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21: Bibliography B

… ►

P. Baldwin (1991) Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.

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External Links:: ISSN 0025-5793, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

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Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt
Referenced by:: §3.11(v).
Encodings:: BibTeX

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P. Bleher and A. Its (1999) Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

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External Links:: ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

A. A. Bogush and V. S. Otchik (1997) Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §31.13.
Encodings:: BibTeX

…

22: 16.23 Mathematical Applications

… ►A variety of problems in classical mechanics and mathematical physics lead to Picard–Fuchs equations. … ►In the proof of this conjecture de Branges (1985) uses the inequality …

23: 27.11 Asymptotic Formulas: Partial Sums

§27.11 Asymptotic Formulas: Partial Sums

… ►

27.11.2

\sum_{n\leq x}d\left(n\right)=x\ln x+(2\gamma-1)x+O\left(\sqrt{x}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $x$ : real number
A&S Ref:: 24.3.3 III
Referenced by:: §27.11, §27.11, Erratum (V1.1.8) for Section 27.11
Permalink:: http://dlmf.nist.gov/27.11.E2
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

… ►Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number

\theta_{0}

such that the error term in (27.11.2) is

O\left(x^{\theta}\right)

for all

\theta>\theta_{0}

. … ►The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if

\left(h,k\right)=1

, then the number of primes

p\leq x

with

p\equiv h\pmod{k}

is asymptotic to

x/(\phi\left(k\right)\ln x)

x\to\infty

24: Bibliography V

… ►

G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:: ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt
Referenced by:: §31.10(i).
Encodings:: BibTeX

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H. C. van de Hulst (1957) Light Scattering by Small Particles. John Wiley and Sons. Inc., New York.

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External Links:: MathReview (P. M. Morse)
Referenced by:: §9.16.
Encodings:: BibTeX

►

H. C. van de Hulst (1980) Multiple Light Scattering. Vol. 1, Academic Press, New York.

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Referenced by:: §6.17.
Encodings:: BibTeX

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J. van de Lune, H. J. J. te Riele, and D. T. Winter (1986) On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (174), pp. 667–681.

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External Links:: ISSN 0025-5718, FullText, Document, MathReview (Jürgen G. Hinz), ZentralBlatt
Referenced by:: §25.10(ii), §25.18(ii).
Encodings:: BibTeX

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H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.12, §19.5.
Encodings:: BibTeX

…

25: Bibliography S

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R. Shail (1978) Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.

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External Links:: ISSN 0020-7225, Document, MathReview, ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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J. A. Shohat and J. D. Tamarkin (1970) The Problem of Moments. 4th edition, American Mathematical Society Mathematical Surveys, Vol. 1, American Mathematical Society, Providence, RI.

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External Links:: MathReview (R. P. Boas, Jr.), ZentralBlatt
Referenced by:: §1.16(ix), §1.4(v), §18.2(viii), (18.40.7), §18.40(ii), §18.40(ii).
Encodings:: BibTeX

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R. Sips (1949) Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales. Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).

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External Links:: FullText, MathReview (M. Strutt), ZentralBlatt
Referenced by:: §28.8(ii).
Encodings:: BibTeX

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B. D. Sleeman (1966a) Some Boundary Value Problems Associated with the Heun Equation. Ph.D. Thesis, London University.

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Notes:: Available from the library, University of Surrey, UK
External Links:: LibraryRecord
Referenced by:: §31.9(i), §31.9(ii), Ch.31, Profile Brian D. Sleeman.
Encodings:: BibTeX

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I. N. Sneddon (1966) Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.

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External Links:: MathReview (R. Shail), ZentralBlatt
Referenced by:: §10.22(iv).
Encodings:: BibTeX

…

26: Bibliography G

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R. D. M. Garashchuk and J. C. Light (2001) Quasirandom distributed bases for bound problems. J. Chem. Phys. 114 (9), pp. 3929–3939.

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Referenced by:: §18.39(iii).
Encodings:: BibTeX

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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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A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:: Problem proposed by P. Smith.
External Links:: Document
Referenced by:: (9.10.14), (9.10.15), (9.10.16), §9.10(v).
Encodings:: BibTeX

… ►

D. Gómez-Ullate, N. Kamran, and R. Milson (2009) An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 359 (1), pp. 352–367.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Li Zhong Peng)
Referenced by:: §18.36(vi).
Encodings:: BibTeX

… ►

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

…

27: Bibliography

… ►

A. R. Ahmadi and S. E. Widnall (1985) Unsteady lifting-line theory as a singular-perturbation problem. J. Fluid Mech 153, pp. 59–81.

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External Links:: Document, ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

… ►

H. Airault, H. P. McKean, and J. Moser (1977) Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:: MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §23.21(ii).
Encodings:: BibTeX

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N. I. Akhiezer (2021) The classical moment problem and some related questions in analysis. Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:: Reprint of the 1965 edition [ 0184042], Translated by N. Kemmer, With a foreword by H. J. Landau
External Links:: ISBN 978-1-611976-38-0, MathReview Entry
Referenced by:: §18.40(ii).
Encodings:: BibTeX

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S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

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U. M. Ascher, R. M. M. Mattheij, and R. D. Russell (1995) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:: Corrected reprint of the 1988 original
External Links:: ISBN 0-89871-354-4, MathReview, ZentralBlatt
Referenced by:: §3.7(iii).
Encodings:: BibTeX

…

28: Bibliography I

… ►

Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of

J_{0}(z)-iJ_{1}(z)

and of Bessel functions

J_{m}(z)

of any real order

m

. Linear Algebra Appl. 194, pp. 35–70.

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External Links:: ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

… ►

A. Iserles, S. P. Nørsett, and S. Olver (2006) Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

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Notes:: Proceedings of ENuMath, Santiago de Compostela (2005)
External Links:: ISBN 3-540-34287-7, MathReview, ZentralBlatt
Referenced by:: §3.5(vii).
Encodings:: BibTeX

…

29: 18.38 Mathematical Applications

… ►also the case

\beta=0

of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. See de Branges (1985). … ►

Riemann–Hilbert Problems

…

30: 16.25 Methods of Computation

… ►Instead a boundary-value problem needs to be formulated and solved. …