Gamma Function

In this research Riemann hypothesis is investigated for a proof. A functional extension of the Riemann zeta function is proposed for which non trivial zeroes can be generated. It found that non trivial zeroes can also be generated outside the critical strip. Thus it is found that the Riemann hypothesis is found to be incomplete.

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks . The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant, γ.

This study demonstrated that the Standardised Flow Index (SFI) was a simple and useful tool to research, monitor and manage hydrologic drought in a highly regulated river system, the Murrumbidgee River in southeast Australia. To validate the applicability of the theory underlining the widely used Standardised Precipitation Index (SPI) to river discharge data, we investigated the probability distribution of the time series of monthly river discharge month by month using long-term (over 100 years) river flow records. Our results showed that the Gamma probability distribution function was adequate to describe and model the skewed river flow data. The generalised additive models (GAM) with Locally Estimated Scattersplot Smoothing (LOESS) additive terms were applied to the computed SFI and SPI sequences to investigate the impacts of river regulation and water diversion on the duration and magnitude of hydrologic droughts in Lower Murrumbidgee River from 1890. The results revealed that upstream regulations had successfully reduced the drought severity at Wagga Wagga, a weir located downstream of the two major dams but immediately upstream of the major irrigation areas. However, the hydrological benefits of river regulation gradually disappeared as the river travels downstream and more and more water abstracted. At Balranald, the end valley weir, hydrologic drought was progressively aggravated during the modelling period, and the impacts were greater during drier periods. The results of the study highlighted the importance of balancing the needs between upstream and downstream water users in river management.

In 1859 Riemann defined the zeta function () s . From Gamma function he derived the zeta function with Gamma function () s . () s  and () s  are the two different functions. It is false that () s  replaces () s . After him later mathematicians put forward Riemann hypothesis(RH) which is false. The Jiang function () n J  can replace RH.

We examined two subjectively distinct memory states that are elicited during recognition memory in humans and compared them in terms of the gamma oscillations (20-60 Hz) in the electroencepahalogram (EEG) that they induced. These subjective states, 'recollection' and 'familiarity' both entail correct recognition but one involves a clear and conscious recollection of the event including memory for contextual detail whilst the other involves a sense of familiarity without clear recollection. Here we show that during a verbal recognition memory test, the subjective experience of 'recollection' induced higher amplitude gamma oscillations than the subjective experience of 'familiarity' in the time period 300-500 ms after stimulus presentation. Recollection, but not familiarity, was also associated with greater functional connectivity in the gamma frequency range between frontal and parietal sites. Furthermore, the magnitude of the gamma functional connectivity varied over time and was modulated at 3 Hz. Previous studies in animals have shown local theta frequency modulation (3-7 Hz) of gamma-oscillations but this is the first time that a similar effect has been reported in the human EEG. ᮊ

We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincaré density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean n-gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic n-gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author.

The aim of this paper is to introduce an approximations family of the factorial function that contains Stirling's formula, Burnside's formula and Gosper's formula. The parameters which provide the best approximations are indicated. Finally, numerical computations are made to show the superiority of our formulas over other known formulas.

In this research Riemann hypothesis is investigated for a proof. A functional extension of the Riemann zeta function is proposed for which non trivial zeroes can be generated. It found that non trivial zeroes can also be generated outside the critical strip. Thus it is found that the Riemann hypothesis is found to be incomplete.

