The gamma function with varying difference interval.
Gamma function concept for varying difference interval and complex argument
GAMMA FUNCTION↗
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High accuracy gamma function values for some rational arguments.
Tabulation of high accuracy gamma function values for some rational arguments, noting applications to Gaussian quadrature formulas, fractional Bessel functions and Airy functions
Byrd, P. F.↗
A comparison of computational methods and algorithms for the complex gamma function
A survey and comparison of some computational methods and algorithms for gamma and log-gamma functions of complex arguments are presented. Methods and algorithms reported include Chebyshev approximations, Pade expansion and Stirling's asymptotic series. The comparison leads to the conclusion that Algorithm 421 published in the Communications of ACM by H. Kuki is the best program either for individual application or for the inclusion in subroutine libraries.
Ng, E. W.↗
On Stieltjes' continued fraction for the gamma function
Explore the source record for details and available documents.
Char, B. W.↗
Irradiation of MOS-FET devices to provide desired logic functions
Gamma, X-ray, electron, or other radiation is used to shift threshold potentials of MOS devices on logic circuits. Before irradiation MOS gates to be shifted are biased positive and other gates are grounded to substrate. Threshold lasts 10 years. Thermal annealing brings circuit back to original configuration.
Danchenko, V.↗
The blazar gamma-ray luminosity function and the diffuse extragalactic gamma-ray background
We have used the data from the new EGRET catalog on 'grazars' (blazers which are observed to be high-energy gamma-ray sources), together with radio data, to construct a new relation between radio and gamma-ray luminosity for these sources. Using this relation to construct a grazar gamma-ray luminosity function, we then calculate the contribution of unresolved grazars to the cosmic gamma-ray background radiation. We derive the energy spectrum of this background component above 100 MeV and the angular fluctuations in this background implied by our model.
Salamon, M. H.↗
Improvement of Stratospheric Aerosol Extinction Retrieval from OMPS/LP Using a New Aerosol Model
The Ozone Mapping and Profiler Suite Limb Profiler (OMPS/LP) has been flying on the Suomi National Polar-orbiting Partnership (S-NPP) satellite since October 2011. It is designed to produce ozone and aerosol vertical profiles at 2-kilometer vertical resolution over the entire sunlit globe. Aerosol extinction profiles are computed with Mie theory using radiances measured at 675 nanometers. The operational Version 1.0 (V1.0) aerosol extinction retrieval algorithm assumes a bimodal lognormal aerosol size distribution (ASD) whose parameters were derived by combining an in situ measurement of aerosol microphysics with the Stratospheric Aerosol and Gas Experiment (SAGE II) aerosol extinction climatology. Internal analysis indicates that this bimodal lognormal ASD does not sufficiently explain the spectral dependence of LP-measured radiances. In this paper we describe the derivation of an improved aerosol size distribution, designated Version 1.5 (V1.5), for the LP retrieval algorithm. The new ASD uses a gamma function distribution that is derived from Community Aerosol and Radiation Model for Atmospheres (CARMA)-calculated results. A cumulative distribution fit derived from the gamma function ASD gives better agreement with CARMA results at small particle radii than bimodal or unimodal functions. The new ASD also explains the spectral dependence of LP-measured radiances better than the V1.0 ASD. We find that the impact of our choice of ASD on the retrieved extinctions varies strongly with the underlying reflectivity of the scene. Initial comparisons with collocated extinction profiles retrieved at 676 nanometers from the SAGE III instrument on the International Space Station (ISS) show a significant improvement in agreement for the LP V1.5 retrievals. Zonal mean extinction profiles agree to within 10 percent between 19 and 29 kilometers, and regression fits of collocated samples show improved correlation and reduced scatter compared to the V1.0 product. This improved agreement will motivate development of more sophisticated ASDs from CARMA results that incorporate latitude, altitude and seasonal variations in aerosol properties.
