Cauchy integral formula

(0.005 seconds)

1—10 of 15 matching pages

2: 3.4 Differentiation
If $f$ can be extended analytically into the complex plane, then from Cauchy’s integral formula1.9(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 integralThen … In the integralThe integral (2.3.24) transforms into …
5: 19.2 Definitions
Formulas involving $\Pi\left(\phi,\alpha^{2},k\right)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_{C}\left(x,y\right)$. …
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 (2000, 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 (2000, 3.168), and its cubic cases similarly replace the $18+36+18=72$ formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.142, and 3.141(1-18)). … The first formula replaces (19.14.4)–(19.14.10). …
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 $\left\langle\mathscr{F}\left(\operatorname{sign}\right),\phi\right\rangle=% \mathrm{i}\sqrt{\frac{2}{\pi}}\pvint^{\infty}_{-\infty}\frac{\phi(x)}{x}\,% \mathrm{d}x.$
1.16.51 $\left\langle\mathscr{F}\left(H\right),\phi\right\rangle=\sqrt{\frac{\pi}{2}}% \phi(0)+\frac{\mathrm{i}}{\sqrt{2\pi}}\pvint^{\infty}_{-\infty}\frac{\phi(x)}{% x}\,\mathrm{d}x.$
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(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 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ 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
• F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
• L. Gårding (1947) 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.
• G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
• W. Gautschi (1968) Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.
• H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.