Wallis's method of interpolation attracted the attention of the young Euler, who obtained some important results. The problem of interpolation led Euler to formulate the problem of integration, i.e., to express the general term of a series by means of an integral. The latter problem was connected to the question of expressing the sum of a series using an integral. The outcome of this research was Euler's derivation of what would later become known as the Euler–Maclaurin formula. Euler subsequently returned to interpolation and formulated the theory of inexplicable functions including the gamma function.The methods used by Euler illustrate well the principles of 18th-century analysis. Eulerian procedures are based upon the notion of geometric quantity. A function is actually conceived as the expression of a quantity and, for this reason, it intrinsically possesses properties we can term continuity, differentiability, Taylor expansion. These correspond to the usual properties of a curve which has “regular” characteristics (lack of jumps, presence of tangents, curvature radius, etc.). They have a “figural” clarity. Although Eulerian analysis remains rooted in geometry, it dispenses with figural representation: it is substantially nonfigural geometry. Reasoning with figures (which integrates the proof in classical geometry) is replaced by reasoning with analytic symbols. These are general because they do not represent a particular quantity and are not subjected to restrictions, but are an abstract representation of quantity.Copyright 1998 Academic Press.Il problema dell'interpolazione wallisiana attrasse l'attenzione del giovane Euler. Egli ottenne rapidamente risultati di grande interesse e proseguı̀ le sue ricerche formulando il problema dell'integrazione consistente nell'esprimere il termine generale di una serie mediante un integrale. Quest'ultimo problema ben presto si evolse nella questione di esprimere la somma di una serie mediante un integrale che guidò Euler nella derivazione della formula sommatoria oggi detta di Euler-Maclaurin. Euler ritornò in seguito sul problema dell'interpolazione sviluppando la teoria delle funzioni inesplicabili che comprendono come caso particolare la gamma.I metodi usati da Euler sono di grande interesse per una comprensione dell'analisi settecen- tesca. Infatti alla base delle procedure euleriane vi è la nozione di quantità geometrica, la quale comporta che la funzione, intesa come espressione della quantità, abbia intrinsecamente proprietà che, con linguaggio moderno, possiamo chiamare continuità, differenziabilità, sviluppabilità in serie de Taylor, e che corrispondono alle proprietà tipiche di una curva dotata di “regolarità” (non fare salti, esistenza della tangente, del raggio di curvatura, ecc.). Queste proprietà hanno un'immediata evidenza nelle curve comunemente studiate, un'evidenza, per cosi dire, figurale. L'analisi euleriana conserva un forte contenuto geometrico, ma elimina la rappresentazione figurale: sostanzialmente essa appare come una geometria non figurale. Il ragionamento sulla figura (che nella geometria classica integrava la dimostrazione) viene ora sostitutito dal ragionamento sui simboli analitici, i quali sono generali, perché non rappresentano questa o quella particolare quantità e non sone soggetti a limitazioni di sorta, ma sono una rappresentazione astratta della quantità.Copyright 1998 Academic Press.Le problème d'interpolation de Wallis attira l'attention du jeune Euler, qui obtint rapidement des résultats de grand intérêt. Il amena Euler à formuler le problème d'intégration consistant à exprimer le terme général d'une série par une intégrale. Ce dernier problème se transforma rapidement en celui d'exprimer la somme d'une série par une intégrale, menant à la formule appelée de nos jours la formule d'Euler-Mac Laurin. Euler reprit ultérieurement le problème d'interpolation, formulant la théorie des fonctions “inexplicables” qui incluent en particulier la fonction Gamma.Les méthodes utilisées par Euler ont un grand intérêt pour comprendre l'analyse du XVIIIe siècle. En effet, à la base des procédures eulériennes, on trouve la notion de quantité géométrique, qui implique que la fonction, comprise comme une expression de cette quantité, possède intrinsèquement certaines propriétés—en langage moderne la continuité, la différentiabilité, le développement en séries de Taylor—correspondant aux propriétés typiques d'une courbe dotée de régularité—ne pas faire de sauts, avoir une tangente, un rayon de courbure. Ces propriétés étaient évidentes pour les courbes communément étudiées, une évidence pour ainsi dire figurale. L'analyse eulérienne conserve un important contenu géométrique, mais élimine la représentation figurale: concrêtement, elle apparaı̂t comme une géométrie non figurale. Le raisonnement sur la figure, qui, dans la géométrie classique, intégrait la démonstration, est ici remplacé par le raisonnement sur les symboles analytiques: ceux-ci sont généraux parce qu'ils ne représentent pas une quantité particulière et ne sont pas soumis à des restrictions, mais sont une représentation abstraite de la quantité.Copyright 1998 Academic Press.AMS 1991 subject classifications: 01A50

In this article, we study lower and upper triangular factorizations of the complex Wishart matrix. Further, using these factorizations, we obtain several expected values of scalar and matrix valued functions of the complex Wishart matrix. We also generalize Muirhead’s identity for the complex case which gives a number of interesting special cases.

The batch flotation process has been commonly characterized assuming a flotation rate distribution function F(k), e.g.: Dirac delta, Rectangular, Gamma or Weibull functions. The identification of F(k) for the collection zone of continuous industrial cells is more complex and to the authors knowledge, has not been reported yet.

The frequency-length distribution of the San Andreas fault system was analyzed and compared with theoretical distributions. Both.density and cumulative distributions were calculated, and errors were estimated. Neither exponential functions nor power laws are consistent with the calculated distributions over the range of studied lengths. The best fit on both density and cumulative distributions was achieved with a gamma function which mixes a power law and an exponential function. At small lengths, the gamma function behaves as a power law with an exponent of-1.3_-_4-0.3. At large lengths (above 10 km), the distribution is a mixed exponential-power law function with a characteristic length scale of about 23+6 km. The gamma distribution is proposed to result from a length-dependent segmentation of a fractal fault pattern. This study shows the importance of comparing both cumulative and density distributions. It also shows that the studied range of lengths (1-100 km) is not appropriate for measuring power law exponents. 12,141 P. Davy, G6oSciences Rennes, UPR 4661 CNRS

The batch flotation process has been commonly characterized assuming a flotation rate distribution function F(k), e.g.: Dirac delta, Rectangular, Gamma or Weibull functions. The identification of F(k) for the collection zone of continuous industrial cells is more complex and to the authors knowledge, has not been reported yet.

In this paper we analyze the different generating  eta − and zeta functions, which in return produce the ME numbers that are directly linked to the Bernoulli −. We establish the characteristics of these numbers, their computation, their values and the domain of the eta − and zeta functions in the complex plain by analytic continuation. Further, we derive formulas for specific values of each function. Then we will analyze their connection with the Γ − and the Riemann ζ function. We also take a look at root series and infinite products representations of periodic polynomials.

Disdrometer data measured during the passage of tropical continental squall lines in Darwin, Australia, are analyzed to study characteristics of raindrop size distribution (DSD). Fifteen continental squall lines were selected for the DSD analysis. An observed squall line was partitioned into three regions based on radar reflectivity pattern, namely, convective line, stratiform, and reflectivity trough. A convective line was further

Models used to predict digestibility and fill of the dietary insoluble fibre (NDF) treat the ruminoreticular particulate mass as a single pool. The underlying assumption is that escape of particles follows firstorder kinetics. In this paper, we proposed and evaluated a model of two ruminoreticular sequential NDF pools. The first pool is formed by buoyant particles (raft pool) and the second one by fluid dispersed particles (escapable pool) ventrally to the raft. The transference of particles between these two pools results from several processes that reduce particles buoyancy, assuming the gamma distribution. The exit of escapable pool particles from the ruminoreticulum is exponentially distributed. These concepts were evaluated by comparing ruminoreticular NDF masses as 43 and 27 means from cattle and sheep, respectively, to the same predicted variable using single-and two-pools models. Predictions of the single-pool model were based on lignin turnover and the turnover associated to the descending phase of the elimination of Yb-labelled forage particles in the faeces of sheep. Predictions of the twopool model were obtained by estimating fractional passage rates associated to the ascending and descending phases of the same Yb excretion profiles in sheep faeces. All turnovers were scaled to the power 0.25 of body mass for interspecies comparisons. Predictions based on lignin turnover (single pool) and the two-pool model presented similar trends, accuracies and precisions. The single-pool approach based solely on the descending phase of the marker yielded biased estimates of the ruminoreticular NDF mass.

Many authors introduced and studied positive linear operators, using Euler's gamma function Γp, p > 0. We shall define a more general linear transform Γ (a,b) p , a, b ∈ R, from which we obtain as particular cases the gamma first-kind transform and the gamma second-kind transform.

We deal here with a class of integral transformations with respect to parameters of hypergeometric functions or the index transforms. In particular, we treat the familiar Olevskii transform, which is associated with the Gauss hypergeometric function as a kernel. It involves, in turn, as particular cases index transforms of the Mehler-Fock type which are used in the mathematical theory of elasticity. Boundedness L2properties for the Olevskii transform are investigated. The Plancherel theorem is proved. It shows that the Olevskii transform is an isometric isomorphism between two weighted L2 -spaces. More examples of such isomorphisms are exhibited for the Mehler-Fock type transforms.

From the characterisation of geometrically convex and geometrically concave functions defined on (0, A] or [A, ∞) with A > 0, by means of their multiplicative conditions, we obtain unified proofs of some known and new inequalities. Functions of class C 2 and strictly increasing on (a, b) fulfil some kind of supermultiplicativity and superadditivity. We have obtained a new constant determining the intervals of sub-and supermultiplicativity for the log function.

We discuss q-analogues of the Euler reflection formula and the Euler gamma integral. The central role here is played by the Ramanujan q-extension of the Euler integral representation for the gamma function, which allows deriving the Mellin integral transformations for the q-polynomials of Stieltjes-Wigert, Rogers-Szegö, Laguerre, and Wall, for the alternative q-polynomials of Charlier, and for the little q-polynomials of Jacobi.

