New bounds for the function involving incomplete gamma function

We denote by and the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to , a lower bound established by Elbert and Laforgia (2000) for the function , with , only for .

در این تحقیق به منظور تعیین بهترین منحنی تولید شیر در یک دوره شیردهی از 260709 رکورد روز آزمون تولید شیر متعلق به 30903 رأس گاو هلشتاین شکم اول استان خراسان استفاده شد. اطلاعات مورد استفاده مربوط به گاو‌های 216 گله بود که طی سالهای 1367 الی 1385 زایش داشتند. توابع مورد استفاده شامل تابع گامای ناقص (وود)، تابع ویلمینک، تابع علی‌شفر و تابع لژاندر (با درجه برازش پنجم) بود. معیارهای مورد نظر برای مقایسه توابع شامل مجموع مربعات خطا، میانگین مجموع مربعات خطا، ضریب تعیین، ضریب تعیین تصحیح شده ، ضریب همبستگی و فاکتور تورم واریانس بود. براساس تمام معیارها به استثناء فاکتور تورم واریانس تابع علی- شفر و براساس فاکتور تورم واریانس تابع لژاندر به نحو مطلوب‌تری با داده‌های تولید شیر مطابقت داشتند.

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.

A new analytical formula for the Arrhenius integral has been proposed by using incomplete gamma functions, which is simple, accurate and reliable. The proposed formula has compared with several published the Arrhenius integral approaches, and is in agreement with the other approaches. Compared with the other approximate formulas proposed in the literature, our proposed method gives more accurate values in the precision of the activation energy as a function of x and temperature.

This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable s. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated.

Chapter 4 - Special functions

In this work the neutral meson properties have been investigated in the presence of thermo-magnetic background using two-flavor Nambu--Jona-Lasinio model. Mass, spectral function and dispersion relations are obtained in the scalar ($\sigma$) and pseudo-scalar ($\pi^0$) channels as well as in the vector ($\rho^0$) and axial vector ($a^0_1$) channels. The general Lorentz structures for the vector and axial-vector meson polarization functions have been considered in detail. The ultra-violet divergences appearing in this work have been regularized using a mixed regularization technique where the gamma functions arising in dimensional regularization are replaced with incomplete gamma functions as usually done in the proper time regularization procedure. The meson spectral functions obtained in the presence of a magnetic field possess nontrivial oscillatory structure. Similar to the scalar and pseudo-scalar channel, the spectral functions for each of the modes of $\rho^0$ are observed to overlap with the corresponding modes of its chiral partner $a_1^0$ mesons in the chiral symmetry restored phase. We observe discontinuities in the masses of all the mesonic excitations for a non-zero external magnetic field.

We construct asymptotic expansions for the normalised incomplete gamma function Q ( a , z ) = Γ ( a , z ) / Γ ( a ) Q(a,z)=\Gamma (a,z)/\Gamma (a) that are valid in the transition regions, including the case z ≈ a z\approx a , and have simple polynomial coefficients. For Bessel functions, these types of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as a → − ∞ a\to -\infty of the unique negative zero of the regularised incomplete gamma function γ ∗ ( a , x ) \gamma ^*(a,x) .

The object of this study was to test the incomplete gamma function previously developed as a descriptor for grapite disintegration. Samples of disintegrating granite were taken from several sites at three locations in a hot semiarid environment in south‐central Africa (southern Rhodesia). These samples were from surface rock, profile corestones, and disintegrating rock at the base of the soil profile. Particlesize distributions were obtained using the standard sieving and pipette method. For materials > 10 mm, Martin's diameter was measured directly. The incomplete gamma function was fitted to these data, and also to U.S. and French data from the literature, for granite in an apparently similar state of disintegration. The values for the degree‐of‐disintegration parameter were fairly uniform. The mode of disintegration appeared the same regardless of site. The two‐parameter incomplete gamma distribution was found to be a reasonably good descriptor of the particle‐size distribution of disintegrating granite. The mean size appeared to be a sensitive and useful single‐valued descriptor for these distributions.

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeros of the Hurwitz zeta function ζ(s,z), including the sum over the inverse first power of its appropriately defined non-trivial zeros. We also study some related properties of the Hurwitz zeta function zeros. In particular, we show that, for any natural N and small real ε, when z tends to n = 0, −1, −2… we can find at least N zeros of ζ(s,z) in the ε neighborhood of 0 for sufficiently small |z+n|, as well as one simple zero tending to 1, etc.

In this paper we use a contour integral method to derive a bilateral generating function in the form of a double series involving Chebyshev polynomials expressed in terms of the incomplete gamma function. Generating functions for the Chebyshev polynomial are also derived and summarized. Special cases are evaluated in terms of composite forms of both Chebyshev polynomials and the incomplete gamma function.

New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function. In each case the expansion contains the complementary error function and an asymptotic series. The expansions are uniformly valid with respect to certain domains of the parameters.

Traditionally, the moments of the Weibull distribution have been calculated using the standard Weibull (Johnson and Kotz, 1970) . This article will expand on that idea and cover the truncated cases for the standard Weibull distributions. Also, the same techniques used for the standard form will be used to derive the moment expressions for the three-parameter complete and truncated Weibull distributions. The summary statistics are then calculated from the moment expressions. Weibull moments involve the gamma and incomplete gamma functions.

We examine the asymptotic nature as k → ∞ of the coefficientsck(�) appearing in the uniform asymptotic expansion of the incomplete gamma function ( a,z) whereis a variable that depends on the ratio z/a. It is shown that this expansion diverges like the familiar factorial divided by a power dependence multiplied by a function fk(�). For values ofnear the real axis, fk(�) is a slowly varying function, but in the left half-plane, there are two lobes situated symmetrically about the negative realaxis in which fk(�) becomes large. The asymptotic expansion of fk(�) as k → ∞ is found to reveal a resurgence-type structure in which the high-order coefficients are related tothe low-order coeffi- cients. Numerical examples are given to illustrate the growth of the coefficients ck(�).