DLMF: Untitled Document

§28.7 Analytic Continuation of Eigenvalues

… ►The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …See also Mulholland and Goldstein (1929), Bouwkamp (1948), Meixner et al. (1980), Hunter and Guerrieri (1981), Hunter (1981), and Shivakumar and Xue (1999). … ► … ►

28.7.4

\sum_{n=0}^{\infty}\left(b_{2n+2}\left(q\right)-(2n+2)^{2}\right)=0.

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and $n$ : integer
Permalink:: http://dlmf.nist.gov/28.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.7 and Ch.28

… ►

L. Gårding (1947) The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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External Links:: FullText, MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §35.2, §35.3(ii).
Encodings:: BibTeX

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K. Goldberg, F. T. Leighton, M. Newman, and S. L. Zuckerman (1976) Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.

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External Links:: MathReview (M. S. Cheema), ZentralBlatt
Referenced by:: §26.21.
Encodings:: BibTeX

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S. Goldstein (1927) Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.

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External Links:: ZentralBlatt
Referenced by:: §28.26(i), §28.26(i), §28.8(iii).
Encodings:: BibTeX

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H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

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B. N. Gupta (1970) On Mill’s ratio. Proc. Cambridge Philos. Soc. 67, pp. 363–364.

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External Links:: MathReview (C. G. Esseen), ZentralBlatt
Referenced by:: §7.8.
Encodings:: BibTeX

…

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H. Maass (1971) Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:: Document, MathReview (K.-B. Gundlach), ZentralBlatt
Referenced by:: §35.4(ii).
Encodings:: BibTeX

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P. Martín, R. Pérez, and A. L. Guerrero (1992) Two-point quasi-fractional approximations to the Airy function

{\rm Ai}(x)

. J. Comput. Phys. 99 (2), pp. 337–340.

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Notes:: There is an error in Eq. (5): the second term in parentheses needs to be multiplied by $x$ .
External Links:: ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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P. Midy (1975) An improved calculation of the general elliptic integral of the second kind in the neighbourhood of

x=0

. Numer. Math. 25 (1), pp. 99–101.

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External Links:: ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

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D. S. Moak (1984) The

q

-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:: ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §5.18(ii).
Encodings:: BibTeX

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H. P. Mulholland and S. Goldstein (1929) The characteristic numbers of the Mathieu equation with purely imaginary parameter. Phil. Mag. Series 7 8 (53), pp. 834–840.

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External Links:: Document, ZentralBlatt
Referenced by:: §28.7.
Encodings:: BibTeX

…

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W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:: FullText, MathReview (D. P. Banerjee), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

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D. E. Amos (1990) Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:: Double- and single-precision Fortran, maximum accuracy 18S or 14S.
External Links:: ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt
Referenced by:: 1st item, 1st item.
Encodings:: BibTeX

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G. E. Andrews (1974) Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §17.16, Ch.17.
Encodings:: BibTeX

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T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:: §24.4(iii).
Encodings:: BibTeX

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M. J. Atia, A. Martínez-Finkelshtein, P. Martínez-González, and F. Thabet (2014) Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt
Referenced by:: §18.15(vi), §18.15(vi).
Encodings:: BibTeX

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10.31.2

K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z\right)+\gamma\right)I_{0}% \left(z\right)+\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}+(1+\tfrac{1}{2})\frac{(% \tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1% }{4}z^{2})^{3}}{(3!)^{2}}+\dotsi.

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Symbols:: $\gamma$ : Euler’s constant, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 9.6.13
Referenced by:: §10.45
Permalink:: http://dlmf.nist.gov/10.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.31 and Ch.10

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10.31.3

I_{\nu}\left(z\right)I_{\mu}\left(z\right)=(\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}% ^{\infty}\frac{{\left(\nu+\mu+k+1\right)_{k}}(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma% \left(\nu+k+1\right)\Gamma\left(\mu+k+1\right)}.

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.31
Permalink:: http://dlmf.nist.gov/10.31.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §10.31 and Ch.10

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K. W. J. Kadell (1988) A proof of Askey’s conjectured

q

-analogue of Selberg’s integral and a conjecture of Morris. SIAM J. Math. Anal. 19 (4), pp. 969–986.

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External Links:: ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.13.
Encodings:: BibTeX

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K. W. J. Kadell (1994) A proof of the

q

-Macdonald-Morris conjecture for

BC_{n}

. Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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External Links:: ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

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D. E. Knuth (1992) Two notes on notation. Amer. Math. Monthly 99 (5), pp. 403–422.

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External Links:: ISSN 0002-9890, FullText, MathReview (W. Dörfler), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations, Notations.
Encodings:: BibTeX

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K. S. Kölbig (1986) Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17 (5), pp. 1232–1258.

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External Links:: ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §25.12(ii).
Encodings:: BibTeX

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Koornwinder (website) Tom Koornwinder’s Personal Collection of Maple Procedures

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External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

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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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P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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

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P. I. Hadži (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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

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P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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D. R. Hartree (1936) Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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External Links:: ZentralBlatt
Referenced by:: §7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).
Encodings:: BibTeX

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D. F. Lawden (1989) Elliptic Functions and Applications. Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.

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External Links:: ISBN 0-387-96965-9, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.2(ii), §19.36(iii), §20.13, §20.15, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.5(i), §20.5(ii), §20.7(i), §20.7(ii), §20.7(iv), §20.7(vi), §20.7(vii), §20.7(vii), §20.7(viii), Ch.20, §22.10(i), §22.10(ii), §22.12, §22.13(i), §22.14(i), §22.14(ii), §22.14(iii), §22.14(iii), §22.15(i), §22.15(ii), §22.15(ii), §22.16(ii), §22.16(iii), §22.17(i), §22.18(i), §22.18(iii), §22.19(ii), §22.19(i), §22.19(i), §22.19(ii), §22.19(ii), §22.19(iv), §22.19(v), §22.2, §22.20(vii), §22.21, §22.4(i), §22.4(iii), §22.5(i), §22.5(ii), §22.6(ii), §22.6(iii), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), Ch.22, §23.10(i), §23.10(i), §23.10(iii), §23.10(iv), §23.12, §23.2(ii), §23.2(iii), §23.2(iii), §23.21(i), §23.3(i), §23.3(ii), §23.6(iv), §23.6(i), §23.6(i), §23.6(ii), §23.6(ii), §23.6(iii), §23.7, §23.8(i), §23.8(ii), §23.8(iii), Ch.23.
Encodings:: BibTeX

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D. H. Lehmer (1943) Ramanujan’s function

\tau(n)

. Duke Math. J. 10 (3), pp. 483–492.

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External Links:: Document, MathReview (H. S. Zuckerman), ZentralBlatt
Referenced by:: §27.20, §27.20, §27.21.
Encodings:: BibTeX

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D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

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S. Lewanowicz (1985) Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.

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External Links:: ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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L.-W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong (1998b) Microwave specific attenuation by oblate spheroidal raindrops: An exact analysis of TCS’s in terms of spheroidal wave functions. J. Electromagn. Waves Appl. 12 (6), pp. 709–711.

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External Links:: ISSN 0920-5071, Document, MathReview
Referenced by:: §30.14(v).
Encodings:: BibTeX

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\mathfrak{S}_{n}

denotes the set of permutations of

\{1,2,\ldots,n\}

.

\sigma\in\mathfrak{S}_{n}

is a one-to-one and onto mapping from

\{1,2,\ldots,n\}

to itself. … ►The number of elements of

\mathfrak{S}_{n}

with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

is given by (26.4.7). … ►The derangement number,

d(n)

, is the number of elements of

\mathfrak{S}_{n}

with no fixed points: … ►Given a permutation

\sigma\in\mathfrak{S}_{n}

, the inversion number of

\sigma

, denoted

\mathop{\mathrm{inv}}(\sigma)

, is the least number of adjacent transpositions required to represent

\sigma

. …

… ►

A. J. E. M. Janssen (2021) Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:: ISSN 1389-2355
External Links:: FullText
Referenced by:: §7.8, §7.8.
Encodings:: BibTeX

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M. Jimbo, T. Miwa, Y. Môri, and M. Sato (1980) Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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External Links:: ISSN 0167-2789, Document, MathReview, ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

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D. S. Jones and B. D. Sleeman (2003) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:: A second edition was published in 2010.
External Links:: ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt
Referenced by:: D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).
Encodings:: BibTeX

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D. S. Jones (1972) Asymptotic behavior of integrals. SIAM Rev. 14 (2), pp. 286–317.

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External Links:: Document, MathReview (A. Erdelyi), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

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B. R. Judd (1976) Modifications of Coulombic interactions by polarizable atoms. Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.

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External Links:: ISSN 0305-0041, Document, MathReview (Cataldo Agostinelli)
Referenced by:: §34.12.
Encodings:: BibTeX

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Goldstein%E2%80%99s

11—20 of 609 matching pages

11: 28.7 Analytic Continuation of Eigenvalues

§28.7 Analytic Continuation of Eigenvalues

12: Bibliography G

13: Bibliography M

14: Bibliography

15: 10.31 Power Series

16: Bibliography K

17: Bibliography H

18: Bibliography L

19: 26.13 Permutations: Cycle Notation

20: Bibliography J