discrete%20Fourier%20transform

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21: 8 Incomplete Gamma and Related
Functions

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22: 28 Mathieu Functions and Hill’s Equation

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23: 28.35 Tables

§28.35 Tables

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Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for $n=0$ or $1(1)6$ , $q=0(1)10(2)20(4)40$ ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for $q=0(1)10$ , $x=1(1)90$ , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

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Kirkpatrick (1960) contains tables of the modified functions $\operatorname{Ce}_{n}\left(x,q\right)$ , $\operatorname{Se}_{n+1}\left(x,q\right)$ for $n=0(1)5$ , $q=1(1)20$ , $x=0.1(.1)1$ ; 4D or 5D.

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National Bureau of Standards (1967) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$ , and $n=4(1)15$ with $q=0(2)100$ ; Fourier coefficients for $\operatorname{ce}_{n}\left(x,q\right)$ and $\operatorname{se}_{n}\left(x,q\right)$ for $n=0(1)15$ , $n=1(1)15$ , respectively, and various values of $q$ in the interval $[0,100]$ ; joining factors $g_{\mathit{e},n}(\sqrt{q})$ , $f_{\mathit{e},n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$ . Precision is generally 8D.

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n+1}\left(q\right)$ for $n=0(1)4$ , $q=0(1)50$ ; $n=0(1)20$ ( $a$ ’s) or 19 ( $b$ ’s), $q=1,3,5,10,15,25,50(50)200$ . Fourier coefficients for $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , $n=0(1)7$ . Mathieu functions $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $x=0(5^{\circ})90^{\circ}$ . Modified Mathieu functions ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n+1}}\left(x,\sqrt{10}\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $j=1,2$ , $x=0(.2)4$ . Precision is mostly 9S.

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24: 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.

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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.

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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.

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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.

25: 23 Weierstrass Elliptic and Modular
Functions

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26: Bibliography H

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P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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N. Hale and A. Townsend (2016) A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

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External Links:: ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)
Referenced by:: §18.16(iii).
Encodings:: BibTeX

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E. W. Hansen (1985) Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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External Links:: ISSN 0096-3518, FullText, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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P. Henrici (1986) Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.

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Notes:: Reprinted in 1993.
External Links:: ISBN 0-471-08703-3; 0-471-58986-1, MathReview (A. E. Heins), ZentralBlatt
Referenced by:: §1.14(v), Ch.3.
Encodings:: BibTeX

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F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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External Links:: ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(i).
Encodings:: BibTeX

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27: 6.16 Mathematical Applications

… ►Consider the Fourier series … ►Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … … ►

See accompanying text — Figure 6.16.2: The logarithmic integral $\operatorname{li}\left(x\right)$ , together with vertical bars indicating the value of $\pi(x)$ for $x=10,20,\dots,1000$ . Magnify

28: Bibliography Z

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D. Zeilberger and D. M. Bressoud (1985) A proof of Andrews’

q

-Dyson conjecture. Discrete Math. 54 (2), pp. 201–224.

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External Links:: ISSN 0012-365X, Document, MathReview (Ira Gessel), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

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A. H. Zemanian (1987) Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.

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Notes:: Reprint of 1965 McGraw-Hill Publication
External Links:: ISBN 0-486-65479-6, MathReview Entry
Referenced by:: §1.16(ix).
Encodings:: BibTeX

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Q. Zheng (1997) Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

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Referenced by:: §35.7(ii).
Encodings:: BibTeX

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Ya. M. Zhileĭkin and A. B. Kukarkin (1995) A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).

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Notes:: English translation in Comput. Math. Math. Phys. 35(7), pp. 901–905
External Links:: ISSN 0044-4669, MathReview (Vítĕzslav Veselý), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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29: Bibliography O

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F. Oberhettinger and T. P. Higgins (1961) Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

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Referenced by:: §1.14(viii).
Encodings:: BibTeX

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F. Oberhettinger (1990) Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).
Encodings:: BibTeX

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F. Oberhettinger (1972) Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.

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Notes:: Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.
External Links:: ISBN 0-387-05997-0, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(v), §10.43(v), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §13.10(v), §13.23(iii), §15.14, §18.17(ix), §8.14, §8.6(iii).
Encodings:: BibTeX

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F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §1.8(v), §22.11, §4.27.
Encodings:: BibTeX

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J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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30: 36 Integrals with Coalescing Saddles

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