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  • L. Saalschütz (1893)
    Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen,
    Springer-Verlag, Berlin (German).
    Show Annotations
    External Links:
    ZentralBlatt.
    Referenced by:
    ‣ §24.1.
    Encodings:
    BibTeX
  • A. Sachse (1882)
    Über die Darstellung der Bernoullischen und Eulerschen Zahlen durch Determinanten,
    Archiv für Mathematik und Physik 68, pp. 427–432 (German).
    Show Annotations
    External Links:
    ZentralBlatt.
    Referenced by:
    §24.14(ii).
    Encodings:
    BibTeX
  • B. E. Sagan (2001)
    The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions,
    2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.
    Show Annotations
    Notes:
    First edition published by Wadsworth & Brooks/Cole Advanced Books & Software, 1991.
    External Links:
    ISBN 0-387-95067-2, MathReview, ZentralBlatt.
    Referenced by:
    §26.19.
    Encodings:
    BibTeX
  • R. P. Sagar (1991a)
    A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals,
    Comput. Phys. Comm. 66 (2-3), pp. 271–275.
    Show Annotations
    External Links:
    ISSN 0010-4655, Document, MathReview, ZentralBlatt.
    Referenced by:
    §25.18(i).
    Encodings:
    BibTeX
  • R. P. Sagar (1991b)
    On the evaluation of the Fermi-Dirac integrals,
    Astrophys. J. 376 (1, part 1), pp. 364–366.
    Show Annotations
    External Links:
    ISSN 0004-637X, Document, MathReview (De Hui Chen).
    Referenced by:
    §25.18(i).
    Encodings:
    BibTeX
  • SAGE (free interactive system)
    Show Annotations
    Notes:
    SAGE is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface.
    External Links:
    Home Page.
    Referenced by:
    § ‣ § Software Cross Index.
    Encodings:
    BibTeX
  • H. Sakai (2001)
    Rational surfaces associated with affine root systems and geometry of the Painlevé equations,
    Comm. Math. Phys. 220 (1), pp. 165–229.
    Show Annotations
    External Links:
    ISSN 0010-3616, Document, MathReview (I. Dolgachev), ZentralBlatt.
    Referenced by:
    §32.7(viii).
    Encodings:
    BibTeX
  • K. L. Sala (1989)
    Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean,
    SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
    Show Annotations
    External Links:
    ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt.
    Referenced by:
    §22.19(i), §22.20(vi).
    Encodings:
    BibTeX
  • L. Z. Salchev and V. B. Popov (1976)
    A property of the zeros of cross-product Bessel functions of different orders,
    Z. Angew. Math. Mech. 56 (2), pp. 120–121.
    Show Annotations
    External Links:
    Document, MathReview (S. Saran), ZentralBlatt.
    Referenced by:
    §10.21(x).
    Encodings:
    BibTeX
  • H. E. Salzer (1955)
    Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms,
    Math. Tables Aids Comput. 9 (52), pp. 164–177.
    Show Annotations
    External Links:
    FullText, MathReview (S. C. van Veen), ZentralBlatt.
    Referenced by:
    §3.5(viii).
    Encodings:
    BibTeX
  • S. Sandström and C. Ackrén (2007)
    Note on the complex zeros of H^{{\prime}}_{\nu}(x)+i\zeta H_{\nu}(x)=0,
    J. Comput. Appl. Math. 201 (1), pp. 3–7.
    Show Annotations
    External Links:
    ISSN 0377-0427, Document, FullText, MathReview Entry, ZentralBlatt.
    Referenced by:
    §10.21(xiv), §10.21(xiv).
    Encodings:
    BibTeX
  • P. Sarnak (1999)
    Quantum Chaos, Symmetry and Zeta Functions. Lecture I, Quantum Chaos,
    in Current Developments in Mathematics, 1997 (Cambridge, MA),
    pp. 127–144.
    Show Annotations
    External Links:
    MathReview (Shin-ya Koyama), ZentralBlatt.
    Referenced by:
    §25.17.
    Encodings:
    BibTeX
  • C. W. Schelin (1983)
    Calculator function approximation,
    Amer. Math. Monthly 90 (5), pp. 317–325.
    Show Annotations
    External Links:
    ISSN 0002-9890, FullText, MathReview (J. Albrycht), ZentralBlatt.
    Referenced by:
    ‣ §4.45(i).
    Encodings:
    BibTeX
  • J. L. Schiff (1999)
    The Laplace Transform: Theory and Applications,
    Undergraduate Texts in Mathematics, Springer-Verlag, New York.
    Show Annotations
    External Links:
    ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt.
    Referenced by:
    §1.14(iii), §1.14(vii).
    Encodings:
    BibTeX
  • T. Schmelzer and L. N. Trefethen (2007)
    Computing the gamma function using contour integrals and rational approximations,
    SIAM J. Numer. Anal. 45 (2), pp. 558–571.
    Show Annotations
    External Links:
    ISSN 0036-1429, Document, MathReview (Amparo Gil), ZentralBlatt.
    Referenced by:
    §5.21, §5.23(iii).
    Encodings:
    BibTeX
  • D. Schmidt and G. Wolf (1979)
    A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations,
    SIAM J. Math. Anal. 10 (4), pp. 823–838.
    Show Annotations
    External Links:
    ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt.
    Referenced by:
    §28.32(i).
    Encodings:
    BibTeX
  • D. Schmidt (1979)
    Die Lösung der linearen Differentialgleichung 2. Ordnung um zwei einfache Singularitäten durch Reihen nach hypergeometrischen Funktionen,
    J. Reine Angew. Math. 309, pp. 127–148.
    Show Annotations
    External Links:
    ISSN 0075-4102, MathReview (J. Nikolaus), ZentralBlatt.
    Referenced by:
    §31.11(v), §31.6, §31.9(ii).
    Encodings:
    BibTeX
  • I. J. Schoenberg (1971)
    Norm inequalities for a certain class of {C}^{{\infty}} functions,
    Israel J. Math. 10, pp. 364–372.
    Show Annotations
    External Links:
    Document, MathReview (V. Ganapathy Iyer), ZentralBlatt.
    Referenced by:
    ‣ §24.17(ii).
    Encodings:
    BibTeX
  • I. J. Schoenberg (1973)
    Cardinal Spline Interpolation,
    Society for Industrial and Applied Mathematics, Philadelphia, PA.
    Show Annotations
    Notes:
    Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12
    External Links:
    MathReview (H. L. Loeb), ZentralBlatt.
    Referenced by:
    ‣ §24.17(ii).
    Encodings:
    BibTeX
  • L. Schoenfeld (1976)
    Sharper bounds for the Chebyshev functions \theta(x) and \psi(x). II,
    Math. Comp. 30 (134), pp. 337–360.
    Show Annotations
    Notes:
    For corrigendum see same journal v. 30 (1976), no. 136, p. 900
    External Links:
    FullText, Document, MathReview (R. P. Brent), ZentralBlatt.
    Referenced by:
    §27.12.
    Encodings:
    BibTeX
  • J. L. Schonfelder (1978)
    Chebyshev expansions for the error and related functions,
    Math. Comp. 32 (144), pp. 1232–1240.
    Show Annotations
    External Links:
    FullText, Document, MathReview (L. Fox), ZentralBlatt.
    Referenced by:
    §7.24(ii).
    Encodings:
    BibTeX
  • J. L. Schonfelder (1980)
    Very high accuracy Chebyshev expansions for the basic trigonometric functions,
    Math. Comp. 34 (149), pp. 237–244.
    Show Annotations
    External Links:
    ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt.
    Referenced by:
    §4.47(i).
    Encodings:
    BibTeX
  • F. Schottky (1903)
    Über die Moduln der Thetafunctionen,
    Acta Math. 27 (1), pp. 235–288.
    Show Annotations
    External Links:
    ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt.
    Referenced by:
    §21.9.
    Encodings:
    BibTeX
  • M. R. Schroeder (2006)
    Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity,
    4th edition, Springer-Verlag, Berlin.
    Show Annotations
    Notes:
    Volume 7 in Springer Series in Information Sciences.
    External Links:
    ZentralBlatt.
    Referenced by:
    §27.16, §27.17.
    Encodings:
    BibTeX
  • L. S. Schulman (1981)
    Techniques and Applications of Path Integration,
    John Wiley & Sons Inc., New York.
    Show Annotations
    Notes:
    Reprinted by Dover in 2005.
    External Links:
    ISBN 0-471-76450-7, MathReview (Aubrey Truman), ZentralBlatt.
    Referenced by:
    §20.13, §22.19(ii).
    Encodings:
    BibTeX
  • K. Schulten and R. G. Gordon (1976)
    Recursive evaluation of 3j- and 6j- coefficients,
    Comput. Phys. Comm. 11 (2), pp. 269–278.
    Show Annotations
    Notes:
    Double-precision Fortran provided in 3 parts, j_{1}-recursion for 3j-coefficients, m_{2}-recursion for 3j-coefficients, and j_{1}-recursion for 6j-coefficients
    External Links:
    Document, Code 1, Code 2, Code 3.
    Referenced by:
    §34.15.
    Encodings:
    BibTeX
  • K. Schulten and R. G. Gordon (1975a)
    Exact recursive evaluation of 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta,
    J. Mathematical Phys. 16 (10), pp. 1961–1970.
    Show Annotations
    External Links:
    Document, MathReview (J.-L. Calais).
