Cauchy

SearchAdvancedHelp

(0.000 seconds)

21—30 of 36 matching pages

21: 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). …

22: 19.21 Connection Formulas

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

23: 18.17 Integrals

… ►

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

ⓘ

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

ⓘ

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)). …

24: 3.4 Differentiation

… ►If $f$ can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …

25: 19.8 Quadratic Transformations

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

26: 19.16 Definitions

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

27: 19.22 Quadratic Transformations

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

28: 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)). …

29: 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

…

30: 19.25 Relations to Other Functions

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