gamma distribution
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11: 19.31 Probability Distributions
§19.31 Probability Distributions
► and occur as the expectation values, relative to a normal probability distribution in or , of the square root or reciprocal square root of a quadratic form. … ►
19.31.2
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12: 25.10 Zeros
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§25.10(i) Distribution
… ►In the region , called the critical strip, has infinitely many zeros, distributed symmetrically about the real axis and about the critical line . … ►
25.10.2
►is chosen to make real, and assumes its principal value.
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13: 9.12 Scorer Functions
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9.12.6
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9.12.24
►where the integration contour separates the poles of from those of .
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►where is Euler’s constant (§5.2(ii)).
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►For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c).
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14: Bibliography J
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Distributions of matrix variates and latent roots derived from normal samples.
Ann. Math. Statist. 35 (2), pp. 475–501.
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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles.
Eurandom Preprint Series
Technical Report 14, Eurandom, Eindhoven, The Netherlands.
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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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Continuous Univariate Distributions.
2nd edition, Vol. I, John Wiley & Sons Inc., New York.
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Continuous Univariate Distributions.
2nd edition, Vol. II, John Wiley & Sons Inc., New York.
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15: 10.21 Zeros
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§10.21(i) Distribution
… ►Some information on the distribution of and for real values of and is given in Muldoon and Spigler (1984). … ►§10.21(ix) Complex Zeros
… ►For describing the distribution of complex zeros by methods based on the Liouville–Green (WKB) approximation for linear homogeneous second-order differential equations, see Segura (2013). … ► …16: Errata
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►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions.
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Rearrangement
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Subsection 25.2(ii) Other Infinite Series
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Subsection 1.16(vii)
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Subsection 1.16(viii)
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In previous versions of the DLMF, in §8.18(ii), the notation was used for the scaled gamma function . Now in §8.18(ii), we adopt the notation which was introduced in Version 1.1.7 (October 15, 2022) and correspondingly, Equation (8.18.13) has been removed. In place of Equation (8.18.13), it is now mentioned to see (5.11.3).
Several changes have been made to
An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.
17: Bibliography L
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Monotonicity results and inequalities for the gamma and error functions.
J. Comput. Appl. Math. 23 (1), pp. 25–33.
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Further inequalities for the gamma function.
Math. Comp. 42 (166), pp. 597–600.
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Algorithm 344: Student’s t-distribution [S14].
Comm. ACM 12 (1), pp. 37–38.
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Sur la distribution des nombres premiers.
Comptes Rendus de l’Academie des Sciences, Paris 158, pp. 1869–1872 (French).
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Inequalities for ultraspherical polynomials and the gamma function.
J. Approx. Theory 40 (2), pp. 115–120.
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18: 12.11 Zeros
19: 25.16 Mathematical Applications
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§25.16(i) Distribution of Primes
►In studying the distribution of primes , Chebyshev (1851) introduced a function (not to be confused with the digamma function used elsewhere in this chapter), given by … ►
25.16.6
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20: Bibliography D
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The computation of lattice frequency distribution functions by means of continued fractions.
Physica 39 (3), pp. 413–423.
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Computation of the incomplete gamma function ratios and their inverses.
ACM Trans. Math. Software 12 (4), pp. 377–393.
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Algorithm 654: Fortran subroutines for computing the incomplete gamma function ratios and their inverses.
ACM Trans. Math. Software 13 (3), pp. 318–319.
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Algorithm 322. F-distribution.
Comm. ACM 11 (2), pp. 116–117.
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On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function.
Methods Appl. Anal. 5 (3), pp. 223–247.
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