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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

… ►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

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§4.37(i) General Definitions

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§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 $$\mathrm{arctanh}z=\frac{1}{2}\ln \left(\frac{1+z}{1-z}\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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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 ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ 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

► ►►►►

$ℂ$	complex plane (excluding infinity).
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$\equiv$	equals by definition.
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$[a,b]$	closed interval in $ℝ$, or closed straight-line segment joining $a$ and $b$ in $ℂ$.

► ►►

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

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7: 2.1 Definitions and Elementary Properties

… ►If the set $𝐗$ 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

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18.40.4 $$\underset{N\to \mathrm{\infty }}{lim}{F}_N(z)=F(z)\equiv \frac{1}{{\mu }_0}{\int }_a^b\frac{w(x)dx}{z-x},$$ $z\in ℂ\backslash [a,b]$,

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Symbols:

$[a,b]$: closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\equiv$: equals by definition, $\int$: integral, $N$: positive integer, $w(x)$: weight function, $z$: complex variable and $x$: real variable

Proved:

Ismail (2009, p. 36)(proved)

Permalink:

http://dlmf.nist.gov/18.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

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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 …