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21: 6.20 Approximations

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

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Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\operatorname{Ein}\left(ax\right)$ , $\operatorname{Si}\left(ax\right)$ , and $\operatorname{Cin}\left(ax\right)$ for $-1\leq x\leq 1$ , $a\in\mathbb{C}$ . The coefficients are given in terms of series of Bessel functions.

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Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for $\operatorname{Ein}\left(z\right)$ , $\operatorname{Si}\left(z\right)$ , $\operatorname{Cin}\left(z\right)$ (valid near the origin), and $E_{1}\left(z\right)$ (valid for large $|z|$ ); approximate errors are given for a selection of $z$ -values.

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22: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

23: Bibliography R

… ►

H. A. Ragheb, L. Shafai, and M. Hamid (1991) Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric. IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.

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External Links:: Document
Referenced by:: 6th item.
Encodings:: BibTeX

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M. Rahman (1981) A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:: ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:: ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt
Referenced by:: §29.20(ii).
Encodings:: BibTeX

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G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

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24: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►and if

\phi\left(n\right)

is the smallest positive integer

f

such that

a^{f}\equiv 1\pmod{n}

, then

a

is a primitive root mod

n

. …Such a set is a reduced residue system modulo

n

. … ►where

p^{a}

is a prime power with

a\geq 1

; otherwise

\Lambda\left(n\right)=0

. …

25: 7.24 Approximations

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Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

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Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$ , $\operatorname{erf}z$ , $\operatorname{erfc}z$ , $C\left(z\right)$ , and $S\left(z\right)$ ; approximate errors are given for a selection of $z$ -values.

26: Bibliography V

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Van Buren (website) Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation

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External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).
Encodings:: BibTeX

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N. Ja. Vilenkin and A. U. Klimyk (1991) Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.

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Notes:: Translated from the Russian by V. A. Groza and A. A. Groza
External Links:: ISBN 0-7923-1466-2, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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N. Ja. Vilenkin and A. U. Klimyk (1992) Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.

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Notes:: Translated from the Russian by V. A. Groza and A. A. Groza
External Links:: ISBN 0-7923-1493-X, MathReview (P. D. F. Ion), ZentralBlatt
Referenced by:: §18.38(iii), Ch.35.
Encodings:: BibTeX

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N. Ja. Vilenkin and A. U. Klimyk (1993) Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

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Notes:: Translated from the Russian by V. A. Groza and A. A. Groza
External Links:: ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

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27: 25.12 Polylogarithms

… ►The notation

\operatorname{Li}_{2}\left(z\right)

was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … ►In the complex plane

\operatorname{Li}_{2}\left(z\right)

has a branch point at

z=1

. … ►valid when

\Re s>0

\Im a>0

\Re s>1

\Im a=0

. When

s=2

and

e^{2\pi ia}=z

, (25.12.13) becomes (25.12.4). … ►For a uniform asymptotic approximation for

F_{s}(x)

see Temme and Olde Daalhuis (1990).

28: Bibliography M

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

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W. Magnus and S. Winkler (1966) Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:: Reprinted by Dover Publications, Inc., New York, 1979.
External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.
Encodings:: BibTeX

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Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

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D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

…

29: 9.18 Tables

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•

Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ ; 8D.

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30: Tom M. Apostol

… ►Apostol was born on August 20, 1923. He received his bachelor of science in chemical engineering in 1944 and a master’s degree in mathematics in 1946, both from the University of Washington, Seattle. … ►He was a visiting professor at the University of Patras in Greece in 1978, and was elected a Corresponding Member of the Academy of Athens in 2001 (where he delivered his inaugural lecture in Greek). … ►He was also a coauthor of three textbooks written to accompany the physics telecourse The Mechanical Universe …and Beyond. … ►He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …