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11: 10.73 Physical Applications

… ► … ► ►The Helmholtz equation, $({\nabla }^2+k^2)\psi =0$, follows from the wave equation … … ►In the theory of plates and shells, the oscillations of a circular plate are determined by the differential equation …

12: 10.68 Modulus and Phase Functions

… ►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. … ►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).

13: Bonita V. Saunders

… ►in computational and applied mathematics from Old Dominion University, Norfolk, Virginia. Her research interests include numerical grid generation, numerical solution of partial differential equations, and visualization of special functions. …

14: Ronald F. Boisvert

… ►in computer science from Purdue University in 1979 and has been at NIST since then. His research interests include numerical solution of partial differential equations, mathematical software, and information services that support computational science. … ►Department of Commerce Gold Medal for Distinguished Achievement in the Federal Service in 2011, and an Outstanding Alumni Award from the Purdue University Department of Computer Science in 2012. …

15: 14.20 Conical (or Mehler) Functions

… ►

14.20.1 $$\left(1-x^2\right)\frac{d^{2}w}{{dx}^2}-2x\frac{dw}{dx}-\left({\tau }^2+\frac{1}{4}+\frac{{\mu }^2}{1-x^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i), §14.20(i), 2nd item

Permalink:

http://dlmf.nist.gov/14.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

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14.20.9 $${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)=\frac{2}{\pi }{\int }_0^{\theta }\frac{\cosh \left(\tau \varphi \right)}{\sqrt{2(\cos \varphi -\cos \theta )}}d\varphi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\tau$: real variable and : variable

A&S Ref:

8.12.3

Referenced by:

§14.20(iv), §14.20(v)

Permalink:

http://dlmf.nist.gov/14.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(iv), §14.20 and Ch.14

… ►From (14.20.9) or (14.20.10) it is evident that ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)$ is positive for real $\theta$. … ►

14.20.12 $$g(x)={\int }_0^{\mathrm{\infty }}{P}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)f(\tau )d\tau .$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $x$: real variable, $\tau$: real variable and $\mu$: general order

Permalink:

http://dlmf.nist.gov/14.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(vi), §14.20 and Ch.14

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14.20.13 $$P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)=\frac{\cosh \left(\tau \pi \right)}{\pi }{\int }_1^{\mathrm{\infty }}\frac{P_{-\frac{1}{2}+\mathrm{i}\tau }\left(t\right)}{x+t}dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable and $\tau$: real variable

Permalink:

http://dlmf.nist.gov/14.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(vi), §14.20 and Ch.14

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16: 11.2 Definitions

… ►(11.2.17) applies when $|\mathrm{ph}z|\le \frac{1}{2}\pi$ with $z$ bounded away from the origin.

17: T. Mark Dunster

… ►in applied mathematics in 1986 from Bristol University, U. …He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. …

18: Errata

… ►

The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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22.14.16 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{sn}(t,k)\right)dt=-\frac{\pi }{4}{K}^{\prime }\left(k\right)-\frac{1}{2}K\left(k\right)\ln k,$$

22.14.17 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{cn}(t,k)\right)dt=-\frac{\pi }{4}{K}^{\prime }\left(k\right)+\frac{1}{2}K\left(k\right)\ln \left(k^{\prime }/k\right)$$

Originally, a factor of $\pi$ was missing from the terms containing the $-\frac{1}{4}{K}^{\prime }\left(k\right)$.

Reported by Fred Hucht on 2020-08-06

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33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}$$

The definite integral, originally written as ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was clarified and rewritten as an orthogonality relation. This follows from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

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9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

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9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

…

19: 3.7 Ordinary Differential Equations

… ►The eigenvalues ${\lambda }_k$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …

20: Howard S. Cohl

… ► in astronomy and astrophysics from Indiana University, Bloomington, Indiana, a M. … in physics from Louisiana State University, Baton Rouge, Louisiana, and a Ph. …in mathematics from the University of Auckland in New Zealand. ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …