DLMF: Bibliography F ‣ Bibliography (original) (raw)

• Computation of complex Airy functions and their zeros using asymptotics and the differential equation. ACM Trans. Math. Software 30 (4), pp. 471–490.

• B. R. Fabijonas, D. W. Lozier, and J. M. Rappoport (2003) Algorithms and codes for the Macdonald function: Recent progress and comparisons. J. Comput. Appl. Math. 161 (1), pp. 179–192.

• B. R. Fabijonas (2004) Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

• B. R. Fabijonas and F. W. J. Olver (1999) On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.

• V. N. Faddeeva and N. M. Terent’ev (1954) Tablicy značeniĭ funkcii w⁢(z)=e−z2⁢(1+2⁢iπ⁢∫0zet2⁢𝑑t) ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).

• V. N. Faddeyeva and N. M. Terent’ev (1961) Tables of Values of the Function w⁢(z)=e−z2⁢(1+2⁢i⁢π−1/2⁢∫0zet2⁢𝑑t) for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

• M. Faierman (1992) Generalized parabolic cylinder functions. Asymptotic Anal. 5 (6), pp. 517–531.

• P. Falloon (2001) Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.

• D. F. Fang and J. F. Shriner (1992) A computer program for the calculation of angular-momentum coupling coefficients. Comput. Phys. Comm. 70 (1), pp. 147–153.

• J. Faraut and A. Korányi (1994) Analysis on Symmetric Cones. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford-New York.

• J. Faraut (1982) Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

• S. Farid Khwaja and A. B. Olde Daalhuis (2014) Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

• R. H. Farrell (1985) Multivariate Calculation. Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York.

• J. D. Fay (1973) Theta Functions on Riemann Surfaces. Springer-Verlag, Berlin.

• FDLIBM (free C library)

• M. V. Fedoryuk (1989) The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

• M. V. Fedoryuk (1991) Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).

• S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

• C. Ferreira, J. L. López, and E. Pérez Sinusía (2013a) The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.

• C. Ferreira, J. L. López, and E. Pérez Sinusía (2005) Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.

• C. Ferreira, J. L. López, and E. P. Sinusía (2013b) The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.

• C. Ferreira and J. L. López (2001) An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.

• C. Ferreira and J. L. López (2004) Asymptotic expansions of the Hurwitz-Lerch zeta function. J. Math. Anal. Appl. 298 (1), pp. 210–224.

• E. M. Ferreira and J. Sesma (2008) Zeros of the Macdonald function of complex order. J. Comput. Appl. Math. 211 (2), pp. 223–231.

• H. E. Fettis, J. C. Caslin, and K. R. Cramer (1973) Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.

• H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

• H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

• H. E. Fettis and J. C. Caslin (1973) Table errata; Complex zeros of Fresnel integrals. Math. Comp. 27 (121), pp. 219.

• H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

• H. E. Fettis (1970) On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

• H. E. Fettis (1976) Complex roots of sin⁡z=a⁢z, cos⁡z=a⁢z, and cosh⁡z=a⁢z. Math. Comp. 30 (135), pp. 541–545.

• F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

• J. L. Fields and Y. L. Luke (1963a) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

• J. L. Fields and Y. L. Luke (1963b) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

• J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

• J. L. Fields (1965) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.

• J. L. Fields (1966) A note on the asymptotic expansion of a ratio of gamma functions. Proc. Edinburgh Math. Soc. (2) 15, pp. 43–45.

• J. L. Fields (1973) Uniform asymptotic expansions of certain classes of Meijer G-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

• J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer G-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

• A. M. S. Filho and G. Schwachheim (1967) Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

• S. Fillebrown (1992) Faster computation of Bernoulli numbers. J. Algorithms 13 (3), pp. 431–445.

• S. R. Finch (2003) Mathematical Constants. Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, Cambridge.

• N. J. Fine (1988) Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.

• G. D. Finn and D. Mugglestone (1965) Tables of the line broadening function H⁢(a,v). Monthly Notices Roy. Astronom. Soc. 129, pp. 221–235.

• P. Flajolet and A. Odlyzko (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

• P. Flajolet and B. Salvy (1998) Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.

• C. Flammer (1957) Spheroidal Wave Functions. Stanford University Press, Stanford, CA.

• H. Flaschka and A. C. Newell (1980) Monodromy- and spectrum-preserving deformations. I. Comm. Math. Phys. 76 (1), pp. 65–116.

• A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie (1962) An Index of Mathematical Tables. Vols. I, II. 2nd edition, Published for Scientific Computing Service Ltd., London, by Addison-Wesley Publishing Co., Inc., Reading, MA.

• A. Fletcher (1948) Guide to tables of elliptic functions. Math. Tables and Other Aids to Computation 3 (24), pp. 229–281.

