repeated%20integrals%20of%20error%20functions

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1: 7.23 Tables

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Abramowitz and Stegun (1964, Chapter 7) includes $\mathrm{erf}x$, $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [0,2]$, 10D; $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [2,10]$, 8S; $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$, $x^{-2}\in [0,0.25]$, 7D; $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=1(1)6,10,11$, $x\in [0,5]$, 6S; $F\left(x\right)$, $x\in [0,2]$, 10D; $xF\left(x\right)$, $x^{-2}\in [0,0.25]$, 9D; $C\left(x\right)$, $S\left(x\right)$, $x\in [0,5]$, 7D; $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in [0,1]$, $x^{-1}\in [0,1]$, 15D.

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Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $\mathrm{erf}x$, $x=0(.02)1(.04)3$, 8D; $C\left(x\right)$, $S\left(x\right)$, $x=0(.2)10(2)100(100)500$, 8D.

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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Fettis et al. (1973) gives the first 100 zeros of $\mathrm{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$, not for $\mathrm{erfc}z$), 11S.

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Zhang and Jin (1996, p. 642) includes the first 10 zeros of $\mathrm{erf}z$, 9D; the first 25 distinct zeros of $C\left(z\right)$ and $S\left(z\right)$, 8S.

2: Bibliography G

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Evaluation of the repeated integrals of the coerror function. ACM Trans. Math. Software 3, pp. 240–252.

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Notes:

For software see Gautschi (1977b).

External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§7.22(iii).

Encodings:

BibTeX

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Algorithm 521: Repeated integrals of the coerror function. ACM Trans. Math. Software 3, pp. 301–302.

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Notes:

Single-precision Fortran, maximum accuracy 14D

External Links:

ISSN 0098-3500, Document, GAMS (module 521; package TOMS)

Referenced by:

3rd item, W. Gautschi (1977a).

Encodings:

BibTeX

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Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.

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External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§7.22(iii).

Encodings:

BibTeX

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Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.

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Notes:

Includes two MATLAB programs claiming 12D and 30D accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

4th item, §7.25(ii).

Encodings:

BibTeX

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A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.

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Notes:

See for the first part same journal, Sect. B 73, 1-20 (1969)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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3: Software Index

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	Open Source													With Book						Commercial
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7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	NMS
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

4: 2.11 Remainder Terms; Stokes Phenomenon

… ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable $x$ that is intended to be used. … ►Here $\mathrm{erfc}$ is the complementary error function (§7.2(i)), and … ►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15). … ►For error bounds see Dunster (1996c). … ►Often the process of re-expansion can be repeated any number of times. …

5: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … ►If the iteration of (19.36.6) and (19.36.12) is stopped when ($M$ and $T$ being approximated by $a_s$ and $t_s$, and the infinite series being truncated), then the relative error in $R_F$ and $R_G$ is less than $r$ if we neglect terms of order $r^2$. … ►The function $\mathrm{cel}(k_c,p,a,b)$ is computed by successive Bartky transformations (Bulirsch and Stoer (1968), Bulirsch (1969b)). … ►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). … ►Similarly, §19.26(ii) eases the computation of functions such as $R_F(x,y,z)$ when $x$ ($>0$) is small compared with $\min (y,z)$. …