Abel%E2%80%93Plana%20formula

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21: 22.7 Landen Transformations

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22: 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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23: 18.5 Explicit Representations

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§18.5(ii) Rodrigues Formulas

… ►Related formula: …See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials. … ►and two similar formulas by symmetry; compare the second row in Table 18.6.1. … ►For corresponding formulas for Chebyshev, Legendre, and the Hermite

\mathit{He}_{n}

polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …

24: 10.31 Power Series

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25: Bibliography M

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L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §14.17(i).
Encodings:: BibTeX

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S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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S. C. Milne (1997) Balanced

\sideset{{}_{3}}{{}_{2}}{\mathop{\Theta}}

summation theorems for

U(n)

basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:: ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt
Referenced by:: §17.15.
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

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L. J. Mordell (1917) On the representation of numbers as a sum of

2r

squares. Quarterly Journal of Math. 48, pp. 93–104.

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External Links:: ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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26: Bibliography O

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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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F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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External Links:: Document, MathReview (Anthony Leung), ZentralBlatt
Referenced by:: §2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).
Encodings:: BibTeX

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F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

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External Links:: Document, MathReview (W. Wasow), ZentralBlatt
Referenced by:: §2.8(v).
Encodings:: BibTeX

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C. Osácar, J. Palacián, and M. Palacios (1995) Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.

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External Links:: ISSN 0923-2958, Document, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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27: 12.14 The Function $W\left(a,x\right)$

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§12.14(iv) Connection Formula

… ►Other expansions, involving

\cos\left(\tfrac{1}{4}x^{2}\right)

and

\sin\left(\tfrac{1}{4}x^{2}\right)

, can be obtained from (12.4.3) to (12.4.6) by replacing

a

-ia

and

z

xe^{\ifrac{\pi i}{4}}

; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). …

28: 1.11 Zeros of Polynomials

… ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. …

29: Bibliography C

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L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

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External Links:: ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

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R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:: ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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D. Colton and R. Kress (1998) Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-62838-X, MathReview, ZentralBlatt
Referenced by:: §10.73(i).
Encodings:: BibTeX

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R. Cools (2003) An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.

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Notes:: Numerical integration and its complexity (Oberwolfach, 2001)
External Links:: ISSN 0885-064X, Document, Web Page, MathReview, ZentralBlatt
Referenced by:: §3.5(x).
Encodings:: BibTeX

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30: 10.34 Analytic Continuation

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