    Referenced by:
    §34.13, §34.3(iii).
    Encodings:
    BibTeX
  • K. Schulten and R. G. Gordon (1975b)
    Semiclassical approximations to 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta,
    J. Mathematical Phys. 16 (10), pp. 1971–1988.
    Show Annotations
    External Links:
    Document, MathReview (J.-L. Calais).
    Referenced by:
    §34.8.
    Encodings:
    BibTeX
  • Z. Schulten, D. G. M. Anderson and R. G. Gordon (1979)
    An algorithm for the evaluation of the complex Airy functions,
    J. Comput. Phys. 31 (1), pp. 60–75.
    Show Annotations
    External Links:
    ISSN 0021-9991, Document, MathReview, ZentralBlatt.
    Referenced by:
    §9.10(vii), §9.17(iii).
    Encodings:
    BibTeX
  • L. L. Schumaker (1981)
    Spline Functions: Basic Theory,
    John Wiley & Sons Inc., New York.
    Show Annotations
    Notes:
    Pure and Applied Mathematics, A Wiley-Interscience Publication. Reprinted by Robert E. Krieger Publishing Co. Inc., 1993.
    External Links:
    ISBN 0-471-76475-2, MathReview (D. D. Stancu), ZentralBlatt.
    Referenced by:
    ‣ §24.17(ii), §3.11(vi).
    Encodings:
    BibTeX
  • I. J. Schwatt (1962)
    An Introduction to the Operations with Series,
    2nd edition, Chelsea Publishing Co., New York.
    Show Annotations
    Notes:
    This is a reprint, with corrections, of the first edition [University of Pennsylvania Press, Philadelphia, Pa., 1924].
    External Links:
    MathReview, ZentralBlatt.
    Referenced by:
    ‣ §24.1, §24.6.
    Encodings:
    BibTeX
  • T. Schäfer and E. V. Shuryak (1998)
    Instantons in QCD,
    Rev. Modern Phys. 70 (2), pp. 323–425.
    Show Annotations
    External Links:
    ISSN 0034-6861, Document, MathReview.
    Referenced by:
    §22.19(ii).
    Encodings:
    BibTeX
  • F. W. Schäfke and A. Finsterer (1990)
    On Lindelöf’s error bound for Stirling’s series,
    J. Reine Angew. Math. 404, pp. 135–139.
    Show Annotations
    External Links:
    ISSN 0075-4102, MathReview (Bruce C. Berndt), ZentralBlatt.
    Referenced by:
    §5.11(ii).
    Encodings:
    BibTeX
  • F. W. Schäfke and D. Schmidt (1966)
    Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung III,
    Numer. Math. 8 (1), pp. 68–71.
    Show Annotations
    Notes:
    Includes Algol program
    External Links:
    ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt.
    Referenced by:
    §28.36(ii).
    Encodings:
    BibTeX
  • F. W. Schäfke (1983)
    Über einige Integrale mit Produkten von Mathieu-Funktionen,
    Arch. Math. (Basel) 41 (2), pp. 152–162.
    Show Annotations
    External Links:
    ISSN 0003-889X, Document, MathReview (J. Meixner), ZentralBlatt.
    Referenced by:
    §28.28(iii), §28.28(iv).
    Encodings:
    BibTeX
  • F. W. Schäfke and H. Groh (1962)
    Zur Berechnung der Eigenwerte der Sphäroiddifferentialgleichung.,
    Numer. Math. 4, pp. 310–312 (German).
    Show Annotations
    External Links:
    ISSN 0029-599X, Document, MathReview (I. Marx), ZentralBlatt.
    Referenced by:
    §30.16(i).
    Encodings:
    BibTeX
  • F. W. Schäfke (1960)
    Reihenentwicklungen analytischer Funktionen nach Biorthogonalsystemen spezieller Funktionen. I,
    Math. Z. 74, pp. 436–470.
    Show Annotations
    External Links:
    ISSN 0025-5874, Document, MathReview (C. J. Bouwkamp).
    Referenced by:
    ‣ §10.23(iii), §28.30(ii).
    Encodings:
    BibTeX
  • F. W. Schäfke (1961a)
    Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung I,
    Numer. Math. 3 (1), pp. 30–38.
    Show Annotations
    External Links:
    ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt.
    Referenced by:
    §28.34(i).
    Encodings:
    BibTeX
  • F. W. Schäfke (1961b)
    Reihenentwicklungen analytischer Funktionen nach Biorthogonalsystemen spezieller Funktionen. II,
    Math. Z. 75, pp. 154–191.
    Show Annotations
    External Links:
    ISSN 0025-5874, Document, MathReview (C. J. Bouwkamp), ZentralBlatt.
    Referenced by:
    ‣ §10.23(iii), §13.24(i), §14.18(i), §28.30(ii).
    Encodings:
    BibTeX
  • R. Schürer (2004)
    Adaptive Quasi-Monte Carlo Integration Based on MISER and VEGAS,
    in Monte Carlo and Quasi-Monte Carlo Methods 2002,
    pp. 393–406.
    Show Annotations
    External Links:
    MathReview Entry, ZentralBlatt.
    Referenced by:
    §3.5(x).
    Encodings:
    BibTeX
  • R. S. Scorer (1950)
    Numerical evaluation of integrals of the form I=\int _{{x_{1}}}^{{x_{2}}}f(x)e^{{i\phi(x)}}dx and the tabulation of the function \mathop{\mathrm{Gi}\/}\nolimits(z)=(1/\pi)\int _{0}^{{\infty}}\mathop{\sin\/}\nolimits(uz+\tfrac{1}{3}u^{3})du,
    Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.
    Show Annotations
    External Links:
    Document, MathReview (S. C. van Veen), ZentralBlatt.
    Referenced by:
    §9.18(vi).
    Encodings:
    BibTeX
  • J. B. Seaborn (1991)
    Hypergeometric Functions and Their Applications,
    Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.
    Show Annotations
    External Links:
    ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt.
    Referenced by:
    §18.39(i), §18.39(i), §18.39(i).
    Encodings:
    BibTeX
  • M. J. Seaton and G. Peach (1962)
    The determination of phases of wave functions,
    Proc. Phys. Soc. 79 (6), pp. 1296–1297.
    Show Annotations
    External Links:
    Document, ZentralBlatt.
    Referenced by:
    §33.23(vii).
    Encodings:
    BibTeX
  • M. J. Seaton (1982)
    Coulomb functions analytic in the energy,
    Comput. Phys. Comm. 25 (1), pp. 87–95.
    Show Annotations
    Notes:
    Double-precision Fortran code.
    External Links:
    Document, Code.
    Referenced by:
    §33.1, §33.14(iv), §33.26(ii).
    Encodings:
    BibTeX
  • M. J. Seaton (1983)
    Quantum defect theory,
    Rep. Prog. Phys. 46 (2), pp. 167–257.
    Show Annotations
    External Links:
    Document.
    Referenced by:
    §33.14(i), ‣ §33.22(ii).
    Encodings:
    BibTeX
  • M. J. Seaton (1984)
    The accuracy of iterated JWBK approximations for Coulomb radial functions,
    Comput. Phys. Comm. 32 (2), pp. 115–119.
    Show Annotations
    External Links:
    Document.
    Referenced by:
    §33.23(vii).
    Encodings:
    BibTeX
  • M. J. Seaton (2002a)
    Coulomb functions for attractive and repulsive potentials and for positive and negative energies,
    Comput. Phys. Comm. 146 (2), pp. 225–249.
  • M. J. Seaton (2002b)
    FGH, a code for the calculation of Coulomb radial wave functions from series expansions,
    Comput. Phys. Comm. 146 (2), pp. 250–253.
    Show Annotations
    Notes:
    Single-precision Fortran code.
    External Links:
    Document, Code, ZentralBlatt.
    Referenced by:
    §33.26(ii).
    Encodings:
    BibTeX
  • M. J. Seaton (2002c)
    NUMER, a code for Numerov integrations of Coulomb functions,
    Comput. Phys. Comm. 146 (2), pp. 254–260.
    Show Annotations
    Notes:
    Single-precision Fortran code.
    External Links:
    Document, Code, ZentralBlatt.
    Referenced by:
    §33.26(ii).
    Encodings:
    BibTeX
  • J. D. Secada (1999)
    Numerical evaluation of the Hankel transform,
    Comput. Phys. Comm. 116 (2-3), pp. 278–294.
    Show Annotations
    External Links:
    ISSN 0010-4655, Document, MathReview Entry, ZentralBlatt.
    Referenced by:
    ‣ §10.74(vii).
    Encodings:
    BibTeX
  • H. Segur and M. J. Ablowitz (1981)
    Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent,
    Phys. D 3 (1-2), pp. 165–184.
    Show Annotations
    External Links:
    Document.
    Referenced by:
    §32.11(ii).
    Encodings:
    BibTeX
  • J. Segura, P. Fernández de Córdoba and Yu. L. Ratis (1997)
    A code to evaluate modified Bessel functions based on the continued fraction method,
    Comput. Phys. Comm. 105 (2-3), pp. 263–272.
    Show Annotations
    External Links:
    ISSN 0010-4655, Document, Code, MathReview.
    Referenced by:
    §10.77(ii).
    Encodings:
    BibTeX
  • J. Segura and A. Gil (1998)
    Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments,
    Comput. Phys. Comm. 115 (1), pp. 69–86.
    Show Annotations
    Notes:
    Double-precision Fortran, minimum accuracy 14S, maximum accuracy 18D.
    External Links:
    ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt.
    Referenced by:
    §12.21(ii).
    Encodings:
    BibTeX
  • J. Segura and A. Gil (1999)
    Evaluation of associated Legendre functions off the cut and parabolic cylinder functions,
    Electron. Trans. Numer. Anal. 9, pp. 137–146.
    Show Annotations
    Notes:
    Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)
    External Links:
    ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt.
    Referenced by:
    §14.30(i), §14.32.
    Encodings:
    BibTeX
  • J. Segura (1998)
    A global Newton method for the zeros of cylinder functions,
    Numer. Algorithms 18 (3-4), pp. 259–276.
    Show Annotations
    External Links:
    ISSN 1017-1398, Document, MathReview, ZentralBlatt.
    Referenced by:
    §10.74(vi).
    Encodings:
    BibTeX
  • J. Segura (2001)
    Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros,
    Math. Comp. 70 (235), pp. 1205–1220.
    Show Annotations
    External Links:
    ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt.
    Referenced by:
    §10.74(vi).
    Encodings:
    BibTeX
  • J. Segura (2002)
    The zeros of special functions from a fixed point method,
    SIAM J. Numer. Anal. 40 (1), pp. 114–133.
    Show Annotations
    External Links:
    ISSN 1095-7170, Document, MathReview, ZentralBlatt.
    Referenced by:
    §3.8(v).
    Encodings:
    BibTeX
  • J. Segura (2008)
    Interlacing of the zeros of contiguous hypergeometric functions,
    Numer. Algorithms 49 (1-4), pp. 387–407.
    Show Annotations
    External Links:
    ISSN 1017-1398, Document, FullText, MathReview Entry, ZentralBlatt.
    Referenced by:
    §13.9(i), §13.9(i), §13.9(ii), §13.9(ii), §15.13, §15.13.
    Encodings:
    BibTeX
  • N. Seiberg and D. Shih (2005)
    Flux vacua and branes of the minimal superstring,
    J. High Energy Phys..
    Show Annotations
    Notes:
    Electronic. E-print number: hep-th/0412315. 38 pp.
    External Links:
    ISSN 1126-6708, Document, MathReview (S. Yu. Alexandrov).
    Referenced by:
    ‣ §32.16.
    Encodings:
    BibTeX
  • R. G. Selfridge and J. E. Maxfield (1958)
    A Table of the Incomplete Elliptic Integral of the Third Kind,
    Dover Publications Inc., New York.
    Show Annotations
    External Links:
    MathReview (C. J. Bouwkamp).
    Referenced by:
    ‣ §19.37(iii).
    Encodings:
    BibTeX
  • J.-P. Serre (1973)
    A Course in Arithmetic,
    Graduate Texts in Mathematics, Vol. 7, Springer-Verlag, New York.
    Show Annotations
    Notes:
    Translated from the French.
    External Links:
    MathReview, ZentralBlatt.
    Referenced by:
    §20.12(i), §20.7(viii), §20.9(i), §20.9(ii), §22.18(iv), §23.15(i), §23.18, §23.19, §23.20(v).
    Encodings:
    BibTeX
  • R. Shail (1978)
    Lamé polynomial solutions to some elliptic crack and punch problems,
    Internat. J. Engrg. Sci. 16 (8), pp. 551–563.
    Show Annotations
    External Links:
    ISSN 0020-7225, Document, MathReview, ZentralBlatt.
    Referenced by:
    §29.19(ii).
    Encodings:
    BibTeX
  • R. Shail (1980)
    On integral representations for Lamé and other special functions,
    SIAM J. Math. Anal. 11 (4), pp. 702–723.
    Show Annotations
    External Links:
    ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt.
    Referenced by:
    §29.17(i), §29.8.
    Encodings:
    BibTeX
  • H. Shanker (1939)
    On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions,
    J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
    Show Annotations
    External Links:
    ZentralBlatt.
    Referenced by:
    §12.13(i).
    Encodings:
    BibTeX
  • H. Shanker (1940a)
    On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series,
    J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
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  • H. Shanker (1940b)
    On certain integrals and expansions involving Weber’s parabolic cylinder functions,
    J. Indian Math. Soc. (N. S.) 4, pp. 158–166.
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    External Links:
    MathReview (M. C. Gray), ZentralBlatt.
    Referenced by:
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  • H. Shanker (1940c)
    On the expansion of the product of two parabolic cylinder functions of non integral order,
    Proc. Benares Math. Soc. (N. S.) 2, pp. 61–68.
