Haar%20measure

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11: 3.1 Arithmetics and Error Measures

§3.1 Arithmetics and Error Measures

… ►

§3.1(v) Error Measures

… ►The relative precision is … ►The mollified error is … ►

12: 8 Incomplete Gamma and Related
Functions

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13: 28 Mathieu Functions and Hill’s Equation

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14: 18.39 Applications in the Physical Sciences

… ►Orthogonality and normalization of eigenfunctions of this form is respect to the measure

r^{2}\,\mathrm{d}r\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi

. … ►where the orthogonality measure is now

\,\mathrm{d}r

r\in[0,\infty).

… ►Orthogonality, with measure

\,\mathrm{d}r

for

r\in[0,\infty)

, for fixed

l

… ►thus recapitulating, for

Z=1

, line 11 of Table 18.8.1, now shown with explicit normalization for the measure

\,\mathrm{d}r

. …normalized with measure

r^{2}\,\mathrm{d}r

r\in[0,\infty)

. …

15: 8.26 Tables

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•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

16: 23 Weierstrass Elliptic and Modular
Functions

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17: Bibliography G

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M. J. Gander and A. H. Karp (2001) Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

ⓘ

Referenced by:: §18.39(iii).
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.

ⓘ

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

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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

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18: 1.4 Calculus of One Variable

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Absolutely Continuous Stieltjes Measure

… ► … ►

Stieltjes Measure with $\alpha(x)$ Discontinuous

… ► See Riesz and Sz.-Nagy (1990, Ch. 3). … ►Definite integrals over the Stieltjes measure

\,\mathrm{d}\alpha(x)

could represent a sum, an integral, or a combination of the two. …

19: 36 Integrals with Coalescing Saddles

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20: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …