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21: 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

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22: 18.1 Notation

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Classical OP’s in Two Variables

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23: 19.18 Derivatives and Differential Equations

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19.18.1 $$\frac{\partial R_F(x,y,z)}{\partial z}=-\frac{1}{6}{R}_D(x,y,z),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Referenced by:

§19.18(i), §19.28

Permalink:

http://dlmf.nist.gov/19.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(i), §19.18 and Ch.19

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24: Bibliography K

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Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.

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External Links:

MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.37(ii).

Encodings:

BibTeX

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25: 1.9 Calculus of a Complex Variable

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Continuity

… ►That is, given any positive number $ϵ$, however small, we can find a positive number $\delta$ such that for all $z$ in the open disk . ►A function of two complex variables $f(z,w)$ is continuous at $(z_0,w_0)$ if $\underset{(z,w)\to (z_0,w_0)}{lim}f(z,w)=f(z_0,w_0)$; compare (1.5.1) and (1.5.2). …

26: Mathematical Introduction

… ►Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. … ►With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …

27: 18.37 Classical OP’s in Two or More Variables

§18.37 Classical OP’s in Two or More Variables

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28: 36.2 Catastrophes and Canonical Integrals

… ►For more extensive lists of normal forms of catastrophes (umbilic and beyond) involving two variables (“corank two”) see Arnol’d (1972, 1974, 1975). …

29: 19.24 Inequalities

… ►The condition $y\le z$ for (19.24.1) and (19.24.2) serves only to identify $y$ as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity. …

30: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►An alternant is a determinant function of $n$ variables which changes sign when two of the variables are interchanged. …