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  • G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
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    External Links:
    ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
    Referenced by:
    §8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).
    Encodings:
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  • G. Nemes (2013b) Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.
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    External Links:
    ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt
    Referenced by:
    §5.11(ii), §5.11(ii).
    Encodings:
    BibTeX
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  • G. Nemes (2015c) The resurgence properties of the incomplete gamma function II. Stud. Appl. Math. 135 (1), pp. 86–116.
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    External Links:
    ISSN 0022-2526, Document, Link, MathReview Entry, ZentralBlatt
    Referenced by:
    §8.11(iii).
    Encodings:
    BibTeX
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  • G. Nemes (2016) The resurgence properties of the incomplete gamma function, I. Anal. Appl. (Singap.) 14 (5), pp. 631–677.
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  • E. Neuman (2013) Inequalities and bounds for the incomplete gamma function. Results Math. 63 (3-4), pp. 1209–1214.
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    External Links:
    ISSN 1422-6383, Document, FullText, MathReview (Pierpaolo Natalini), ZentralBlatt
    Referenced by:
    §8.10, §8.10.
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    2: Bibliography F
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  • FDLIBM (free C library)
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    Notes:
    The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
    External Links:
    GAMS (package FDLIBM)
    Referenced by:
    § β€£ Software Cross Index, § β€£ Software Cross Index.
    Encodings:
    BibTeX
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  • 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.
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    External Links:
    ISSN 0196-8858, Document, MathReview (Boro Döring), ZentralBlatt
    Referenced by:
    §8.12.
    Encodings:
    BibTeX
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  • J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.
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    External Links:
    FullText, MathReview (L. J. Slater), ZentralBlatt
    Referenced by:
    §16.10.
    Encodings:
    BibTeX
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  • W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.
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    Notes:
    Reprint with corrections of two books published originally in 1916 and 1936.
    External Links:
    MathReview
    Referenced by:
    §2.10(iii).
    Encodings:
    BibTeX
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  • L. W. Fullerton (1972) Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.
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    Notes:
    Includes Fortran code. For a Remark see ACM Trans. Math. Software 4 (1978), no. 3, pp. 296–304.
    External Links:
    ISSN 0001-0782, Document, Remark
    Referenced by:
    §8.28(ii).
    Encodings:
    BibTeX
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    3: Bibliography S
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  • 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.
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    Notes:
    Single-precision Fortran code.
    External Links:
    Document, Code, ZentralBlatt
    Referenced by:
    10th item.
    Encodings:
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  • 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.
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    External Links:
    ZentralBlatt
    Referenced by:
    §12.13(i).
    Encodings:
    BibTeX
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  • 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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    External Links:
    MathReview (M. C. Gray), ZentralBlatt
    Referenced by:
    §12.13(ii).
    Encodings:
    BibTeX
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  • 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.
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    External Links:
    Document, MathReview (F. M. Arscott), ZentralBlatt
    Referenced by:
    §29.17(i).
    Encodings:
    BibTeX
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  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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    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.
    External Links:
    ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt
    Referenced by:
    §3.3(vi), §3.4(i), §3.5(i).
    Encodings:
    BibTeX
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    4: Bibliography M
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  • T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.
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    External Links:
    MathReview (A. Erdélyi), ZentralBlatt
    Referenced by:
    Ch.14.
    Encodings:
    BibTeX
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  • G. F. Miller (1966) On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.
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    External Links:
    ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt
    Referenced by:
    §3.11(ii).
    Encodings:
    BibTeX
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  • S. C. Milne (1985d) A q -analog of hypergeometric series well-poised in π‘†π‘ˆ ⁒ ( n ) and invariant G -functions. Adv. in Math. 58 (1), pp. 1–60.
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    External Links:
    ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt
    Referenced by:
    §17.15.
    Encodings:
    BibTeX
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  • R. J. Moore (1982) Algorithm AS 187. Derivatives of the incomplete gamma integral. Appl. Statist. 31 (3), pp. 330–335.
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    Notes:
    Single-precision Fortran.
    External Links:
    FullText, Code, Document, ZentralBlatt
    Referenced by:
    4th item.
    Encodings:
    BibTeX
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  • H. J. W. Müller (1966b) Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.
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    External Links:
    Document, MathReview (M. J. O. Strutt), ZentralBlatt
    Referenced by:
    §29.11, §29.7(ii).
    Encodings:
    BibTeX
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    5: Bibliography P
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  • V. I. Pagurova (1963) Tablitsy nepolnoi gamma-funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).
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    Notes:
    In Russian. Translated title: Tables of the Incomplete Gamma Function.
    External Links:
    MathReview (John Todd)
    Referenced by:
    2nd item.
    Encodings:
    BibTeX
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  • V. I. Pagurova (1965) An asymptotic formula for the incomplete gamma function. Ε½. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).
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    Notes:
    English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166
    External Links:
    Document, MathReview (E. Lukacs), ZentralBlatt
    Referenced by:
    §8.11(iv).
    Encodings:
    BibTeX
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  • R. B. Paris (2002a) Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.
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    External Links:
    ISSN 0377-0427, Document, MathReview, ZentralBlatt
    Referenced by:
    §8.12.
    Encodings:
    BibTeX
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  • R. B. Paris (2002b) A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.
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  • R. B. Paris (2003) The asymptotic expansion of a generalised incomplete gamma function. J. Comput. Appl. Math. 151 (2), pp. 297–306.
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    External Links:
    ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt
    Referenced by:
    §8.16.
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    6: Bibliography O
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  • A. B. Olde Daalhuis (1994) Asymptotic expansions for q -gamma, q -exponential, and q -Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.
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    External Links:
    ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt
    Referenced by:
    §5.18(ii).
    Encodings:
    BibTeX
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  • A. B. Olde Daalhuis (1998c) On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.
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    External Links:
    ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt
    Referenced by:
    §8.12.
    Encodings:
    BibTeX
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  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
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    External Links:
    Document, MathReview (David L. Elliott), ZentralBlatt
    Referenced by:
    §3.11(ii).
    Encodings:
    BibTeX
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  • F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.
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    External Links:
    ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
    Referenced by:
    §2.7(ii), §2.7(ii).
    Encodings:
    BibTeX
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  • F. W. J. Olver (1995) On an asymptotic expansion of a ratio of gamma functions. Proc. Roy. Irish Acad. Sect. A 95 (1), pp. 5–9.
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    External Links:
    ISSN 0035-8975, MathReview (Richard B. Paris), ZentralBlatt
    Referenced by:
    §5.11(iii).
    Encodings:
    BibTeX
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    7: Bibliography C
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  • B. C. Carlson (1961b) Some series and bounds for incomplete elliptic integrals. J. Math. and Phys. 40, pp. 125–134.
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  • M. A. Chaudhry, N. M. Temme, and E. J. M. Veling (1996) Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.
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    External Links:
    ISSN 0377-0427, Document, MathReview, ZentralBlatt
    Referenced by:
    §8.16.
    Encodings:
    BibTeX
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  • M. A. Chaudhry and S. M. Zubair (1994) Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.
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    External Links:
    ISSN 0377-0427, Document, MathReview, ZentralBlatt
    Referenced by:
    §8.16.
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    BibTeX
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  • M. A. Chaudhry and S. M. Zubair (2001) On a Class of Incomplete Gamma Functions with Applications. Chapman & Hall/CRC, Boca Raton, FL.
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    External Links:
    ISBN 1-58488-143-7, MathReview (R. K. Raina), ZentralBlatt
    Referenced by:
    §8.16.
    Encodings:
    BibTeX
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  • T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.
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    External Links:
    MathReview (I. I. Hirschman, Jr.), ZentralBlatt
    Referenced by:
    §12.16.
