Cauchy integral formula

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1: 1.9 Calculus of a Complex Variable

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Cauchy’s Integral Formula

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2: 3.4 Differentiation

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

3: 2.10 Sums and Sequences

… ►These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula …

4: 2.3 Integrals of a Real Variable

§2.3 Integrals of a Real Variable

… ►For the Fourier integral … ►Then … ►In the integral … ►The integral (2.3.24) transforms into …

5: 19.2 Definitions

… ►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 $R_C(x,y)$. …

6: 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. Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1. ►The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the $8+8+12=28$ formulas in Gradshteyn and Ryzhik (2015, 3.147, 3.131, 3.152) after taking $x^2$ as the variable of integration in 3. …(19.29.7) subsumes all 72 formulas in Gradshteyn and Ryzhik (2015, 3.168), and its cubic cases similarly replace the $18+36+18=72$ formulas in Gradshteyn and Ryzhik (2015, 3.133, 3.142, and 3.141(1-18)). … ►The first formula replaces (19.14.4)–(19.14.10). …

7: 3.5 Quadrature

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Gauss–Laguerre Formula

… ► … ►a complex Gauss quadrature formula is available. … ►where the integral is the Cauchy principal value. …

8: 1.16 Distributions

… ►Since ${\delta }_{x_0}$ is the Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ corresponding to $\alpha (x)=H\left(x-x_0\right)$ (see §1.4(v)), formula (1.16.16) is a special case of (1.16.3_5), (1.16.9_5) for that choice of $\alpha$. … ►The second to last equality follows from the Fourier integral formula (1.17.8). … ►

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

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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

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

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

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

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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

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See also:

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

►For more detailed discussions of the formulas in this section, see Kanwal (1983) and Debnath and Bhatta (2015). …

9: 19.21 Connection Formulas

§19.21 Connection Formulas

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§19.21(i) Complete Integrals

… ►The complete cases of $R_F$ and $R_G$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►Connection formulas for $R_{-a}(𝐛;𝐳)$ are given in Carlson (1977b, pp. 99, 101, and 123–124). … ►The latter case allows evaluation of Cauchy principal values (see (19.20.14)). …

10: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

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BibTeX

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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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External Links:

FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

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BibTeX

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

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BibTeX

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

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BibTeX

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

Encodings:

BibTeX

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