• N. Fleury and A. Turbiner (1994) Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

• J. P. M. Flude (1998) The Edmonds asymptotic formulas for the 3⁢j and 6⁢j symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

• FN (free Fortran library)

• V. A. Fock (1965) Electromagnetic Diffraction and Propagation Problems. International Series of Monographs on Electromagnetic Waves, Vol. 1, Pergamon Press, Oxford.

• V. Fock (1945) Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.

• A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

• A. S. Fokas, B. Grammaticos, and A. Ramani (1993) From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

• A. S. Fokas, A. R. Its, and A. V. Kitaev (1991) Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.

• A. S. Fokas, A. R. Its, and X. Zhou (1992) Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

• A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

• A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Yu. Novokshënov (2006) Painlevé Transcendents: The Riemann-Hilbert Approach. Mathematical Surveys and Monographs, Vol. 128, American Mathematical Society, Providence, RI.

• K. W. Ford and J. A. Wheeler (1959a) Semiclassical description of scattering. Ann. Physics 7 (3), pp. 259–286.

• K. W. Ford and J. A. Wheeler (1959b) Application of semiclassical scattering analysis. Ann. Physics 7 (3), pp. 287–322.

• W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

• B. Fornberg and J. A. C. Weideman (2011) A numerical methodology for the Painlevé equations. J. Comput. Phys. 230 (15), pp. 5957–5973.

• P. J. Forrester and N. S. Witte (2001) Application of the τ-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Comm. Math. Phys. 219 (2), pp. 357–398.

• P. J. Forrester and N. S. Witte (2002) Application of the τ-function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.

• P. J. Forrester and N. S. Witte (2004) Application of the τ-function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

• R. C. Forrey (1997) Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.

• T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

• L. Fox and I. B. Parker (1968) Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London.

• L. Fox (1960) Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

• P. A. Fox, A. D. Hall, and N. L. Schryer (1978) The PORT mathematical subroutine library. ACM Trans. Math. Software 4 (2), pp. 104–126.

• C. H. Franke (1965) Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

• C. K. Frederickson and P. L. Marston (1992) Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.

• C. K. Frederickson and P. L. Marston (1994) Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

• D. Frenkel and R. Portugal (2001) Algebraic methods to compute Mathieu functions. J. Phys. A 34 (17), pp. 3541–3551.

• C. L. Frenzen and R. Wong (1985a) A note on asymptotic evaluation of some Hankel transforms. Math. Comp. 45 (172), pp. 537–548.

• C. L. Frenzen and R. Wong (1985b) A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Canad. J. Math. 37 (5), pp. 979–1007.

• C. L. Frenzen and R. Wong (1986) Asymptotic expansions of the Lebesgue constants for Jacobi series. Pacific J. Math. 122 (2), pp. 391–415.

• C. L. Frenzen and R. Wong (1988) Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal. 19 (5), pp. 1232–1248.

• C. L. Frenzen (1987a) Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18 (3), pp. 890–896.

• C. L. Frenzen (1987b) On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

• C. L. Frenzen (1990) Error bounds for a uniform asymptotic expansion of the Legendre function Qn−m⁢(cosh⁢z). SIAM J. Math. Anal. 21 (2), pp. 523–535.

• C. L. Frenzen (1992) Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.

• A. Fresnel (1818) Mémoire sur la diffraction de la lumière. Mém. de l’Académie des Sciences, pp. 247–382.

• G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

• G. Freud (1976) On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

• B. D. Fried and S. D. Conte (1961) The Plasma Dispersion Function: The Hilbert Transform of the Gaussian. Academic Press, London-New York.

• B. R. Frieden (1971) Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions. In Progress in Optics, E. Wolf (Ed.), Vol. 9, pp. 311–407.

• F. G. Friedlander (1958) Sound Pulses. Cambridge University Press, Cambridge-New York.

• B. Friedman (1990) Principles and Techniques of Applied Mathematics. Dover, New York.

• F. N. Fritsch, R. E. Shafer, and W. P. Crowley (1973) Solution of the transcendental equation w⁢ew=x. Comm. ACM 16 (2), pp. 123–124.

• C. Fröberg (1955) Numerical treatment of Coulomb wave functions. Rev. Mod. Phys. 27 (4), pp. 399–411.

• R. Fuchs (1907) Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.

• Y. Fukui and T. Horiguchi (1992) Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.

• T. Fukushima (2010) Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.

• T. Fukushima (2012) Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

• L. W. Fullerton and G. A. Rinker (1986) Generalized Fermi-Dirac integrals—FD, FDG, FDH. Comput. Phys. Comm. 39 (2), pp. 181–185.

• L. W. Fullerton (1972) Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.

• L. W. Fullerton (1977) Portable Special Function Routines. In Portability of Numerical Software (Oak Brook, Illinois, 1976), W. R. Cowell (Ed.), Lecture Notes in Computer Science, Vol. 57, pp. 452–483.

• Y. V. Fyodorov (2005) Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.