Ozone Mapping Profiler Suite Limb Profiler (OMPS/L↗
On Study of Air/Space-borne Dual-Wavelength Radar for Estimates of Rain Profiles
In this study, a framework is discussed to apply air/space-borne dual-wavelength radar for the estimation of characteristic parameters of hydrometeors. The focus of our study is on the Global Precipitation Measurements (GPM) precipitation radar, a dual-wavelength radar that operates at Ku (13.8 GHz) and Ka (35 GHz) bands. As the droplet size distributions (DSD) of rain are expressed as the Gamma function, a procedure is described to derive the median volume diameter (D(sub 0)) and particle number concentration (N(sub T)) of rain. The correspondences of an important quantity of dual-wavelength radar, defined as deferential frequency ratio (DFR), to the D(sub 0) in the melting region are given as a function of the distance from the 0 C isotherm. A self-consistent iterative algorithm that shows a promising to account for rain attenuation of radar and infer the DSD without use of surface reference technique (SRT) is examined by applying it to the apparent radar reflectivity profiles simulated from the DSD model and then comparing the estimates with the model (true) results. For light to moderate rain the self-consistent rain profiling approach converges to unique and correct solutions only if the same shape factors of Gamma functions are used both to generate and retrieve the rain profiles, but does not converges to the true solutions if the DSD form is not chosen correctly. To further examine the dual-wavelength techniques, the self-consistent algorithm, along with forward and backward rain profiling algorithms, is then applied to the measurements taken from the 2nd generation Precipitation Radar (PR-2) built by Jet Propulsion Laboratory. It is found that rain profiles estimated from the forward and backward approaches are not sensitive to shape factor of DSD Gamma distribution, but the self-consistent method is.
Liao, Liang↗
Further Development of a Function Generator Technique
Further features and properties of functions generated by generalization of the process originally uncovered to describe the evolution of pressure in a rigid volume due to outgassing or desorption are explored. Properties presented include development of a general Maclaurin series associated with these functions, a limitation in using integration by parts to produce asymptotic expansions, and a general description of an implicit Adams-Moulton method for numerical quadrature of transformed functions. These developments are then applied to develop and explore functions that solve the Sievert integral and the modified Bessel function of the first kind, order zero. Features of the error function and the incomplete lower gamma function are also considered.
Michael S. Woronowicz↗
On computing closed forms for summations
The problem of finding closed forms for a summation involving polynomials and rational functions is considered. A method closely related to Hermite's method for integration of rational functions derived. The method expresses the sum of a rational function as a rational function part and a transcendental part involving derivatives of the gamma function.
Moenck, R.↗
Reactant conversion in homogeneous turbulence: Mathematical modeling, computational validations and practical applications
Closed form analytical expressions are obtained for predicting the limited rate of reactant conversion in a binary reaction of the type F + rO yields (1 + r) Product in unpremixed homogeneous turbulence. These relations are obtained by means of a single point Probability Density Function (PDF) method based on the Amplitude Mapping Closure. It is demonstrated that with this model, the maximum rate of the reactants' decay can be conveniently expressed in terms of definite integrals of the Parabolic Cylinder Functions. For the cases with complete initial segregation, it is shown that the results agree very closely with those predicted by employing a Beta density of the first kind for an appropriately defined Shvab-Zeldovich scalar variable. With this assumption, the final results can also be expressed in terms of closed form analytical expressions which are based on the Incomplete Beta Functions. With both models, the dependence of the results on the stoichiometric coefficient and the equivalence ratio can be expressed in an explicit manner. For a stoichiometric mixture, the analytical results simplify significantly. In the mapping closure, these results are expressed in terms of simple trigonometric functions. For the Beta density model, they are in the form of Gamma Functions. In all the cases considered, the results are shown to agree well with data generated by Direct Numerical Simulations (DNS). Due to the simplicity of these expressions and because of nice mathematical features of the Parabolic Cylinder and the Incomplete Beta Functions, these models are recommended for estimating the limiting rate of reactant conversion in homogeneous reacting flows. These results also provide useful insights in assessing the extent of validity of turbulence closures in the modeling of unpremixed reacting flows. Some discussions are provided on the extension of the model for treating more complicated reacting systems including realistic kinetics schemes and multi-scalar mixing with finite rate chemical reactions in more complex configurations.
Madnia, C. K.↗
Standardized Pearson type 3 density function area tables
Tables constituting extension of similar tables published in 1936 are presented in report form. Single and triple parameter gamma functions are discussed. Report tables should interest persons concerned with development and use of numerical analysis and evaluation methods.
Cohen, A. C.↗
Results of APL rain gauge network measurements in mid-Atlantic coast region and comparisons of distributions with CCIR models
In this effort are described cumulative rain rate distributions for a network of nine tipping bucket rain gauge systems located in the mid-Atlantic coast region in the vicinity of the NASA Wallops Flight Facility, Wallops Island, Virginia. The rain gauges are situated within a gridded region of dimensions of 47 km east-west by 70 km north-south. Distributions are presented for the individual site measurements and the network average for the year period June 1, 1986 through May 31, 1987. A previous six year average distribution derived from measurements at one of the site locations is also presented. Comparisons are given of the network average, the CCIR (International Radio Consultative Committee) climatic zone, and the CCIR functional model distributions, the latter of which approximates a log normal at the lower rain rate and a gamma function at the higher rates.
Goldhirsh, Julius↗
Independent Pixel and Two Dimensional Estimates of LANDSAT-Derived Cloud Field Albedo
A theoretical study has been conducted on the effects of cloud horizontal inhomogeneity on cloud albedo bias. A two-dimensional (2D) version of the Spherical Harmonic Discrete Ordinate Method (SHDOM) is used to estimate the albedo bias of the plane parallel (PP-IPA) and independent pixel (IPA-2D) approximations for a wide range of 2D cloud fields obtained from LANDSAT. They include single layer trade cumulus, open and closed cell broken stratocumulus, and solid stratocumulus boundary layer cloud fields over ocean. Findings are presented on a variety of averaging scales and are summarized as a function of cloud fraction, mean cloud optical depth, cloud aspect ratio, standard deviation of optical depth, and the gamma function parameter Y (a measure of the width of the optical depth distribution). Biases are found to be small for small cloud fraction or mean optical depth, where the cloud fields under study behave linearly. They are large (up to 0.20 for PP-IPA bias, -0.12 for IPA-2D bias) for large v. On a scene average basis PP-IPA bias can reach 0.30, while IPA-2D bias reaches its largest magnitude at -0.07. Biases due to horizontal transport (IPA-2D) are much smaller than PP-IPA biases but account for 20% RMS of the bias overall. Limitations of this work include the particular cloud field set used, assumptions of conservative scattering, constant cloud droplet size, no gas absorption or surface reflectance, and restriction to 2D radiative transport. The LANDSAT data used may also be affected by radiative smoothing.
Chambers, L. H.↗
"Novel Techniques in Non-Stationary Analysis of Rotorcraft Vibration Signitures"
This research effort produced new methods to analyze the performance of linear predictors that track non-stationary processes. Specifically, prediction methods have been applied to the vibration pattern of rotorcraft drivetrains. This analysis is part or a larger rotorcraft Health and Usage Monitoring System (HUMS) that can diagnose immediate failures of the subsystems, as indicated by abrupt change in the vibration signature, and prognosticate future health, by examining the vibration patterns against long-term trends. This problem is described by a earlier joint paper co-authored by members of the funding agency and the recipient institutions prior to this grant effort. Specific accomplishments under this grant include the following: (1) Definition of a framework for analysis of non-stationary time-series estimation using the coefficients of an adaptive filter. (2) Description of a novel method of combining short-term predictor error and long-term regression error to analyze the performance of a non-stationary predictor. (3) Formulation of a multi-variate probability density function that quantifies the performance of a adaptive predictor by using the short- and long-term error variables in a Gamma function distribution. and (4) Validation of the mathematical formulations with empirical data from NASA flight tests and simulated data to illustrate the utility beyond the domain of vibrating machinery.
Meng, Teresa↗
The Gamma-Ray Luminosity Function of Radio Pulsars
This final report is a study of gamma-ray luminosity function of radio pulsars. The goal is to constrain certain parameters in order to address such diverse issues as the high energy emission mechanism in pulsars and the fraction of the Galaxy's gamma ray emission attributable to these objects.
Helfand, David J.↗
Computation and analysis
Developing high-speed variable accuracy method for computing ordered sequences of incomplete gamma function
Source record↗
Beta distributions: A computer program for probabilities and fractile points
A beta distribution is specified by range parameters a b, and two shape parameters alpha and beta 0. The computer program presented calculates any desired probability and/or fractile point for specified values of a, b, alpha, and beta. This program additionally computes gamma function values for integer and noninteger arguments.
Brownlow, J. D.↗