Cauchy principal values

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11: 9.12 Scorer Functions

… ►

9.12.23 $$\mathrm{Gi}\left(x\right)=\frac{4x^2}{3^{3/2}{\pi }^2}{⨍}_0^{\mathrm{\infty }}\frac{K_{1/3}\left(t\right)}{{\zeta }^2-t^2}dt,$$ $x>0$,

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Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${⨍}_a^b$: Cauchy principal value, $x$: real variable and $\zeta$: variable

Source:

Gordon (1970, Appendix A)

Permalink:

http://dlmf.nist.gov/9.12.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12 and Ch.9

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

12: 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 of $x$ , $\mathrm{e}$: 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 of $x$ 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 of $x$ 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

…

13: 19.20 Special Cases

… ►Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. If $x,y,z$ are permuted so that , then the Cauchy principal value of $R_J$ is given by …

14: 19.7 Connection Formulas

… ►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when ${\alpha }^2>{\csc }^2\varphi$ (see (19.6.5) for the complete case). …

15: 19.21 Connection Formulas

… ►The latter case allows evaluation of Cauchy principal values (see (19.20.14)). …

16: 18.17 Integrals

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18.17.42 $${⨍}_{-1}^1{T}_n\left(y\right)\frac{{(1-y^2)}^{-\frac{1}{2}}}{y-x}dy=\pi U_{n-1}\left(x\right),$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${⨍}_a^b$: Cauchy principal value, $y$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.13.3

Referenced by:

§18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

►

18.17.43 $${⨍}_{-1}^1{U}_{n-1}\left(y\right)\frac{{(1-y^2)}^{\frac{1}{2}}}{y-x}dy=-\pi T_n\left(x\right).$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${⨍}_a^b$: Cauchy principal value, $y$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.13.4

Referenced by:

§18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

►These integrals are Cauchy principal values (§1.4(v)). …

17: 19.8 Quadratic Transformations

… ►If ${\alpha }^2>1$, then the Cauchy principal value is …

18: 19.16 Definitions

… ►In (19.16.2) the Cauchy principal value is taken when $p$ is real and negative. …

19: 19.22 Quadratic Transformations

… ►If the last variable of $R_J$ is negative, then the Cauchy principal value is …

20: 19.36 Methods of Computation

… ►All cases of $R_F$, $R_C$, $R_J$, and $R_D$ are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). …