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    External Links:
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    Referenced by:
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  • D. Shanks (1955)
    Non-linear transformations of divergent and slowly convergent sequences,
    J. Math. Phys. 34, pp. 1–42.
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    External Links:
    MathReview (J. Kuntzmann), ZentralBlatt.
    Referenced by:
    §17.18.
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  • G. Shanmugam (1978)
    Parabolic Cylinder Functions and their Application in Symmetric Two-centre Shell Model,
    Matscience Rep., Vol. 91, Aarhus, pp. P81–P89.
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    External Links:
    MathReview.
    Referenced by:
    §12.17.
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  • J. Shao and P. Hänggi (1998)
    Decoherent dynamics of a two-level system coupled to a sea of spins,
    Phys. Rev. Lett. 81 (26), pp. 5710–5713.
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    External Links:
    Document.
    Referenced by:
    §11.12.
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  • J. Shapiro (1970)
    Arbitrary 3n-j symbols for \rm{SU}(2),
    Comput. Phys. Comm. 1 (3), pp. 207–215.
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    Notes:
    Single-precision Fortran
    External Links:
    Document, Code.
    Referenced by:
    §34.15.
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  • O. A. Sharafeddin, H. F. Bowen, D. J. Kouri and D. K. Hoffman (1992)
    Numerical evaluation of spherical Bessel transforms via fast Fourier transforms,
    J. Comput. Phys. 100 (2), pp. 294–296.
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    External Links:
    ISSN 0021-9991, Document, MathReview, ZentralBlatt.
    Referenced by:
    ‣ §10.74(vii).
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  • A. Sharples (1967)
    Uniform asymptotic forms of modified Mathieu functions,
    Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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    External Links:
    Document, MathReview (F. W. J. Olver), ZentralBlatt.
    Referenced by:
    ‣ §28.8(iv).
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  • A. Sharples (1971)
    Uniform asymptotic expansions of modified Mathieu functions,
    J. Reine Angew. Math. 247, pp. 1–17.
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    External Links:
    MathReview (J. Meixner), ZentralBlatt.
    Referenced by:
    ‣ §28.8(iv).
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  • I. Shavitt and M. Karplus (1965)
    Gaussian-transform method for molecular integrals. I. Formulation for energy integrals,
    J. Chem. Phys. 43 (2), pp. 398–414.
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    External Links:
    Document, MathReview (R. K. Nesbet).
    Referenced by:
    §8.24(i).
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  • I. Shavitt (1963)
    The Gaussian Function in Calculations of Statistical Mechanics and Quantum Mechanics,
    in Methods in Computational Physics: Advances in Research and Applications,
    (S. Fernbach and M. Rotenberg Eds.), Vol. 2, pp. 1–45.
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    Referenced by:
    §8.24(i), §8.24(iii).
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  • D. C. Shaw (1985)
    Perturbational results for diffraction of water-waves by nearly-vertical barriers,
    IMA J. Appl. Math. 34 (1), pp. 99–117.
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    External Links:
    ISSN 0272-4960, Document, MathReview, ZentralBlatt.
    Referenced by:
    §11.12.
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  • N. T. Shawagfeh (1992)
    The Laplace transforms of products of Airy functions,
    Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.
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    External Links:
    ISSN 0253-424X, MathReview.
    Referenced by:
    §9.10(v).
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  • B. L. Shea (1988)
    Algorithm AS 239. Chi-squared and incomplete gamma integral,
    Appl. Statist. 37 (3), pp. 466–473.
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    Notes:
    Double-precision Fortran.
    External Links:
    FullText, Code.
    Referenced by:
    §8.28(ii).
    Encodings:
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  • L. Shen (1981)
    The elliptical microstrip antenna with circular polarization,
    IEEE Trans. Antennas and Propagation 29 (1), pp. 90–94.
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    External Links:
    FullText.
    Referenced by:
    §28.33(ii).
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  • L. Shen (1998)
    On an identity of Ramanujan based on the hypergeometric series \mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\frac{1}{3},\frac{2}{3};\frac{1}{2};x\right),
    J. Number Theory 69 (2), pp. 125–134.
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    External Links:
    ISSN 0022-314X, Document, MathReview (Bruce C. Berndt), ZentralBlatt.
    Referenced by:
    §20.11(iii).
    Encodings:
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  • M. M. Shepherd and J. G. Laframboise (1981)
    Chebyshev approximation of (1+2x)\,\mathop{\exp\/}\nolimits(x^{{2}})\,\mathrm{erfc}\, x in 0\leq x<\infty,
    Math. Comp. 36 (153), pp. 249–253.
