on a region

SearchAdvancedHelp

(0.004 seconds)

11—20 of 60 matching pages

11: 8.11 Asymptotic Approximations and Expansions

… ►In the case that $a=n$, a positive integer, the $z$-region of validity of (8.11.7) is discussed in Ameur and Cronvall (2023). …

12: 3.10 Continued Fractions

… ►For example, by converting the Maclaurin expansion of $\arctan z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\le 1$, $z\ne \pm \mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in ℂ$ except on the branch cuts from $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$ and $-\mathrm{i}$ to $-\mathrm{i}\mathrm{\infty }$. …

13: 1.6 Vectors and Vector-Valued Functions

… ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►

1.6.44 $${\iint }_S\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA={\int }_C𝐅\cdot d𝐬={\int }_C{F}_{1}dx+F_{2}dy.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝐅(x,y)$: vector, $S$: closed region, $C$: closed curve and $F_j$: vector function components

Referenced by:

§1.6(iv)

Permalink:

http://dlmf.nist.gov/1.6.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iv), §1.6(iv), §1.6 and Ch.1

… ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. … ►

1.6.59 $${\iiint }_V(f{\nabla }^{2}g+\nabla f\cdot \nabla g)dV={\iint }_{S}f\frac{\partial g}{\partial n}dA,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E59

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

… ►

1.6.60 $${\iiint }_V(f{\nabla }^{2}g-g{\nabla }^{2}f)dV={\iint }_S\left(f\frac{\partial g}{\partial n}-g\frac{\partial f}{\partial n}\right)dA,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E60

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

…

14: 10.19 Asymptotic Expansions for Large Order

… ►

§10.19(iii) Transition Region

►As $\nu \to \mathrm{\infty }$, with $a(\in ℂ)$ fixed, … ►

$P_1(a)=-\frac{1}{5}a,$

… ►

$R_1(a)=-\frac{4}{5}a,$

… ►For higher coefficients in (10.19.8) in the case $a=0$ (that is, in the expansions of $J_{\nu }\left(\nu \right)$ and $Y_{\nu }\left(\nu \right)$), see Watson (1944, §8.21), Temme (1997), and Jentschura and Lötstedt (2012). …

15: 16.4 Argument Unity

… ►(16.4.11) provides a partial analytic continuation to the region when the only restrictions on the parameters are $\mathrm{\Re }\left(e-a\right)>0$, and $d,e$, and $d+e-b-c\ne 0,-1,\mathrm{\dots }$. …

16: 35.2 Laplace Transform

… ►Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. …

17: 4.2 Definitions

… ►As a consequence, it has the advantage of extending regions of validity of properties of principal values. …

18: Bibliography W

… ►

Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region. Phys. Rev. B 13, pp. 316–374.

ⓘ

External Links:

Document, Link

Referenced by:

§32.16, §32.16.

Encodings:

BibTeX

…

19: 33.12 Asymptotic Expansions for Large $\eta$

… ►

§33.12(i) Transition Region

… ►

$A_1=\frac{1}{5}{x}^2,$

►

$A_2=\frac{1}{35}(2x^3+6),$

►

$A_3=\frac{1}{15750}(21x^7+370x^4+580x),$

… ►They would include the results of §33.12(i) as a special case. …

20: 3.1 Arithmetics and Error Measures

… ►With this arithmetic the computed result can be proved to lie in a certain interval, which leads to validated computing with guaranteed and rigorous inclusion regions for the results. …