This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are evaluated.

In this paper, we introduce a function , ; ( , , ) z s a     , which is an extension to the general Hurwitz-Lerch Zeta function. Having defined the incomplete generalized beta type-2 and incomplete generalized gamma functions, some differentiation formulae are established for these incomplete functions. We have introduced two new statistical distributions, termed as generalized Hurwitz-Lerch Zeta beta type-2 distribution and generalized Hurwitz-Lerch Zeta gamma distribution and then derived the expressions for the moments, distribution function, the survivor function, the hazard rate function and the mean residue life function for these distributions. Graphs for both these distributions are given, which reflect the role of shape and scale parameters.

From the integration of non-symmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. We show that functions characterizing complex systems can be conveniently written in terms of this generalization of the exponential function. The gamma function is then generalized and we generalize the factorial operation. Also a very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one-and two-tail stretched exponential functions. One obtains, as particular cases, the generalized error function, the Zipf-Mandelbrot probability density function (pdf), the generalized gaussian and Laplace pdf. One can also obtain analytically their cumulative functions and moments.

On the one hand the Fermi-Dirac and Bose-Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand the Fourier transform representation of the gamma and generalized gamma functions proved useful in deriving various integral formulae for these functions. In this paper we use the Fourier transform representation of the extended functions to evaluate integrals of products of these functions. In particular we evaluate some integrals containing the Riemann and Hurwitz zeta functions, which had not been evaluated before.

This paper describes an accurate method to obtain the Tone Reproduction Curve (TRC) of display devices without using a measurement device. It is an improvement of an existing technique based on human observation, solving its problem of numerical instability and resulting in functions in log-log scale which correspond better to the nature of display devices. We demonstrate the effiency of our technique on different monitor technologies, comparing it with direct measurements using a spectrophotometer.

A Rodrigues-type representation for the second kind solutions of a second-order di erential equation of hypergeometric type is given. This representation contains some integrals related with relevant special functions. For these integrals, a general recurrence relation, which only involves the coe cients of the di erential equation, is also presented. Finally, an extension of the Rodrigues type representation for the second solution of a second-order di erence equation of hypergeometric type is indicated.

In this article we study a generalisation of Lalescu's sequence involving the Gamma function. n √ n!. The sequence (Ln)n∈N , also known as Lalescu's sequence, has been studied by many Romanian mathematicians and it has been shown that it converges to 1/e. For a recent study of this sequence as well as related Lalescu-like sequences, we recommend (3) and

This paper continues the work by Molchan ( , 1991b, who has considered earthquake prediction as a problem of the optimization of a certain loss function -y. FUnction -y is defined by specific social, economic, and geophysical goals. This problem can be fully solved for the loss function -y dependent on only two parameters: fraction of alarm time, r, and fraction of failures to predict, v. For such a loss function the tasks of geophysicists and decision makers are clearly defined and can be separated; the requirements of the prediction algorithm can be completely formulated in terms of hazard function. We also consider a more complex model of the loss function, introducing a finite number of various alarms. For each of these alarms, -y depends on three parameters; in addition to r and v, we take into account the total number of alarms, A. For this model we find strategies which optimize the losses for each time unit (locally optimal). Contrary to the simple case of -y(v',y), such strategies may not be optimal globally (for the entire time interval). We determine the conditions under which the locally optimal strategy becomes globally optimal. We illustrate our results using the model of a short-term earthquake prediction for central California. We emphasize that for this complex case, the tasks of geophysicists and decision makers in prediction are intertwined. Prediction and mitigation of many other natural disasters may benefit from the alert strategies discussed here.

There is a large body of evidence that stress-induced DNA damage may be responsible for cell lethality, cancer proneness and/or immune reaction. However, statistical features of their repair rate remain poorly documented. In order to interpret the shape of the radiation-induced DNA damage repair curves with a minimum of biological assumptions, we introduced the concept of repair probability, specific to any individual radiation-induced DNA damage, whatever its biochemical type. We strengthened the apparent paradox that the repair rate of a population of DNA damage is time-dependent even if the repair rate of the individual DNA damage is constant. Hence, the existing models, based on a dual approach of the DNA repair may be insufficient for describing the DNA repair rate over a large range of repair times. Since the repair probability of DNA damage cannot be assessed individually, the measurement of the DNA repair rate is assumed to consist in determining the instantaneous mean of all repair probabilities. The relevance of this model was examined with different endpoints: cell species, genotypes, radiation type and chromatin condensation. The Euler's Gamma function was shown to provide the distribution the most consistent with such hypotheses. Furthermore, formulas, deduced from the Gamma distribution, were found to be compatible with our previous model, empirically defined but based on a variable repair half-time. r

