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1: 18.37 Classical OP’s in Two or More Variables

§18.37 Classical OP’s in Two or More Variables

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2: 1.5 Calculus of Two or More Variables

§1.5 Calculus of Two or More Variables

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§1.5(i) Partial Derivatives

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1.5.1 $$\underset{(x,y)\to (a,b)}{lim}f(x,y)=f(a,b),$$

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Symbols:

$(a,b)$: open interval

Referenced by:

§1.9(ii)

Permalink:

http://dlmf.nist.gov/1.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

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§1.5(iii) Taylor’s Theorem; Maxima and Minima

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3: 35.2 Laplace Transform

… ►where the integration variable $𝐗$ ranges over the space $𝛀$. …

4: 21.8 Abelian Functions

… ►In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …

5: 19.23 Integral Representations

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19.23.3 $$R_D(0,y,z)=3{\int }_0^{\pi /2}{(y{\cos }^2\theta +z{\sin }^2\theta )}^{-3/2}{\sin }^2\theta d\theta .$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $\sin z$: sine function

Referenced by:

§19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

… ►In (19.23.8)–(19.23.10) one or more of the variables may be 0 if the integral converges. …

6: Errata

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Two sentences at the end of §36.15(iii) have been modified to include new references and provide further clarification.

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The wording was changed to make the integration variable more apparent.

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A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).

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Originally used $f(x)$ to represent both $U(-c,x)$ and $\overline{U}(-c,x)$. This has been replaced by two equations giving explicit definitions for the two envelope functions. Some slight changes in wording were needed to make this clear to readers.

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This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

Reported 2015-11-12 by James W. Pitman.

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7: 37.2 General Orthogonal Polynomials of Two Variables

§37.2 General Orthogonal Polynomials of Two Variables

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§37.2(iii) Reproducing Kernels

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§37.2(iv) Zeros

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8: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). Such a solution is given in terms of a Riemann theta function with two phases. …The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).

9: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. … ►In these applications, the $Z$-scaled variables $r$ and $ϵ$ are more convenient. …

10: Sidebar 21.SB1: Periodic Surface Waves

… ►Two-dimensional periodic waves in a shallow water wave tank. Taken from Joe Hammack, Daryl McCallister, Norman Scheffner and Harvey Segur, “Two-dimensional periodic waves in shallow water. …The caption reads “Mosaic of two overhead photographs, showing surface patterns of waves in shallow water”. …