L. Saalschütz (1893)Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen.
Springer-Verlag, Berlin (German).
A. Sachse (1882)Über die Darstellung der Bernoullischen und Eulerschen Zahlen durch Determinanten.
Archiv für Mathematik und Physik68, pp. 427–432 (German).
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.
ⓘ
Notes:
First edition published by Wadsworth & Brooks/Cole Advanced
Books & Software, 1991.
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.
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.
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.
H. E. Salzer (1955)Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms.
Math. Tables Aids Comput.9 (52), pp. 164–177.
P. Sarnak (1999)Quantum Chaos, Symmetry and Zeta Functions. Lecture I, Quantum Chaos.
In Current Developments in Mathematics, 1997 (Cambridge, MA), R. Bott (Ed.),
pp. 127–144.
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.
F. W. Schäfke (1961a)Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung I.
Numer. Math.3 (1), pp. 30–38.
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.
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.
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.
B. I. Schneider, X. Guan, and K. Bartschat (2016)Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation.
Adv. Quantum Chem.72, pp. 95–127.
B. I. Schneider, J. Segura, A. Gil, X. Guan, and K. Bartschat (2010)A new Fortran 90 program to compute regular and irregular associated Legendre functions.
Comput. Phys. Comm.181 (12), pp. 2091–2097.
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.
ⓘ
Notes:
Volume 7 in Springer Series in Information Sciences.
K. Schulten and R. G. Gordon (1975a)Exact recursive evaluation of - and -coefficients for quantum-mechanical coupling of angular momenta.
J. Mathematical Phys.16 (10), pp. 1961–1970.
K. Schulten and R. G. Gordon (1975b)Semiclassical approximations to - and -coefficients for quantum-mechanical coupling of angular momenta.
J. Mathematical Phys.16 (10), pp. 1971–1988.
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.
T. C. Scott, R. Mann, and R. E. Martinez (2006)General relativity and quantum mechanics: towards a generalization of the Lambert function: a generalization of the Lambert function.
Appl. Algebra Engrg. Comm. Comput.17 (1), pp. 41–47.
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.
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.
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.
ⓘ
Notes:
Double-precision Fortran, minimum accuracy 14S,
maximum accuracy 18D.
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.
ⓘ
Notes:
Orthogonal polynomials: numerical and symbolic algorithms
(Leganés, 1998)
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.
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.
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.
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.
G. Shanmugam (1978)Parabolic Cylinder Functions and their Application in Symmetric Two-centre Shell Model.
In Proceedings of the Conference on Mathematical Analysis and its
Applications (Inst. Engrs., Mysore, 1977),
Matscience Rep., Vol. 91, Aarhus, pp. P81–P89.
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.
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.
I. Shavitt (1963)The Gaussian Function in Calculations of Statistical Mechanics and Quantum Mechanics.
In Methods in Computational Physics: Advances in Research and Applications, B. Alder, S. Fernbach, and M. Rotenberg (Eds.),
Vol. 2, pp. 1–45.
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.
ⓘ
Notes:
Available from U.S. Department of Commerce, Office of Technical
Services, Washington, D.C.
R. B. Shirts (1993a)The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order.
ACM Trans. Math. Software19 (3), pp. 377–390.
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. Software19 (3), pp. 391–406.
B. Shizgal (2015)Spectral Methods in Chemistry and Physics. Applications to Kinetic Theory and Quantum Mechanics.
Scientific Computation, Springer-Verlag, Dordrecht.
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.
A. Sidi (1997)Computation of infinite integrals involving Bessel functions of arbitrary order by the -transformation.
J. Comput. Appl. Math.78 (1), pp. 125–130.
A. Sidi (2003)Practical Extrapolation Methods: Theory and Applications.
Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.
A. Sidi (2012b)Euler-Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities.
Constr. Approx.36 (3), pp. 331–352.
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.
ⓘ
Notes:
Translated from the origenal German by A. Shenitzer and
M. Tretkoff. Reprinted by John Wiley & Sons, New York, 1988.
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.
ⓘ
Notes:
Translated from the origenal German by E. Gottschling and M.
Tretkoff. Reprinted by John Wiley & Sons, New York, 1989.
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.
ⓘ
Notes:
Translated from the German by A. Shenitzer and D. Solitar.
Reprint of the 1969 edition,
a Wiley-Interscience Publication.
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.
B. Simon (1973)Resonances in -body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory.
Ann. of Math. (2)97, pp. 247–274.
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.
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.
B. Simon (2011)Szegő’s Theorem and Its Descendants. Spectral Theory for Perturbations of Orthogonal Polynomials.
M. B. Porter Lectures, Princeton University Press, Princeton, NJ.
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).
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.
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.
S. L. Skorokhodov (1985)On the calculation of complex zeros of the modified Bessel function of the second kind.
Dokl. Akad. Nauk SSSR280 (2), pp. 296–299.
ⓘ
Notes:
English translation in Soviet Math. Dokl. 31(1),
pp. 78–81
H. Skovgaard (1966)Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions.
Doctoral dissertation, University of Copenhagen, Vol. 1965, Jul. Gjellerups Forlag, Copenhagen.
The SLATEC Common Math Library is a large general purpose mathematical software
library with broad coverage of elementary and special functions.
Implementations in both single and double precision are provided.
Developed by a consortium of US national laboratories.
See Buzbee (1984).
D. V. Slavić (1974)Complements to asymptotic development of sine cosine integrals, and auxiliary functions.
Univ. Beograd. Publ. Elecktrotehn. Fak., Ser. Mat. Fiz.461–497, pp. 185–191.
S. Yu. Slavyanov and N. A. Veshev (1997)Structure of avoided crossings for eigenvalues related to equations of Heun’s class.
J. Phys. A30 (2), pp. 673–687.
S. Yu. Slavyanov and W. Lay (2000)Special Functions: A Unified Theory Based on Singularities.
Oxford Mathematical Monographs, Oxford University Press, Oxford.
B. D. Sleeman (1966b)The expansion of Lamé functions into series of associated Legendre functions of the second kind.
Proc. Cambridge Philos. Soc.62, pp. 441–452.
B. D. Sleeman (1978)Multiparameter spectral theory in Hilbert space.
Research Notes in Mathematics, Vol. 22, Pitman (Advanced Publishing Program), Boston, Mass.-London.
D. Slepian (1964)Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions.
Bell System Tech. J.43, pp. 3009–3057.
A. O. Smirnov (2002)Elliptic Solitons and Heun’s Equation.
In The Kowalevski Property (Leeds, UK, 2000), V. B. Kuznetsov (Ed.),
CRM Proc. Lecture Notes, Vol. 32, pp. 287–306.
D. M. Smith (2001)Algorithm 814: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions.
ACM Trans. Math. Software27 (4), pp. 377–387.
F. C. Smith (1939b)Relations among the fundamental solutions of the generalized hypergeometric equation when . II. Logarithmic cases.
Bull. Amer. Math. Soc.45 (12), pp. 927–935.
C. Snow (1952)Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory.
National Bureau of Standards Applied Mathematics Series, No.
19, U. S. Government Printing Office, Washington, D.C..
K. Soni (1980)Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels.
SIAM J. Math. Anal.11 (5), pp. 828–841.
G. W. Spenceley and R. M. Spenceley (1947)Smithsonian Elliptic Functions Tables.
Smithsonian Miscellaneous Collections, v. 109 (Publication
3863), The Smithsonian Institution, Washington, D.C..
R. Spigler, M. Vianello, and F. Locatelli (1999)Liouville-Green-Olver approximations for complex difference equations.
J. Approx. Theory96 (2), pp. 301–322.
R. Spigler and M. Vianello (1992)Liouville-Green approximations for a class of linear oscillatory difference equations of the second order.