The shape of the lactation curve for 475 Turkish Holsteins was estimated by fitting a gamma function to daily milk yields from monthly recording of 754 lactations. Lactation curve traits that were analyzed included a scaling factor associated with yield at the beginning of lactation, the inclining and declining slopes before and after peak yield, DIM at peak yield, and peak and lactation yields. Persistency of lactation yield was measured from 1) the gamma function, 2) the coefficient of variation for monthly test-day yields, and 3) the ratio of lactation yield to peak yield. The log-transformed gamma function explained 71% of variation in daily yield. Effects of farm operation, calving year, calving season, parity, and service period were significant for the various lactation curve traits. Peak and lactation yields were higher for cows that calved in fall and winter, and persistency was higher for cows that calved in summer and fall. Peak and lactation yields were lower, but persistency was higher during first lactation. Repeatability estimates were moderate for peak (0.26) and lactation (0.34) yields and lower (0.06 to 0.20) for other lactation curve traits. (Key words: lactation curve, persistency, milk yield, Turkey)

This study compared fullframe fisheye photography and cover photography with destructive leaf area index (L) estimation and the Licor LAI-2000 plant canopy analyser (PCA) in plantations of the vertical leaved species Eucalyptus globulus. Fullframe fisheye photography differs from circular fisheye photography in that the images have reduced field of view such that the zenithal range of 0-908 extends to the corners of the rectangular image, roughly doubling image resolution compared to circular images. Cover images instead are obtained by pointing a 70 mm equivalent focal length lens (in 35 mm format) straight upwards. Measurements of cover and indirect estimates of plant area index (L t ) were made in 12 stands of 6-8 years old Eucalyptus globulus. L was measured using destructive sampling and allometry in nine of these stands and ranged from 2.5 to 6.6. Both foliage cover and L t from the PCA were well correlated with L from allometry, but fullframe fisheye photography provided poor estimates of L despite corrections for foliage clumping. Sampling location had a significant effect on estimates of crown porosity, crown cover and zenithal clumping index from cover photography. The zenithal extinction coefficient (k), calculated from L, crown porosity and cover, ranged from 0.14 to 0.25 and appeared to decrease as L increased; hence, we were unable to obtain an unambiguous estimate of k for E. globulus stands. Nonetheless, the study showed that L can be estimated from foliage cover with similar certainty to that of the PCA. We conclude that the greatest challenge facing indirect estimation of L in forests using photographic methods is to separate the effects of foliage angle from those of foliage clumping. #

In this paper, we continue to study properties of rational approximations to Euler's constant and values of the Gamma function defined by linear recurrences, which were found recently by A. I. Aptekarev and T. Rivoal. Using multiple Jacobi-Laguerre orthogonal polynomials we present rational approximations to the quotient of values of the Gamma function at rational points. As a limit case of our result, we obtain new explicit formulas for numerators and denominators of the Aptekarev approximants to Euler's constant.

We performed a comparison between two source signal extraction algorithms, namely the Wavelet Denoising (WD) by Soft Thresholding and Independent Component Analysis (ICA) on a simulated functional optical imaging data. The simulated data are generated by combining a gamma function superimposed on a very low frequency sine wave as the source data and the additive noise components are chosen as having both Gaussian and non-Gaussian parts. We observed that ICA denoising outperforms significantly wavelet denoising scheme when the signal-to-noise ratio (SNR) decreases to below 0 dB.

We introduce the Stirling's formula in a more general class of approximation formulas to extend the integral representation of Z. Liu [Tamsui Oxf. J. Math. Sci. 4 (2007) 389-392]. Finally, an accurate approximation for the factorial function is established. MSC 2000: 05A10; 41A60; 41A80; 26D15; 40A05

The aim of this paper is to re…ne some recent results stated by Alzer and Grinshpan [Inequalities for the gamma and q-gamma functions J. Approx. Theory 144 67 -83] and Batir [An interesting double inequality for Euler's gamma function J. Inequal. Pure Appl. Math. 5(4) article 97 (2004)] about the problem of approximation of the gamma function in terms of the digamma function.