B. Gabutti and B. Minetti (1981)A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function.
J. Comput. Phys.42 (2), pp. 277–287.
E. A. Galapon and K. M. L. Martinez (2014)Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.470 (2162), pp. 20130529, 16.
B. Gambier (1910)Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes.
Acta Math.33 (1), pp. 1–55.
M. J. Gander and A. H. Karp (2001)Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer.
J. Quant. Spectrosc. Radiat. Transfer68 (2), pp. 213–223.
F. Gao and V. J. W. Guo (2013)Contiguous relations and summation and transformation formulae for basic hypergeometric series.
J. Difference Equ. Appl.19 (12), pp. 2029–2042.
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.
I. Gargantini and P. Henrici (1967)A continued fraction algorithm for the computation of higher transcendental functions in the complex plane.
Math. Comp.21 (97), pp. 18–29.
F. G. Garvan and M. E. H. Ismail (Eds.) (2001)Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics.
Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.
G. Gasper and M. Rahman (1990)Basic Hypergeometric Series.
Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.
ⓘ
Notes:
A revised and updated edition, with three new chapters, was published in 2004 by Cambridge University Press (Encyclopedia of Mathematics and its Applications, vol. 96).
G. Gasper and M. Rahman (2004)Basic Hypergeometric Series.
Second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge.
G. Gasper (1975)Formulas of the Dirichlet-Mehler Type.
In Fractional Calculus and its Applications, B. Ross (Ed.),
Lecture Notes in Math., Vol. 457, pp. 207–215.
L. Gatteschi (1990)New inequalities for the zeros of confluent hypergeometric functions.
In Asymptotic and computational analysis (Winnipeg, MB, 1989),
pp. 175–192.
W. Gautschi (1964b)Algorithm 236: Bessel functions of the first kind.
Comm. ACM7 (8), pp. 479–480.
ⓘ
Notes:
For certification see same journal 8(1965), pp. 105–106,
for remark see ACM Trans. Math. Software, 1(1975),
pp. 282–284. Includes Algol program, maximum accuracy: 11D.
W. Gautschi (1994)Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software20 (1), pp. 21–62.
W. Gautschi (1975)Computational Methods in Special Functions – A Survey.
In Theory and Application of Special Functions (Proc. Advanced
Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis.,
1975), R. A. Askey (Ed.),
pp. 1–98. Math. Res. Center, Univ. Wisconsin Publ., No. 35.
W. Gautschi (1997b)The Computation of Special Functions by Linear Difference Equations.
In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.),
pp. 213–243.
W. Gautschi (1998)The incomplete gamma functions since Tricomi.
In Tricomi’s Ideas and Contemporary Applied Mathematics
(Rome/Turin, 1997),
Atti Convegni Lincei, Vol. 147, pp. 203–237.
W. Gautschi (2002b)Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions.
J. Comput. Appl. Math.139 (1), pp. 173–187.
W. Gautschi (2004)Orthogonal Polynomials: Computation and Approximation.
Numerical Mathematics and Scientific Computation, Oxford University Press, New York.
ⓘ
Notes:
OPQ, a package of Matlab routines for generating classical and Sobolev
orthogonal polynomials, accompanies this book.
W. Gautschi (2016)Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature.
ACM Trans. Math. Softw.42 (1), pp. 9:1–9:10.
ⓘ
Notes:
Includes two MATLAB programs claiming 12D and 30D accuracy
M. Geller and E. W. Ng (1971)A table of integrals of the error function. II. Additions and corrections.
J. Res. Nat. Bur. Standards Sect. B75B, pp. 149–163.
ⓘ
Notes:
See for the first part same journal, Sect. B
73, 1-20 (1969)
V. X. Genest, L. Vinet, and A. Zhedanov (2016)The non-symmetric Wilson polynomials are the Bannai-Ito polynomials.
Proc. Amer. Math. Soc.144 (12), pp. 5217–5226.
K. Germey (1964)Die Beugung einer ebenen elektromanetischen Welle an zwei parallelen unendlich langen idealleitenden Zylindern von elliptischem Querschnitt.
Ann. Physik (7)468, pp. 237–251 (German).
J. S. Geronimo, O. Bruno, and W. Van Assche (2004)WKB and turning point theory for second-order difference equations.
In Spectral Methods for Operators of Mathematical Physics,
Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.
A. Gervois and H. Navelet (1984)Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities.
J. Math. Phys.25 (11), pp. 3350–3356.
P. Gianni, M. Seppälä, R. Silhol, and B. Trager (1998)Riemann surfaces, plane algebraic curves and their period matrices.
J. Symbolic Comput.26 (6), pp. 789–803.
A. Gibbs, D. P. Hewett, and D. Huybrechs (2024)Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation.
J. Comput. Phys.501, pp. Paper No. 112787, 23.
A. Gil and J. Segura (2001)DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics.
Comput. Phys. Comm.139 (2), pp. 186–191.
A. Gil, D. Ruiz-Antolín, J. Segura, and N. M. Temme (2016)Algorithm 969: computation of the incomplete gamma function for negative values of the argument.
ACM Trans. Math. Softw.43 (3), pp. 26:1–26:9.
A. Gil, J. Segura, and N. M. Temme (2002a)Algorithm 819: AIZ, BIZ: two Fortran 77 routines for the computation of complex Airy functions.
ACM Trans. Math. Software28 (3), pp. 325–336.
A. Gil, J. Segura, and N. M. Temme (2002b)Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions.
ACM Trans. Math. Software28 (4), pp. 436–447.
A. Gil, J. Segura, and N. M. Temme (2002d)Evaluation of the modified Bessel function of the third kind of imaginary orders.
J. Comput. Phys.175 (2), pp. 398–411.
A. Gil, J. Segura, and N. M. Temme (2003a)Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion.
J. Comput. Appl. Math.153 (1-2), pp. 225–234.
A. Gil, J. Segura, and N. M. Temme (2004a)Algorithm 831: Modified Bessel functions of imaginary order and positive argument.
ACM Trans. Math. Software30 (2), pp. 159–164.
A. Gil, J. Segura, and N. M. Temme (2004b)Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments.
ACM Trans. Math. Software30 (2), pp. 145–158.
A. Gil, J. Segura, and N. M. Temme (2007a)Numerical Methods for Special Functions.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
A. Gil, J. Segura, and N. M. Temme (2011a)Algorithm 914: parabolic cylinder function and its derivative.
ACM Trans. Math. Software38 (1), pp. Art. 6, 5.
A. Gil, J. Segura, and N. M. Temme (2011b)Fast and accurate computation of the Weber parabolic cylinder function .
IMA J. Numer. Anal.31 (3), pp. 1194–1216.
A. Gil, J. Segura, and N. M. Temme (2012)An improved algorithm and a Fortran 90 module for computing the conical function .
Comput. Phys. Commun.183 (3), pp. 794–799.
A. Gil and J. Segura (2003)Computing the zeros and turning points of solutions of second order homogeneous linear ODEs.
SIAM J. Numer. Anal.41 (3), pp. 827–855.
C. D. Godsil, M. Grötschel, and D. J. A. Welsh (1995)Combinatorics in Statistical Physics.
In Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grötschel, and L. Lovász (Eds.),
pp. 1925–1954.
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. B80B (1), pp. 99–171.
G. H. Golub and G. Meurant (2010)Matrices, moments and quadrature with applications.
Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.
V. V. Golubev (1960)Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point.
Translated from the Russian by J. Shorr-Kon, Office of Technical Services, U. S. Department of Commerce, Washington, D.C..
ⓘ
Notes:
Published for the National Science Foundation by the Israel
Program for Scientific Translations, 1960.
D. Gómez-Ullate, N. Kamran, and R. Milson (2010)Exceptional orthogonal polynomials and the Darboux transformation.
J. Phys. A43 (43), pp. 43016, 16 pp..
D. Gómez-Ullate and R. Milson (2014)Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials.
J. Phys. A47 (1), pp. 015203, 26 pp..
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.
Z. Gong, L. Zejda, W. Dappen, and J. M. Aparicio (2001)Generalized Fermi-Dirac functions and derivatives: Properties and evaluation.
Comput. Phys. Comm.136 (3), pp. 294–309.
D. Gottlieb and S. A. Orszag (1977)Numerical Analysis of Spectral Methods: Theory and Applications.
Society for Industrial and Applied Mathematics, Philadelphia, PA.
ⓘ
Notes:
CBMS-NSF Regional Conference Series in Applied Mathematics,
No. 26
H. P. W. Gottlieb (1985)On the exceptional zeros of cross-products of derivatives of spherical Bessel functions.
Z. Angew. Math. Phys.36 (3), pp. 491–494.
É. Goursat (1881)Sur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique.
Ann. Sci. École Norm. Sup. (2)10, pp. 3–142 (French).
R. L. Graham, D. E. Knuth, and O. Patashnik (1994)Concrete Mathematics: A Foundation for Computer Science.
2nd edition, Addison-Wesley Publishing Company, Reading, MA.
Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992)Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys.217 (1), pp. 1–20.
A. Gray, G. B. Mathews, and T. M. MacRobert (1922)A Treatise on Bessel Functions and their Applications to Physics.
2nd edition, Macmillan and Co., London.
ⓘ
Notes:
Table errata: Math. Comp. v. 24 (1970), p. 240,
v. 31 (1978), pp. 806 and 1046, and v. 32 (1978), pp. 318.
Reprinted by Dover Publications Inc., New York, 1966.
M. B. Green, J. H. Schwarz, and E. Witten (1988a)Superstring Theory: Introduction, Vol. 1.
2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
M. B. Green, J. H. Schwarz, and E. Witten (1988b)Superstring Theory: Loop Amplitudes, Anomalies and Phenomenolgy, Vol. 2.
2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
W. Greiner, B. Müller, and J. Rafelski (1985)Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics.
Texts and Monographs in Physics, Springer.
V. I. Gromak, I. Laine, and S. Shimomura (2002)Painlevé Differential Equations in the Complex Plane.
Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin-New York.
K. I. Gross and D. St. P. Richards (1987)Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions.
Trans. Amer. Math. Soc.301 (2), pp. 781–811.
GSL (free C library)
GNU Scientific Library
The GNU Project.
ⓘ
Notes:
The GNU Scientific Library is a large general purpose numerical software library
with broad coverage of elementary and special functions.
Implementation is in double precision.
X. Guan, O. Zatsarinny, K. Bartschat, B. I. Schneider, J. Feist, and C. J. Noble (2007)General approach to few-cycle intense laser interactions with complex atoms.
Phys. Rev. A76, pp. 053411.
J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda (2003)Mathieu functions, a visual approach.
Amer. J. Phys.71 (3), pp. 233–242.
A. J. Guttmann and T. Prellberg (1993)Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions.
Phys. Rev. E47 (4), pp. R2233–R2236.