principal value (or branch)

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41: 5.12 Beta Function

… ►In this section all fractional powers have their principal values, except where noted otherwise. … ►In (5.12.8) the fractional powers have their principal values when $w>0$ and $z>0$, and are continued via continuity. … ►In (5.12.11) and (5.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning. …

42: 15.6 Integral Representations

… ►In (15.6.1) all functions in the integrand assume their principal values. ►In (15.6.2) the point $1/z$ lies outside the integration contour, $t^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty })$, and ${(1-zt)}^a=1$ at $t=0$. … ►In (15.6.6) the integration contour separates the poles of $\mathrm{\Gamma }\left(a+t\right)$ and $\mathrm{\Gamma }\left(b+t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$, and ${(-z)}^t$ has its principal value. ►In (15.6.7) the integration contour separates the poles of $\mathrm{\Gamma }\left(a+t\right)$ and $\mathrm{\Gamma }\left(b+t\right)$ from those of $\mathrm{\Gamma }\left(c-a-b-t\right)$ and $\mathrm{\Gamma }\left(-t\right)$, and ${(1-z)}^t$ has its principal value. ►In each of (15.6.8) and (15.6.9) all functions in the integrand assume their principal values. …

43: 16.2 Definition and Analytic Properties

… ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$; compare §4.2(i). …Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. …

44: 3.5 Quadrature

… ►

3.5.46 $$\mathscr{H}f\left(x\right)=\frac{1}{\pi }{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{f(t)}{t-x}dt,$$ $x\in ℝ$,

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Symbols:

$\mathscr{H}\left(f\right)\left(x\right)$: Hilbert transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\in$: element of, ${⨍}_a^b$: Cauchy principal value and $ℝ$: real line

Permalink:

http://dlmf.nist.gov/3.5.E46

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Hilbert transform was changed to $\mathscr{H}f\left(x\right)$ from $\mathscr{H}(f;x)$.

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

►where the integral is the Cauchy principal value. …

45: 19.6 Special Cases

… ►If , then the Cauchy principal value satisfies … ►Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_C(x,y)$. … ►For the Cauchy principal value of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when ${\alpha }^2>c$, see §19.7(iii). …

46: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►In this equation (and in (5.17.5) below), the $\mathrm{Ln}$’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i). …

47: 25.10 Zeros

… ►is chosen to make $Z(t)$ real, and $\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)$ assumes its principal value. …

48: 4.16 Elementary Properties

… ►

Table 4.16.3: Trigonometric functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

► ►►

	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
…

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49: 10.69 Uniform Asymptotic Expansions for Large Order

… ►All fractional powers take their principal values. …

50: 14.1 Special Notation

… ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. …