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11: 29.18 Mathematical Applications

§29.18 Mathematical Applications

… ►when transformed to sphero-conal coordinates

r,\beta,\gamma

: … ►

29.18.4

u(r,\beta,\gamma)=u_{1}(r)u_{2}(\beta)u_{3}(\gamma),

ⓘ

Symbols:: $u$ : solution, $r$ : sphero-conal coordinate, $\beta$ : sphero-conal coordinate, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Permalink:: http://dlmf.nist.gov/29.18.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

… ►with separation constants

h

and

\nu

. … ►The wave equation (29.18.1), when transformed to ellipsoidal coordinates

\alpha,\beta,\gamma

: …

12: Bibliography C

… ►

F. Cajori (1929) A History of Mathematical Notations, Volume II. Open Court Publishing Company, Chicago.

ⓘ

Notes:: Reprinted by Cosimo in 2007.
External Links:: ZentralBlatt
Referenced by:: §24.1.
Encodings:: BibTeX

… ►

S. Chandrasekhar (1984) The Mathematical Theory of Black Holes. In General Relativity and Gravitation (Padova, 1983), pp. 5–26.

ⓘ

External Links:: MathReview Entry
Referenced by:: §31.17(ii).
Encodings:: BibTeX

… ►

A. P. Clarke and W. Marwood (1984) A compact mathematical function package. Australian Computer Journal 16 (3), pp. 107–114.

ⓘ

Notes:: Includes Pascal function evaluation package
External Links:: ISSN 0004-8917, ZentralBlatt
Referenced by:: §16.27(ii).
Encodings:: BibTeX

… ►

E. T. Copson (1965) Asymptotic Expansions. Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, New York.

ⓘ

External Links:: MathReview (A. E. Danese), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

… ►

R. Courant and D. Hilbert (1953) Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y..

ⓘ

External Links:: MathReview (J. B. Diaz), ZentralBlatt
Referenced by:: (14.30.8_5).
Encodings:: BibTeX

…

13: 18.39 Applications in the Physical Sciences

… ►All are written in the same form as the product of three factors: the square root of a weight function

w(x)

, the corresponding OP or EOP, and constant factors ensuring unit normalization. … ►and

\hbar=k=m=1

, has eigenfunctions …There is no need for a normalization constant here, as appropriate constants already appear in §18.36(vi). … ►In what follows the radial and spherical radial eigenfunctions corresponding to (18.39.27) are found in four different notations, with identical eigenvalues, all of which appear in the current and past mathematical and theoretical physics and chemistry literatures, regarding this central problem. … ►Mathematical underpinnings appear in Ismail (2009, §5.8), and Ismail and Koelink (2011). …

14: 28.33 Physical Applications

… ►

28.33.2

V_{n}(\xi,\eta)=\left(c_{n}{\operatorname{M}^{(1)}_{n}}\left(\xi,\sqrt{q}% \right)+d_{n}{\operatorname{M}^{(2)}_{n}}\left(\xi,\sqrt{q}\right)\right)% \operatorname{me}_{n}\left(\eta,q\right),

ⓘ

Symbols:: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $q=h^{2}$ : parameter, $n$ : integer, $V(\xi,\eta)$ : solution, $d_{n}$ : constants, $\eta$ : variable, $\xi$ : variable and $c_{n}$ : constants
Permalink:: http://dlmf.nist.gov/28.33.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.33(ii), §28.33 and Ch.28

►where

q=\tfrac{1}{4}c^{2}k^{2}

and

a_{n}(q)

b_{n}(q)

is the separation constant; compare (28.12.11), (28.20.11), and (28.20.12). Here

c_{n}

and

d_{n}

are constants. … ►If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as

\cos\left(2\omega t\right)

. …with

b

f

, and

\omega

positive constants. …

15: 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.

…

16: Bibliography H

… ►

G. H. Hardy (1952) A Course of Pure Mathematics. 10th edition, Cambridge University Press.

ⓘ

Notes:: Numerous reprintings exist, including the Centenary Edition with Foreword by T. W. Körner (Cambridge Univerity Press, 2008).
External Links:: MathReview, ZentralBlatt
Referenced by:: §1.4(ii), §1.4(iii), §1.4(iv), §1.4(v), §1.4(vi), §1.4(vii), §4.4(iii), §4.6(i), §4.6(ii).
Encodings:: BibTeX

… ►

P. W. Hemker, T. H. Koornwinder, and N. M. Temme (1993) Wavelets: mathematical preliminaries. In Wavelets: an elementary treatment of theory and applications, Ser. Approx. Decompos., Vol. 1, pp. 13–26.

ⓘ

External Links:: Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Tom H. Koornwinder.
Encodings:: BibTeX

… ►

H. Hochstadt (1971) The Functions of Mathematical Physics. Wiley-Interscience [John Wiley & Sons, Inc.], New York-London-Sydney.

