function. …

38: Bibliography L

… ►

E. Levin and D. S. Lubinsky (2001) Orthogonal Polynomials for Exponential Weights. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 4, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-98941-2, MathReview (Walter Van Assche), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

►

E. Levin and D. Lubinsky (2005) Orthogonal polynomials for exponential weights

x^{2\rho}e^{-2Q(x)}

[0,d)

. J. Approx. Theory 134 (2), pp. 199–256.

ⓘ

External Links:: ISSN 0021-9045, Document, Link, MathReview (Leonid B. Golinskiĭ)
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

L. Lorch and P. Szegő (1964) Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

ⓘ

External Links:: ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(ii).
Encodings:: BibTeX

… ►

T. A. Lowdon (1970) Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

ⓘ

External Links:: Document, MathReview (T. Erber), ZentralBlatt
Referenced by:: §12.12, §12.16.
Encodings:: BibTeX

… ►

Y. L. Luke (1959) Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

ⓘ

External Links:: FullText, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §13.24(ii).
Encodings:: BibTeX

…

39: Bibliography

… ►

J. Abad and J. Sesma (1995) Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.

ⓘ

External Links:: ISSN 1047-5974, FullText
Referenced by:: §13.32(iii).
Encodings:: BibTeX

… ►

M. Abramowitz (1954) Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

ⓘ

External Links:: MathReview (E. Weber), ZentralBlatt
Referenced by:: §33.9(ii).
Encodings:: BibTeX

… ►

Z. Altaç (1996) Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

ⓘ

External Links:: Document
Referenced by:: §10.73(iv), §8.24(iii).
Encodings:: BibTeX

… ►

Y. Ameur and J. Cronvall (2023) Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.

ⓘ

External Links:: ISSN 0010-3616, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.11(iii), §8.11(iii), Erratum (V1.1.10) for Section 8.11(iii).
Encodings:: BibTeX

… ►

G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992a) Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23 (2), pp. 512–524.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

…

40: Bibliography F

… ►

M. Faierman (1992) Generalized parabolic cylinder functions. Asymptotic Anal. 5 (6), pp. 517–531.

ⓘ

External Links:: ISSN 0921-7134, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: §12.15.
Encodings:: BibTeX

… ►

H. E. Fettis, J. C. Caslin, and K. R. Cramer (1973) Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §7.13(i), §7.13(ii), 1st item.
Encodings:: BibTeX

… ►

J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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External Links:: FullText, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.10.
Encodings:: BibTeX

… ►

C. Flammer (1957) Spheroidal Wave Functions. Stanford University Press, Stanford, CA.

ⓘ

External Links:: MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: §30.1, §30.10, 2nd item, Ch.30.
Encodings:: BibTeX

… ►

G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

…