# closed definition

(0.002 seconds)

## 1—10 of 48 matching pages

##### 1: 4.2 Definitions

…
►We regard this as the

*closed definition of the principal value*. ►In contrast to (4.2.5) the closed definition is symmetric. …For example, with the definition (4.2.5) the identity (4.8.7) is valid only when $$, but with the closed definition the identity (4.8.7) is valid when $\left|\mathrm{ph}z\right|\le \pi $. … ►Again, without the closed definition the $\ge $ and $\le $ signs would have to be replaced by $>$ and $$, respectively.##### 2: 1.4 Calculus of One Variable

##### 3: 1.13 Differential Equations

…
►As the interval $[a,b]$ is mapped, one-to-one, onto $[0,c]$ by the above definition of $t$, the integrand being positive, the inverse of this same transformation allows $\widehat{q}(t)$ to be calculated from $p,q,\rho $ in (1.13.31), $p,\rho \in {C}^{2}(a,b)$ and $q\in C(a,b)$.
…

##### 4: 4.37 Inverse Hyperbolic Functions

…
►

###### §4.37(i) General Definitions

… ►###### §4.37(ii) Principal Values

… ►It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on $(-\mathrm{\infty},1]$. … ►An equivalent definition is … ►
4.37.24
$$\mathrm{arctanh}z=\frac{1}{2}\mathrm{ln}\left(\frac{1+z}{1-z}\right),$$
$z\in \u2102\setminus (-\mathrm{\infty},-1]\cup [1,\mathrm{\infty})$;

…
##### 5: 14.23 Values on the Cut

…
►If cuts are introduced along the intervals $(-\mathrm{\infty},-1]$ and $[1,\mathrm{\infty})$, then (14.23.4) and (14.23.6) could be used to extend the definitions of ${\U0001d5af}_{\nu}^{\mu}\left(x\right)$ and ${\U0001d5b0}_{\nu}^{\mu}\left(x\right)$ to complex $x$.
…

##### 6: Mathematical Introduction

…
►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3).
…
►

►
►►

…

###### Common Notations and Definitions

► ►►►►$\u2102$ | complex plane (excluding infinity). |
---|---|

… | |

$\equiv $ | equals by definition. |

… | |

$[a,b]$ | closed interval in $\mathbb{R}$, or closed straight-line segment joining $a$ and $b$ in $\u2102$. |

$(a,b]$ or $[a,b)$ | half-closed intervals. |
---|---|

… |

##### 7: 2.1 Definitions and Elementary Properties

…
►If the set $\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha \le \mathrm{ph}x\le \beta $, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty}$.
…

##### 8: 3.5 Quadrature

…
►where $h=b-a$, $f\in {C}^{2}[a,b]$, and $$.
…
►Let $h=\frac{1}{2}(b-a)$ and $f\in {C}^{4}[a,b]$.
…
►If $f\in {C}^{2m+2}[a,b]$, then the remainder ${E}_{n}(f)$ in (3.5.2) can be expanded in the form
…
►is computed with $p=1$ on the interval $[0,30]$.
…
►Rules of closed type include the

*Newton–Cotes formulas*such as the trapezoidal rules and*Simpson’s rule*. …##### 9: 18.40 Methods of Computation

…
►

18.40.4
$$\underset{N\to \mathrm{\infty}}{lim}{F}_{N}(z)=F(z)\equiv \frac{1}{{\mu}_{0}}{\int}_{a}^{b}\frac{w(x)dx}{z-x},$$
$z\in \u2102\backslash [a,b]$,

…
##### 10: 2.6 Distributional Methods

…
►Let $f(t)$ be locally integrable on $[0,\mathrm{\infty})$.
The

*Stieltjes transform*of $f(t)$ is defined by …Since $f(t)$ is locally integrable on $[0,\mathrm{\infty})$, it defines a distribution by … ►We again assume $f(t)$ is locally integrable on $[0,\mathrm{\infty})$ and satisfies (2.6.9). … ►of two locally integrable functions on $[0,\mathrm{\infty})$, (2.6.33) can be written …