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    External Links:
    ISSN 0025-5718, FullText, Document, MathReview (John P. Coleman), ZentralBlatt.
    Referenced by:
    §7.24(ii).
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  • M. E. Sherry (1959)
    The zeros and maxima of the Airy function and its first derivative to 25 significant figures,
    Report AFCRC-TR-59-135, ASTIA Document No. AD214568
    Air Research and Development Command, U.S. Air Force, Bedford, MA.
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    Notes:
    Available from U.S. Department of Commerce, Office of Technical Services, Washington, D.C.
    Referenced by:
    §9.18(iv), §9.8(iv).
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  • G. Shimura (1982)
    Confluent hypergeometric functions on tube domains,
    Math. Ann. 260 (3), pp. 269–302.
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    External Links:
    ISSN 0025-5831, Document, MathReview (A. B. Venkov), ZentralBlatt.
    Referenced by:
    §35.6(ii).
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  • T. Shiota (1986)
    Characterization of Jacobian varieties in terms of soliton equations,
    Invent. Math. 83 (2), pp. 333–382.
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    External Links:
    ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt.
    Referenced by:
    §21.9.
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  • R. B. Shirts (1993a)
    The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order,
    ACM Trans. Math. Software 19 (3), pp. 377–390.
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    External Links:
    ISSN 0098-3500, Document, ZentralBlatt.
    Referenced by:
    §28.34(ii).
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  • R. B. Shirts (1993b)
    Algorithm 721: MTIEU1 and MTIEU2: Two subroutines to compute eigenvalues and solutions to Mathieu’s differential equation for noninteger and integer order,
    ACM Trans. Math. Software 19 (3), pp. 391–406.
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    Notes:
    Double-precision Fortran, maximum accuracy 18D
    External Links:
    ISSN 0098-3500, Document, TOMS module 721 (in GAMS), ZentralBlatt.
    Referenced by:
    §28.36(ii), §28.36(iii).
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  • P. N. Shivakumar and R. Wong (1988)
    Error bounds for a uniform asymptotic expansion of the Legendre function \mathop{P^{{-m}}_{{n}}\/}\nolimits\!\left({\rm cosh}\ z\right),
    Quart. Appl. Math. 46 (3), pp. 473–488.
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    External Links:
    ISSN 0033-569X, MathReview (F. W. J. Olver), ZentralBlatt.
    Referenced by:
    §14.26.
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  • P. N. Shivakumar and J. Xue (1999)
    On the double points of a Mathieu equation,
    J. Comput. Appl. Math. 107 (1), pp. 111–125.
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    External Links:
    ISSN 0377-0427, Document, MathReview, ZentralBlatt.
    Referenced by:
    §28.7.
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  • B. W. Shore and D. H. Menzel (1968)
    Principles of Atomic Spectra,
    John Wiley & Sons Ltd., New York.
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    External Links:
    ISBN 0-471-78835-X.
    Referenced by:
    §34.12.
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  • A. Sidi (1997)
    Computation of infinite integrals involving Bessel functions of arbitrary order by the \overline{D}-transformation,
    J. Comput. Appl. Math. 78 (1), pp. 125–130.
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    External Links:
    ISSN 0377-0427, Document, MathReview, ZentralBlatt.
    Referenced by:
    ‣ §10.74(vii).
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  • A. Sidi (2003)
    Practical Extrapolation Methods: Theory and Applications,
    Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.
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    External Links:
    ISBN 0-521-66159-5, MathReview (Thomas A. Atchison), ZentralBlatt.
    Referenced by:
    §3.9(vi).
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  • A. Sidi (2004)
    Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective,
    Numer. Math. 98 (2), pp. 371–387.
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    External Links:
    ISSN 0029-599X, Document, MathReview (Iulian Coroian), ZentralBlatt.
    Referenced by:
    ‣ §2.10(i).
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  • C. L. Siegel (1971)
    Topics in Complex Function Theory. Vol. II: Automorphic Functions and Abelian Integrals,
    Interscience Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience [John Wiley & Sons Inc.], New York.
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    Notes:
    Translated from the original German by A. Shenitzer and M. Tretkoff. Reprinted by John Wiley & Sons, New York, 1988.
    External Links:
    ISBN 0-471-79080-X, MathReview, ZentralBlatt.
    Referenced by:
    Ch.21.
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  • C. L. Siegel (1973)
    Topics in Complex Function Theory. Vol. III: Abelian Functions and Modular Functions of Several Variables,
    Interscience Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience, [John Wiley & Sons, Inc], New York-London-Sydney.
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    Notes:
    Translated from the original German by E. Gottschling and M. Tretkoff. Reprinted by John Wiley & Sons, New York, 1989.
    External Links:
    ISBN 0-471-79090-7, MathReview, ZentralBlatt.