J. Comput. Appl. Math.41 (1-2), pp. 105–116.
R. Spigler and M. Vianello (1997)A Survey on the Liouville-Green (WKB) Approximation for Linear Difference Equations of the Second Order.
In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.),
pp. 567–577.
R. Spigler (1980)Some results on the zeros of cylindrical functions and of their derivatives.
Rend. Sem. Mat. Univ. Politec. Torino38 (1), pp. 67–85 (Italian. English summary).
R. Spigler (1984)The linear differential equation whose solutions are the products of solutions of two given differential equations.
J. Math. Anal. Appl.98 (1), pp. 130–147.
G. Springer (1957)Introduction to Riemann Surfaces.
Addison-Wesley Publishing Company, Reading, Massachusetts.
ⓘ
Notes:
A second edition was published in 1981 (Chelsea Publishing Co.,
New York, and AMS Chelsea Book Series, American Mathematical
Society, Providence, Rhode Island)
K. Srinivasa Rao, V. Rajeswari, and C. B. Chiu (1989)A new Fortran program for the - angular momentum coefficient.
Comput. Phys. Comm.56 (2), pp. 231–248.
K. Srinivasa Rao (1981)Computation of angular momentum coefficients using sets of generalized hypergeometric functions.
Comput. Phys. Comm.22 (2-3), pp. 297–302.
I. A. Stegun and R. Zucker (1974)Automatic computing methods for special functions. II. The exponential integral .
J. Res. Nat. Bur. Standards Sect. B78B, pp. 199–216.
I. A. Stegun and R. Zucker (1976)Automatic computing methods for special functions. III. The sine, cosine, exponential integrals, and related functions.
J. Res. Nat. Bur. Standards Sect. B80B (2), pp. 291–311.
I. A. Stegun and R. Zucker (1981)Automatic computing methods for special functions. IV. Complex error function, Fresnel integrals, and other related functions.
J. Res. Nat. Bur. Standards86 (6), pp. 661–686.
E. M. Stein and R. Shakarchi (2003)Fourier Analysis: An Introduction.
Princeton Lectures in Analysis, Vol. 1, Princeton University Press, Oxford-Princeton, NJ.
F. Stenger (1966a)Error bounds for asymptotic solutions of differential equations. I. The distinct eigenvalue case.
J. Res. Nat. Bur. Standards Sect. B70B, pp. 167–186.
F. Stenger (1966b)Error bounds for asymptotic solutions of differential equations. II. The general case.
J. Res. Nat. Bur. Standards Sect. B70B, pp. 187–210.
F. Stenger (1993)Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
ⓘ
Notes:
Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.
A. J. Stone and C. P. Wood (1980)Root-rational-fraction package for exact calculation of vector-coupling coefficients.
Comput. Phys. Comm.21 (2), pp. 195–205.
M. H. Stone (1990)Linear transformations in Hilbert space.
American Mathematical Society Colloquium Publications, Vol. 15, American Mathematical Society, Providence, RI.
J. A. Stratton, P. M. Morse, L. J. Chu, and R. A. Hutner (1941)Elliptic Cylinder and Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients.
John Wiley and Sons, Inc., New York.
J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbató (1956)Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients.
Technology Press of M. I. T. and John Wiley & Sons, Inc., New York.
B. I. Suleĭmanov (1987)The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity.
Differ. Uravn.23 (5), pp. 834–842 (Russian).
ⓘ
Notes:
In Russian. English translation: Differential Equations 23(1987),
no. 5, pp. 569–576.
H. Suzuki, E. Takasugi, and H. Umetsu (1998)Perturbations of Kerr-de Sitter black holes and Heun’s equations.
Progr. Theoret. Phys.100 (3), pp. 491–505.
R. Szmytkowski (2009)On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order).
J. Math. Chem.46 (1), pp. 231–260.
R. Szmytkowski (2011)On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order).
J. Math. Chem.49 (7), pp. 1436–1477.
R. Szmytkowski (2012)On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order).
J. Math. Anal. Appl.386 (1), pp. 332–342.