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Notes:: Pure and Applied Mathematics, Vol. XXIII. Reprinted with corrections by Dover Publications Inc., New York, 1986.
External Links:: MathReview, ZentralBlatt
Referenced by:: §13.28(i), Ch.13, §15.17(ii), Ch.15.
Encodings:: BibTeX

… ►

E. Hopf (1934) Mathematical Problems of Radiative Equilibrium. Cambridge Tracts in Mathematics and Mathematical Physics No. 31, Cambridge University Press, Cambridge.

ⓘ

Notes:: Reprinted by Stechert-Hafner, Inc., New York, 1964
External Links:: MathReview, ZentralBlatt
Referenced by:: §8.19(ix), §8.24(iii).
Encodings:: BibTeX

… ►

L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

ⓘ

Notes:: Reprinted in 1979.
External Links:: ISBN 0-8218-1556-3, MathReview, ZentralBlatt
Referenced by:: §35.4(i).
Encodings:: BibTeX

…

17: Bibliography S

… ►

I. J. Schoenberg (1973) Cardinal Spline Interpolation. Society for Industrial and Applied Mathematics, Philadelphia, PA.

ⓘ

Notes:: Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12
External Links:: MathReview (H. L. Loeb), ZentralBlatt
Referenced by:: §24.17(ii).
Encodings:: BibTeX

… ►

J. A. Shohat and J. D. Tamarkin (1970) The Problem of Moments. 4th edition, American Mathematical Society Mathematical Surveys, Vol. 1, American Mathematical Society, Providence, RI.

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External Links:: MathReview (R. P. Boas, Jr.), ZentralBlatt
Referenced by:: §1.16(ix), §1.4(v), §18.2(viii), (18.40.7), §18.40(ii), §18.40(ii).
Encodings:: BibTeX

… ►

B. Simon (1982) Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.

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External Links:: Document
Referenced by:: §22.19(ii).
Encodings:: BibTeX

… ►

B. Simon (2005a) Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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External Links:: ISBN 0-8218-3446-0, MathReview (P. L. Duren), ZentralBlatt
Referenced by:: (18.33.17), (18.33.18), (18.33.21), (18.33.22), (18.33.23), (18.33.24), (18.33.25), (18.33.26), (18.33.27), (18.33.28), (18.33.30), (18.33.31), (18.33.32), §18.33(vi), §18.33(i), §18.33(vi).
Encodings:: BibTeX

… ►

G. Szegő (1967) Orthogonal Polynomials. 3rd edition, American Mathematical Society, New York.

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Notes:: American Mathematical Society Colloquium Publications, Vol. 23
External Links:: MathReview (J. Burlak)
Referenced by:: Ch.14, §9.1, Notations A.
Encodings:: BibTeX

…

18: 22.18 Mathematical Applications

§22.18 Mathematical Applications

… ►

22.18.7

ax^{2}y^{2}+b(x^{2}y+xy^{2})+c(x^{2}+y^{2})+2dxy+e(x+y)+f=0,

ⓘ

Symbols:: $x$ : real, $y$ : real, $c$ : real constant, $d$ : real constant, $e$ : real constant, $f$ : real constant, $a$ : real constant and $b$ : real constant
Permalink:: http://dlmf.nist.gov/22.18.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.18(iii), §22.18 and Ch.22

►in which

a, b, c, d, e, f

are real constants, can be achieved in terms of single-valued functions. … ►

19: 7.23 Tables

… ►Lebedev and Fedorova (1960) and Fletcher et al. (1962) give comprehensive indexes of mathematical tables. … ►

•

Abramowitz and Stegun (1964, Chapter 7) includes $\operatorname{erf}x$ , $(2/\sqrt{\pi})e^{-x^{2}}$ , $x\in[0,2]$ , 10D; $(2/\sqrt{\pi})e^{-x^{2}}$ , $x\in[2,10]$ , 8S; $xe^{x^{2}}\operatorname{erfc}x$ , $x^{-2}\in[0,0.25]$ , 7D; $2^{n}\Gamma\left(\frac{1}{2}n+1\right)\mathop{\mathrm{i}^{n}\mathrm{erfc}}% \left(x\right)$ , $n=1(1)6,10,11$ , $x\in[0,5]$ , 6S; $F\left(x\right)$ , $x\in[0,2]$ , 10D; $xF\left(x\right)$ , $x^{-2}\in[0,0.25]$ , 9D; $C\left(x\right)$ , $S\left(x\right)$ , $x\in[0,5]$ , 7D; $\mathrm{f}\left(x\right)$ , $\mathrm{g}\left(x\right)$ , $x\in[0,1]$ , $x^{-1}\in[0,1]$ , 15D.

…

20: 24.17 Mathematical Applications

§24.17 Mathematical Applications

… ►

24.17.7

M_{n}(x)=O\left(|x|^{\gamma}\right),

x\to\pm\infty

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $n$ : integer, $x$ : real or complex and $M_{n}(x)$ : function
Permalink:: http://dlmf.nist.gov/24.17.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

►for some positive constant

\gamma

. … ►