several variables

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31—34 of 34 matching pages

31: 2.7 Differential Equations

… ►The first of these references includes extensions to complex variables and reversions for zeros. … ►From the numerical standpoint, however, the pair

w_{3}(z)

and

w_{4}(z)

has the drawback that severe numerical cancellation can occur with certain combinations of

C

and

D

, for example if

C

and

D

are equal, or nearly equal, and

z

, or

\Re z

, is large and negative. …

32: 36.12 Uniform Approximation of Integrals

… ►In the cuspoid case (one integration variable) ►

36.12.1

I(\mathbf{y},k)=\int_{-\infty}^{\infty}\exp\left(ikf(u;\mathbf{y})\right)g(u,% \mathbf{y})\,\mathrm{d}u,

ⓘ

Defines:: $I(\mathbf{y},k)$ : oscillatory integral (locally)
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $k$ : variable, $g(u,\mathbf{y})$ : function, $f(u,\mathbf{y})$ : function and $u(t;\mathbf{y})$ : mapping
Referenced by:: §36.13
Permalink:: http://dlmf.nist.gov/36.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §36.12(i), §36.12 and Ch.36

… ►

36.12.4

f(u(t,\mathbf{y});\mathbf{y})=A(\mathbf{y})+\Phi_{K}\left(t;\mathbf{x}(\mathbf% {y})\right),

ⓘ

Defines:: $u(t;\mathbf{y})$ : mapping (locally)
Symbols:: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $t$ : variable, $K$ : codimension, $f(u,\mathbf{y})$ : function, $\mathbf{x}(\mathbf{y})$ : function and $A(\mathbf{y})$ : function
Permalink:: http://dlmf.nist.gov/36.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §36.12(i), §36.12 and Ch.36

… ►

36.12.10

G_{n}(\mathbf{y})=g(t_{n}(\mathbf{y}),\mathbf{y})\sqrt{\frac{\ifrac{{\partial}% ^{2}\Phi_{K}\left(t_{n}(\mathbf{x}(\mathbf{y}));\mathbf{x}(\mathbf{y})\right)}% {{\partial t}^{2}}}{\ifrac{{\partial}^{2}f(u_{n}(\mathbf{y}))}{{\partial u}^{2% }}}}.

ⓘ

Symbols:: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $m$ : integer, $t$ : variable, $K$ : codimension, $g(u,\mathbf{y})$ : function, $f(u,\mathbf{y})$ : function, $u(t;\mathbf{y})$ : mapping, $\mathbf{x}(\mathbf{y})$ : function and $t_{j}(\mathbf{x})$ : solutions
Referenced by:: §36.12(i)
Permalink:: http://dlmf.nist.gov/36.12.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §36.12(i), §36.12 and Ch.36

… ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).

33: 22.5 Special Values

§22.5 Special Values

… ►

Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.

► ►►

	$z$
…

► … ►

Table 22.5.2: Other special values of Jacobian elliptic functions.

► ►►

	$z$
…

► …

34: 10.74 Methods of Computation

… ►It should be noted, however, that there is a difficulty in evaluating the coefficients

A_{k}(\zeta)

B_{k}(\zeta)

C_{k}(\zeta)

, and

D_{k}(\zeta)

, from the explicit expressions (10.20.10)–(10.20.13) when

z

is close to

1

owing to severe cancellation. … ►And since there are no error terms they could, in theory, be used for all values of

z

; however, there may be severe cancellation when

|z|

is not large compared with

n^{2}

. … ►, §10.3(i)) with the

x

-axis, or graphical intersection of

3D

complex-variable surfaces (e. …