# 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 $\left|\operatorname{ph}z\right|<\pi$, but with the closed definition the identity (4.8.7) is valid when $\left|\operatorname{ph}z\right|\leq\pi$. … Again, without the closed definition the $\geq$ and $\leq$ signs would have to be replaced by $>$ and $<$, respectively.
##### 2: 1.4 Calculus of One Variable
A similar definition applies to closed intervals $[a,b]$. …
##### 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(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 $(-\infty,1]$. … An equivalent definition is …
4.37.24 $\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;
##### 5: 14.23 Values on the Cut
If cuts are introduced along the intervals $(-\infty,-1]$ and $[1,\infty)$, then (14.23.4) and (14.23.6) could be used to extend the definitions of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ 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
 $\mathbb{C}$ complex plane (excluding infinity). … equals by definition. … closed interval in $\mathbb{R}$, or closed straight-line segment joining $a$ and $b$ in $\mathbb{C}$.
 $(a,b]$ 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 $\alpha\leq\operatorname{ph}x\leq\beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\operatorname{ph}x\in[\alpha,\beta]$ as $|x|\to\infty$. …
where $h=b-a$, $f\in C^{2}[a,b]$, and $a<\xi. … 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. …
18.40.4 $\lim_{N\to\infty}F_{N}(z)=F(z)\equiv\frac{1}{\mu_{0}}\int_{a}^{b}\frac{w(x)\,% \mathrm{d}x}{z-x},$ $z\in\mathbb{C}\backslash[a,b]$,
Let $f(t)$ be locally integrable on $[0,\infty)$. The Stieltjes transform of $f(t)$ is defined by …Since $f(t)$ is locally integrable on $[0,\infty)$, it defines a distribution by … We again assume $f(t)$ is locally integrable on $[0,\infty)$ and satisfies (2.6.9). … of two locally integrable functions on $[0,\infty)$, (2.6.33) can be written …