    Encodings:
    BibTeX
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    8: Bibliography K
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  • D. Karp, A. Savenkova, and S. M. Sitnik (2007) Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.
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    External Links:
    ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt
    Referenced by:
    §19.12, §19.5.
    Encodings:
    BibTeX
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  • S. H. Khamis (1965) Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).
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    Notes:
    In cooperation with W. Rudert
    External Links:
    MathReview, ZentralBlatt
    Referenced by:
    1st item.
    Encodings:
    BibTeX
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  • K. S. Kölbig (1970) Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.
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    External Links:
    FullText, MathReview, ZentralBlatt
    Referenced by:
    §8.13(iii), §8.22(ii).
    Encodings:
    BibTeX
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  • K. S. Kölbig (1972b) On the zeros of the incomplete gamma function. Math. Comp. 26 (119), pp. 751–755.
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    External Links:
    FullText, MathReview, ZentralBlatt
    Referenced by:
    §8.13(i), §8.13(iii), §8.13(iii).
    Encodings:
    BibTeX
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  • C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively q -binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
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    Notes:
    Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
    External Links:
    Code, Document, MathReview (G. Gasper), ZentralBlatt
    Referenced by:
    1st item.
    Encodings:
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    9: Bibliography B
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  • R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.
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    External Links:
    FullText, MathReview (J. C. P. Miller), ZentralBlatt
    Referenced by:
    §8.7.
    Encodings:
    BibTeX
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  • W. G. Bickley and J. Nayler (1935) A short table of the functions Ki n ⁒ ( x ) , from n = 1 to n = 16 . Phil. Mag. Series 7 20, pp. 343–347.
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    External Links:
    Document, ZentralBlatt
    Referenced by:
    §10.43(iii), 2nd item.
    Encodings:
    BibTeX
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  • 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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  • W. G. C. Boyd (1987) Asymptotic expansions for the coefficient functions that arise in turning-point problems. Proc. Roy. Soc. London Ser. A 410, pp. 35–60.
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    External Links:
    ISSN 0080-4630, FullText, MathReview (P. F. Hsieh), ZentralBlatt
    Referenced by:
    §2.8(iii).
    Encodings:
    BibTeX
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  • T. BuriΔ‡ and N. ElezoviΔ‡ (2011) Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions. J. Comput. Appl. Math. 235 (11), pp. 3315–3331.
    β“˜
    External Links:
    ISSN 0377-0427, Document, FullText, MathReview (Roberto S. Costas-Santos), ZentralBlatt
    Referenced by:
    §5.11(iii), §5.11(iii).
    Encodings:
    BibTeX
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    10: Bibliography
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  • G. Allasia and R. Besenghi (1987b) Numerical calculation of incomplete gamma functions by the trapezoidal rule. Numer. Math. 50 (4), pp. 419–428.
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    External Links:
    ISSN 0029-599X, Document, MathReview (F. W. J. Olver), ZentralBlatt
    Referenced by:
    §8.25(ii).
    Encodings:
    BibTeX
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  • H. Alzer (1997b) On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.
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    External Links:
    ISSN 0025-5718, Document, MathReview, ZentralBlatt
    Referenced by:
    §8.10.
    Encodings:
    BibTeX
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  • T. M. Apostol (1990) Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.
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  • H. Appel (1968) Numerical Tables for Angular Correlation Computations in Ξ± -, Ξ² - and Ξ³ -Spectroscopy: 3 ⁒ j -, 6 ⁒ j -, 9 ⁒ j -Symbols, F- and Ξ“ -Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.
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    External Links:
    ISBN 978-3-540-04218-1
    Referenced by:
    §34.14.
    Encodings:
    BibTeX
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  • R. Askey (1975b) Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.
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    External Links:
    ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt
    Referenced by:
    §18.10(ii), §18.18(vii), Profile Richard A. Askey.
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