    Referenced by:
    Ch.21.
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  • C. L. Siegel (1988)
    Topics in Complex Function Theory. Vol. I: Elliptic Functions and Uniformization Theory,
    Wiley Classics Library, John Wiley & Sons Inc., New York.
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    Notes:
    Translated from the German by A. Shenitzer and D. Solitar. Reprint of the 1969 edition, a Wiley-Interscience Publication.
    External Links:
    ISBN 0-471-60844-0, MathReview, ZentralBlatt.
    Referenced by:
    §20.12(ii), ‣ §22.18(i), §22.18(iii), §22.18(iv).
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  • C. L. Siegel (1935)
    Über die analytische Theorie der quadratischen Formen,
    Ann. of Math. (2) 36 (3), pp. 527–606.
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    External Links:
    ISSN 0003-486X, FullText, MathReview Entry, ZentralBlatt.
    Referenced by:
    §35.2.
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  • K. M. Siegel and F. B. Sleator (1954)
    Inequalities involving cylindrical functions of nearly equal argument and order,
    Proc. Amer. Math. Soc. 5 (3), pp. 337–344.
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    External Links:
    ISSN 0002-9939, FullText, MathReview (N. D. Kazarinoff), ZentralBlatt.
    Referenced by:
    §10.14.
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  • K. M. Siegel (1953)
    An inequality involving Bessel functions of argument nearly equal to their order,
    Proc. Amer. Math. Soc. 4 (6), pp. 858–859.
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    ISSN 0002-9939, FullText, MathReview (N. D. Kazarinoff), ZentralBlatt.
    Referenced by:
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  • C. E. Siewert and E. E. Burniston (1973)
    Exact analytical solutions of ze^{z}=a,
    J. Math. Anal. Appl. 43 (3), pp. 626–632.
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    External Links:
    Document, MathReview (Pl. Kannappan), ZentralBlatt.
    Referenced by:
    §4.13.
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  • J. H. Silverman and J. Tate (1992)
    Rational Points on Elliptic Curves,
    Undergraduate Texts in Mathematics, Springer-Verlag, New York.
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    External Links:
    ISBN 0-387-97825-9, MathReview (Andrew Bremner), ZentralBlatt.
    Referenced by:
    §22.18(iv), ‣ §23.1, §23.20(ii), §23.20(ii), §23.20(v).
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  • R. A. Silverman (1967)
    Introductory Complex Analysis,
    Prentice-Hall, Inc., Englewood Cliffs, N.J..
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    Notes:
    Reprinted by Dover Publications Inc. 1972
    External Links:
    MathReview (E. Hille), ZentralBlatt.
    Referenced by:
    Ch.4.
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  • G. F. Simmons (1972)
    Differential Equations with Applications and Historical Notes,
    McGraw-Hill Book Co., New York.
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    Notes:
    International Series in Pure and Applied Mathematics
    External Links:
    MathReview (J. S. Joel), ZentralBlatt.
    Referenced by:
    §1.13(iii).
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  • B. Simon (1982)
    Large orders and summability of eigenvalue perturbation theory: A mathematical overview,
    Int. J. Quantum Chem. 21, pp. 3–25.
    Show Annotations
    External Links:
    Document.
    Referenced by:
    §22.19(ii).
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  • B. Simon (2005a)
    Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory,
    American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
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    External Links:
    ISBN 0-8218-3446-0, MathReview (P. L. Duren), ZentralBlatt.
    Referenced by:
    §18.33(i).
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  • B. Simon (2005b)
    Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory,
    American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
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    External Links:
    ISBN 0-8218-3675-7, MathReview (P. L. Duren), ZentralBlatt.
    Referenced by:
    §18.33(i).
    Encodings:
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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).
    Show Annotations
    External Links:
    FullText, MathReview (M. Strutt), ZentralBlatt.
    Referenced by:
    §28.8(ii).
    Encodings:
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  • R. Sips (1959)
    Représentation asymptotique des fonctions de Mathieu et des fonctions sphéroidales. II,
    Trans. Amer. Math. Soc. 90 (2), pp. 340–368.
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    External Links:
    FullText, MathReview (M. J. O. Strutt), ZentralBlatt.
    Referenced by:
    §28.8(ii).
    Encodings:
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  • R. Sips (1965)
    Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill,
    Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.
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    External Links:
    ZentralBlatt.
    Referenced by:
    §28.8(ii).
    Encodings:
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  • R. Sips (1967)
    Répartition du courant alternatif dans un conducteur cylindrique de section elliptique,
    Acad. Roy. Belg. Bull. Cl. Sci. (5) 53 (8), pp. 861–878.
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    Referenced by:
    §28.33(ii).
    Encodings:
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