connection with incomplete gamma functions

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1: 8.23 Statistical Applications

§8.23 Statistical Applications

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2: 8.22 Mathematical Applications

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§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

►The function $\mathrm{\Gamma }(a,z)$, with $|\mathrm{ph}a|\le \frac{1}{2}\pi$ and $\mathrm{ph}z=\frac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)) on the critical line $\mathrm{\Re }s=\frac{1}{2}$. …

3: Bibliography K

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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

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MathReview, ZentralBlatt

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1st item.

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Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.

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Pdf, Document

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§22.7(iii).

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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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FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

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Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.

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FullText, MathReview, ZentralBlatt

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§8.22(ii).

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On the zeros of the incomplete gamma function. Math. Comp. 26 (119), pp. 751–755.

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FullText, MathReview, ZentralBlatt

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§8.13(i), §8.13(iii), §8.13(iii).

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4: Bibliography M

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Algorithm AS 245. A robust and reliable algorithm for the logarithm of the gamma function. Appl. Statist. 38 (2), pp. 397–402.

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FullText, Code, ZentralBlatt

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5th item.

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Algorithm AS 187. Derivatives of the incomplete gamma integral. Appl. Statist. 31 (3), pp. 330–335.

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Notes:

Single-precision Fortran.

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FullText, Code, Document, ZentralBlatt

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4th item.

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A connection formula for the $q$-confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.

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ISSN 1815-0659, Document, MathReview (Jeremy Lovejoy), ZentralBlatt

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§17.6(ii), §17.6(ii).

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New sharp bounds for gamma and digamma functions. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 57 (1), pp. 57–60.

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ISSN 1221-8421, Document, MathReview (Daniele Ritelli), ZentralBlatt

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§5.6(i).

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A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.

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ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt

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§5.10.

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5: Bibliography J

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Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

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MathReview (Marietta J. Tretter), ZentralBlatt

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§8.25(iv).

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Algorithm 73: Incomplete elliptic integrals. Comm. ACM 4 (12), pp. 543.

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Notes:

Includes Algol program. For certification see same journal 6 (1963), p. 167, for remark see same journal 5 (1962), p. 514

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ISSN 0001-0782, Document

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§19.39(iii).

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Stirling Numbers, Lambert W and the Gamma Function. In Mathematical Aspects of Computer and Information Sciences, J. Blömer, I. S. Kotsireas, T. Kutsia, and D. E. Simos (Eds.), Cham, pp. 275–279.

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ISBN 978-3-319-72453-9, ZentralBlatt

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§4.13.

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On the computation of incomplete gamma functions in the complex domain. J. Comput. Appl. Math. 12/13, pp. 401–417.

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ISSN 0377-0427, Document, MathReview (M. Marsden), ZentralBlatt

Referenced by:

§8.25(iv), §8.9.

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The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.

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ISSN 0022-2526, MathReview (I. P. Martynov), ZentralBlatt

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§32.11(i), §32.12(ii).

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6: Bibliography C

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Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

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§8.16.

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Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

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§8.16.

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On a Class of Incomplete Gamma Functions with Applications. Chapman & Hall/CRC, Boca Raton, FL.

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ISBN 1-58488-143-7, MathReview (R. K. Raina), ZentralBlatt

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§8.16.

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Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

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ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt

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§10.46.

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Efficient and accurate computation of the incomplete Airy functions. Radio Science 28 (4), pp. 441–457.

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ISSN 0048-6604, Document

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§9.14.

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7: Bibliography B

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Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

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FullText, MathReview (J. C. P. Miller), ZentralBlatt

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§8.7.

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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

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Infinitely many Stokes smoothings in the gamma function. Proc. Roy. Soc. London Ser. A 434, pp. 465–472.

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ISSN 0962-8444, FullText, MathReview (E. Riekstiņš), ZentralBlatt

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§5.11(ii).

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The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

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§31.12.

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Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.

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Notes:

Algol 60 programs for the incomplete elliptic integral of the third kind, and for a general complete elliptic integral, are included.

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ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§19.2(iii), §19.36(ii), §19.36(ii), §19.39(ii), §19.39(iii), R. Bulirsch (1965a).

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8: Bibliography T

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The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

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ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

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§8.12.

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On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.

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FullText, MathReview (G. Meinardus), ZentralBlatt

Referenced by:

§8.25(iii).

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Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.

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ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§8.12, §8.12.

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A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.

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Notes:

Includes Pascal program

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FullText

Referenced by:

§5.24(ii), §7.25(ii), §8.28(ii).

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Asymptotics of zeros of incomplete gamma functions. Ann. Numer. Math. 2 (1-4), pp. 415–423.

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Notes:

Special functions (Torino, 1993)

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ISSN 1021-2655, FullText, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§8.13(ii).

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9: Bibliography L

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Further inequalities for the gamma function. Math. Comp. 42 (166), pp. 597–600.

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ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

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§5.6(i).

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A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

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Document, MathReview (R. D. James), ZentralBlatt

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§17.18.

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On incomplete Airy functions and their application to diffraction problems. Radio Sci. 4 (10), pp. 959–969.

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ISSN 0048-6604, Document, MathReview (T. Erber)

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§9.14.

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On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. Constr. Approx. 10 (1), pp. 15–30.

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ISSN 0176-4276, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§8.13(i).

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Inequalities for ultraspherical polynomials and the gamma function. J. Approx. Theory 40 (2), pp. 115–120.

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ISSN 0021-9045, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§18.14(i).

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10: 10.17 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for other ranges of $\mathrm{ph}z$ can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4). … ►

§10.17(iii) Error Bounds for Real Argument and Order

… ►where $\chi (\mathrm{\ell })={\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{\ell }+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{\ell }+\frac{1}{2}\right)$; see §9.7(i). … ►where $\mathrm{\Gamma }(1-p,z)$ is the incomplete gamma function (§8.2(i)). …