Cauchy principal values

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1: 4.10 Integrals

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4.10.7 $${⨍}_0^x\frac{dt}{\ln t}=\mathrm{li}\left(x\right),$$ $x>1$.

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Symbols:

$dx$: differential, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

4.1.57

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

►The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). …

2: 1.4 Calculus of One Variable

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Cauchy Principal Values

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1.4.24 $${⨍}_a^{b}f(x)dx=𝑃{\int }_a^{b}f(x)dx=\underset{ϵ\to 0+}{lim}\left({\int }_a^{c-ϵ}f(x)dx+{\int }_{c+ϵ}^{b}f(x)dx\right),$$

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Defines:

${⨍}_a^b$: Cauchy principal value

Symbols:

$dx$: differential and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.4.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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1.4.25 $${⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=𝑃{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=\underset{b\to \mathrm{\infty }}{lim}{\int }_{-b}^{b}f(x)dx,$$

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Symbols:

$dx$: differential, $\int$: integral and ${⨍}_a^b$: Cauchy principal value

Referenced by:

§1.14(i)

Permalink:

http://dlmf.nist.gov/1.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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3: 19.3 Graphics

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Figure 19.3.2: $R_C(x,1)$ and the Cauchy principal value of $R_C(x,-1)$ for $0\le x\le 5$. … Magnify

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Figure 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ as a function of $k^2$ and ${\alpha }^2$ for , $-2\le {\alpha }^2\le 2$. Cauchy principal values are shown when ${\alpha }^2>1$. … Magnify 3D Help

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . Cauchy principal values are shown when ${\sin }^2\varphi >\frac{1}{2}$. …If ${\sin }^2\varphi =1$ ($>k^2$), then the function reduces to $\mathrm{\Pi }(2,k)$ with Cauchy principal value $K\left(k\right)-\mathrm{\Pi }(\frac{1}{2}{k}^2,k)$, which tends to $-\mathrm{\infty }$ as $k^2\to 1-$. …If ${\sin }^2\varphi =1/k^2$ (), then by (19.7.4) it reduces to $\mathrm{\Pi }(2/k^2,1/k)/k$, $k^2\ne 2$, with Cauchy principal value $(K\left(1/k\right)-\mathrm{\Pi }(\frac{1}{2},1/k))/k$, , by (19.6.5). … Magnify 3D Help

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4: 19.17 Graphics

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Figure 19.17.6: Cauchy principal value of $R_J(x,y,1,-0.5)$ for $0\le x\le 1$, $y=0,\mathrm{\hspace{0.17em}0.1},\mathrm{\hspace{0.17em}0.5},\mathrm{\hspace{0.17em}1}$. … Magnify

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Figure 19.17.7: Cauchy principal value of $R_J(0.5,y,1,p)$ for $y=0,\mathrm{\hspace{0.17em}0.01},\mathrm{\hspace{0.17em}0.05},\mathrm{\hspace{0.17em}0.2},\mathrm{\hspace{0.17em}1}$, . … Magnify

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Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. Cauchy principal values are shown when . … Magnify 3D Help

5: 4.2 Definitions

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6: 19.2 Definitions

… ►The integral for $E(\varphi ,k)$ is well defined if $k^2={\sin }^2\varphi =1$, and the Cauchy principal value (§1.4(v)) of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is taken if $1-{\alpha }^2{\sin }^2\varphi$ vanishes at an interior point of the integration path. … ►If , then the integral in (19.2.11) is a Cauchy principal value. … ►where the Cauchy principal value is taken if . Formulas involving $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_C(x,y)$. … ►The Cauchy principal value is hyperbolic: …

7: 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). …

8: 6.2 Definitions and Interrelations

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6.2.5 $$\mathrm{Ei}\left(x\right)=-{⨍}_{-x}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t}dt={⨍}_{-\mathrm{\infty }}^x\frac{{\mathrm{e}}^t}{t}dt,$$

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Symbols:

$dx$: differential, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

5.1.2

Permalink:

http://dlmf.nist.gov/6.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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6.2.8 $$\mathrm{li}\left(x\right)={⨍}_0^x\frac{dt}{\ln t}=\mathrm{Ei}\left(\ln x\right),$$ $x>1$.

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Defines:

$\mathrm{li}\left(x\right)$: logarithmic integral

Symbols:

$dx$: differential, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

5.1.3

Referenced by:

§27.12, §6.12(i)

Permalink:

http://dlmf.nist.gov/6.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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9: 9.10 Integrals

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9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt,$$ $x>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\mathrm{e}$: base of natural logarithm, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

Proof sketch:

To prove this equation, combine (9.2.10) with (9.10.18).

Referenced by:

§9.5(i), Erratum (V1.0.10) for Equation (9.10.19)

Permalink:

http://dlmf.nist.gov/9.10.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Bi}\left(x\right)=\frac{x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2^{5/2}\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

Annotations for §9.10(vii), §9.10 and Ch.9

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

10: 1.14 Integral Transforms

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1.14.3 $$\frac{1}{2}(f(u+)+f(u-))=\frac{1}{\sqrt{2\pi }}{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}F(x){\mathrm{e}}^{-\mathrm{i}xu}dx,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${⨍}_a^b$: Cauchy principal value and $F(x)$: Fourier transform of $f(t)$

Permalink:

http://dlmf.nist.gov/1.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

►where the last integral denotes the Cauchy principal value (1.4.25). … ►

1.14.41 $$\mathscr{H}\left(f\right)\left(x\right)=\mathscr{H}f\left(x\right)=\frac{1}{\pi }{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{f(t)}{t-x}dt,$$

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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 and ${⨍}_a^b$: Cauchy principal value

Permalink:

http://dlmf.nist.gov/1.14.E41

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

Annotations for §1.14(v), §1.14 and Ch.1

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1.14.44 $$f(x)=-\frac{1}{\pi }{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathscr{H}f\left(u\right)}{u-x}du.$$

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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 and ${⨍}_a^b$: Cauchy principal value

Referenced by:

§1.14(v)

Permalink:

http://dlmf.nist.gov/1.14.E44

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

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