DLMF: Untitled Document

… ►

18.40.5

F_{N}(z)=\frac{1}{\mu_{0}}\sum_{n=1}^{N}\frac{w_{n}}{z-x_{n}}.

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Symbols:: $N$ : positive integer, $w_{x}$ : weights, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.40.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

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18.40.6

\lim_{\varepsilon\to 0{+}}\int_{a}^{b}\frac{w(x)\,\mathrm{d}x}{x^{\prime}+% \mathrm{i}\varepsilon-x}\,\mathrm{d}x=\pvint_{a}^{b}\frac{w(x)\,\mathrm{d}x}{x% ^{\prime}-x}-\mathrm{i}\pi w(x^{\prime}),

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, $w(x)$ : weight function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.40.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►Results of low (

~{}2

to

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

to

20

. … ►

18.40.7

\mu_{N}(x)=\sum_{n=1}^{N}w_{n}H\left(x-x_{n}\right),

x\in(a,b)

,

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Symbols:: $H\left(\NVar{x}\right)$ : Heaviside function, $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $N$ : positive integer, $w_{x}$ : weights, $n$ : nonnegative integer and $x$ : real variable
Source:: Shohat and Tamarkin (1970, pp. 42-43)
Referenced by:: §18.40(ii)
Permalink:: http://dlmf.nist.gov/18.40.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

…

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B_{-2}^{5}=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),

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B_{-3}^{6}=-\tfrac{1}{720}(12-8t-45t^{2}+20t^{3}+15t^{4}-6t^{5}),

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B_{-2}^{6}=\tfrac{1}{60}(9-9t-30t^{2}+20t^{3}+5t^{4}-3t^{5}),

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B_{2}^{6}=-\tfrac{1}{60}(9+9t-30t^{2}-20t^{3}+5t^{4}+3t^{5}),

… ►If

f

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

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►

19.36.2

1-\tfrac{3}{14}E_{2}+\tfrac{1}{6}E_{3}+\tfrac{9}{88}E_{2}^{2}-\tfrac{3}{22}E_{% 4}-\tfrac{9}{52}E_{2}E_{3}+\tfrac{3}{26}E_{5}-\tfrac{1}{16}E_{2}^{3}+\tfrac{3}% {40}E_{3}^{2}+\tfrac{3}{20}E_{2}E_{4}+\tfrac{45}{272}E_{2}^{2}E_{3}-\tfrac{9}{% 68}(E_{3}E_{4}+E_{2}E_{5}).

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Symbols:: $E_{s}(\mathbf{z})$ : symmetric function
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.36.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36 and Ch.19

… ►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)). … ►

19.36.13

2R_{G}\left(t_{0}^{2},t_{0}^{2}+\theta c_{0}^{2},t_{0}^{2}+\theta a_{0}^{2}% \right)=\left(t_{0}^{2}+\theta\sum_{m=0}^{\infty}2^{m-1}c_{m}^{2}\right)R_{C}% \left(T^{2}+\theta M^{2},T^{2}\right)+h_{0}+\sum_{m=1}^{\infty}2^{m}(h_{m}-h_{% m-1}).

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $m$ : nonnegative integer, $M$ : limit, $T$ : limit and $h_{n}$
Permalink:: http://dlmf.nist.gov/19.36.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(ii), §19.36(ii), §19.36 and Ch.19

… ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

… ►

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.

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External Links:: FullText, MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §35.2, §35.3(ii).
Encodings:: BibTeX

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W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

E. Grosswald (1985) Representations of Integers as Sums of Squares. Springer-Verlag, New York.

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External Links:: ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

…

Cauchy%20sum

4 matching pages

1: 18.40 Methods of Computation

2: 3.4 Differentiation

3: 19.36 Methods of Computation

4: Bibliography G