spherical coordinates
(0.002 seconds)
10 matching pages
1: 14.31 Other Applications
2: 1.5 Calculus of Two or More Variables
Spherical Coordinates
… βΊ3: 14.30 Spherical and Spheroidal Harmonics
4: 18.39 Applications in the Physical Sciences
5: 10.73 Physical Applications
6: 31.10 Integral Equations and Representations
7: Errata
The following additions were made in Chapter 18:

§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),

§18.3
A new introduction, minor changes in the presentation, and three new paragraphs.
 §18.5

§18.8
Line numbers and two extra rows were added to Table 18.8.1.
 §18.9
 §18.12
 §18.14

§18.15
Equation (18.15.4_5).

§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).

§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).

§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”.

§18.19
A new introduction.

§18.21
Equation (18.21.13).
 §18.25
 §18.26

§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).
 §18.28

§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).

§18.32
This short section has been expanded, with Equation (18.32.2).
 §18.33
 §18.34

§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).

§18.36
This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on nonclassical 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”.

§18.38
The paragraphs of Subsection 18.38(i) have been reordered 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).

§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 JMatrix 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 timedependent Schrödinger equation is solved by separation of variables. Consider, for example, the onedimensional form of this equation for a particle of mass $m$ with potential energy $V\beta \x81\u2018(x)$:
errata.1 $$\left(\frac{{\mathrm{\beta \x84\x8f}}^{2}}{2\beta \x81\u2019m}\beta \x81\u2019\frac{{\partial}^{2}}{{\partial x}^{2}}+V\beta \x81\u2018(x)\right)\beta \x81\u2019\mathrm{{\rm O}\x88}\beta \x81\u2018(x,t)=i\beta \x81\u2019\mathrm{\beta \x84\x8f}\beta \x81\u2019\frac{\partial}{\partial t}\beta \x81\u2018\mathrm{{\rm O}\x88}\beta \x81\u2018(x,t),$$where $\mathrm{\beta \x84\x8f}$ is the reduced Planck’s constant. On substituting $\mathrm{{\rm O}\x88}\beta \x81\u2018(x,t)=\mathrm{\Xi \xb7}\beta \x81\u2018(x)\beta \x81\u2019\mathrm{\Xi \u0386}\beta \x81\u2018(t)$, we obtain two ordinary differential equations, each of which involve the same constant $E$. The equation for $\mathrm{\Xi \xb7}\beta \x81\u2018(x)$ is
errata.2 $$\frac{{d}^{2}\mathrm{\Xi \xb7}}{{dx}^{2}}+\frac{2\beta \x81\u2019m}{{\mathrm{\beta \x84\x8f}}^{2}}\beta \x81\u2019\left(EV\beta \x81\u2018(x)\right)\beta \x81\u2019\mathrm{\Xi \xb7}=0.$$For a harmonic oscillator, the potential energy is given by
errata.3 $$V\beta \x81\u2018(x)=\frac{1}{2}\beta \x81\u2019m\beta \x81\u2019{\mathrm{{\rm O}\x89}}^{2}\beta \x81\u2019{x}^{2},$$where $\mathrm{{\rm O}\x89}$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval $$, the constant $E$ (the total energy of the particle) must satisfy
errata.4 $$E={E}_{n}=\left(n+\frac{1}{2}\right)\beta \x81\u2019\mathrm{\beta \x84\x8f}\beta \x81\u2019\mathrm{{\rm O}\x89},$$ $n=0,1,2,\mathrm{\dots}$.The corresponding eigenfunctions are
errata.5 $${\mathrm{\Xi \xb7}}_{n}\beta \x81\u2018(x)={\mathrm{{\rm O}\x80}}^{\frac{1}{4}}\beta \x81\u2019{2}^{\frac{1}{2}\beta \x81\u2019n}\beta \x81\u2019{(n!\beta \x81\u2019b)}^{\frac{1}{2}}\beta \x81\u2019{H}_{n}\beta \x81\u2018\left(x/b\right)\beta \x81\u2019{\mathrm{e}}^{{x}^{2}/2\beta \x81\u2019{b}^{2}},$$where $b={(\mathrm{\beta \x84\x8f}/m\beta \x81\u2019\mathrm{{\rm O}\x89})}^{1/2}$, and ${H}_{n}$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 7172).
A second example is provided by the threedimensional timeindependent Schrödinger equation
errata.6 $${\nabla}^{2}\mathrm{{\rm O}\x88}+\frac{2\beta \x81\u2019m}{{\mathrm{\beta \x84\x8f}}^{2}}\beta \x81\u2019\left(EV\beta \x81\u2018(\mathrm{\pi \x9d\x90\pm})\right)\beta \x81\u2019\mathrm{{\rm O}\x88}=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. 6975).
For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 8793) and Nikiforov and Uvarov (1988, pp. 7680 and 320323).”
 Section 18.40