Cauchy principal values

SearchAdvancedHelp

(0.002 seconds)

21—28 of 28 matching pages

21: 1.16 Distributions

… ►

1.16.49 $$⟨ℱ\left(\mathrm{sign}\right),\varphi ⟩=\mathrm{i}\sqrt{\frac{2}{\pi }}{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\varphi (x)}{x}dx.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\mathrm{i}$: imaginary unit, ${⨍}_a^b$: Cauchy principal value, $\mathrm{sign}x$: sign of and $\varphi (x)$: test function

Permalink:

http://dlmf.nist.gov/1.16.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(viii), §1.16 and Ch.1

… ►

1.16.51 $$⟨ℱ\left(H\right),\varphi ⟩=\sqrt{\frac{\pi }{2}}\varphi (0)+\frac{\mathrm{i}}{\sqrt{2\pi }}{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\varphi (x)}{x}dx.$$

ⓘ

Symbols:

$H\left(x\right)$: Heaviside function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\mathrm{i}$: imaginary unit, ${⨍}_a^b$: Cauchy principal value and $\varphi (x)$: test function

Permalink:

http://dlmf.nist.gov/1.16.E51

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(viii), §1.16 and Ch.1

…

22: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.34 $$\frac{1}{{\pi }^2}{⨍}_0^{2\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{\sin t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_1(\nu ,\nu +2m+1,0),$$

ⓘ

Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, ${⨍}_a^b$: Cauchy principal value, $\sin z$: sine function, $m$: integer, $h$: parameter, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

►where the integral is a Cauchy principal value (§1.4(v)). …

23: 19.25 Relations to Other Functions

… ►with Cauchy principal value … ►If ${\alpha }^2>c$, then the Cauchy principal value is …

24: 3.5 Quadrature

… ►

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

ⓘ

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. …

25: 2.10 Sums and Sequences

… ►For an extension to integrals with Cauchy principal values see Elliott (1998). …

26: 19.29 Reduction of General Elliptic Integrals

… ►The Cauchy principal value is taken when $U_{\alpha 5}^2$ or $Q_{\alpha 5}^2$ is real and negative. …

27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

1.18.54 $$\underset{ϵ\to 0+}{lim}{\int }_X\frac{f(y)}{x\pm \mathrm{i}ϵ-y}dy=P{⨍}_X\frac{f(y)}{x-y}dy\mp \mathrm{i}\pi f(x).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and ${⨍}_a^b$: Cauchy principal value

Permalink:

http://dlmf.nist.gov/1.18.E54

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

…

28: 2.3 Integrals of a Real Variable

… ►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$. … ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by … ►provided that the integral on the left-hand side of (2.3.9) converges for all sufficiently large values of $x$. … ► $\kappa =\kappa (\alpha )$ being the value of $w$ at $t=k$. We now expand $f(\alpha ,w)$ in a Taylor series centered at the peak value $w=a$ of the exponential factor in the integrand: …