Complementary Incomplete Gamma Function Research Articles

Certain inequalities for the modified Bessel-type function

We establish some new inequalities for the modified Bessel-type function lambda _{nu ,sigma }^{(beta )} (x ) studied by Glaeske et al. [in J. Comput. Appl. Math. 118(1–2):151–168, 2000] as the kernel of an integral transformation that modifies Krätzel’s integral transformation. The inequalities obtained are closely related to the generalized Hurwitz–Lerch zeta function and complementary incomplete gamma function. We also deduce some useful inequalities for the modified Bessel function of the second kind K_{nu } (x ) and Mills’ ratio mathsf{M} (x ) as worthwhile applications of our main results.

The incomplete gamma functions

Recall the integral definition of the gamma function: for a > 0. By splitting this integral at a point x ⩾ 0, we obtain the two incomplete gamma functions:(1)(2)Γ(a, x)is sometimes called the complementary incomplete gamma function. These functions were first investigated by Prym in 1877, and Γ(a, x) has also been called Prym's function. Not many books give these functions much space. Massive compilations of results about them can be seen stated without proof in [1, chapter 9] and [2, chapter 8]. Here we offer a small selection of these results, with proofs and some discussion of context. We hope to convince some readers that the functions are interesting enough to merit attention in their own right.

Uniform bounds for the complementary incomplete Gamma function

We prove upper and lower bounds for the complementary incomplete gamma function $\G(a,z)$ with complex parameters $a$ and $z$. Our bounds are refined within the circular hyperboloid of one sheet $\{(a,z):|z|>c|a-1|\}$ with $a$ real and $z$ complex. Our results show that within the hyperboloid, $|\G(a,z)|$ is of order $|z|^{a-1}e^{-\Re(z)}$, and extends an upper estimate of Natalini and Palumbo to complex values of $z$.

Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems

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. Several specific differential equations are presented, and their initialization responses are found for a variety of initializations. Both the time-domain and Laplace-domain solutions are obtained and compared. The complementary incomplete gamma function is shown to be essential in finding the Laplace-domain solution of a fractional-order differential equation.

A General Method for Calculating Error Probabilities Over Fading Channels

Signal fading is a ubiquitous problem in mobile and wireless communications. In digital systems, fading results in bit errors, and evaluating the average error rate under fairly general fading models and multichannel reception is often required. Predominantly to date, most researchers perform the averaging using the probability density function method or the moment generating function (MGF) method. This paper presents a third method, called the characteristic function (CHF) method, for calculating the average error rates and outage performance of a broad class of coherent, differentially coherent, and noncoherent communication systems, with or without diversity reception, in a myriad of fading environments. Unlike the MGF technique, the proposed CHF method (based on Parseval's theorem) enables us to unify the average error-rate analysis of different modulation formats and all commonly used predetection diversity techniques (i.e., maximal-ratio combining, equal-gain combining, selection diversity, and switched diversity) within a single common framework. The CHF method also lends itself to the averaging of the conditional error probability involving the complementary incomplete Gamma function and the confluent hypergeometric function over fading amplitudes, which heretofore resisted to a simple form. As an aside, we show some previous results as special cases of our unified framework.

Some expansions of the exponential integral in series of the incomplete Gamma function

In a recent paper in this journal, Gautschi et al. [W. Gautschi, F.E. Harris, N.M. Temme, Expansions of the exponential integral in incomplete Gamma functions, Appl. Math. Lett. 16 (2003) 1095–1099] presented an interesting expansion formula for the exponential integral E 1 ( z ) in a series of the incomplete Gamma function γ ( α , z ) . Their investigation was motivated by a search for better methods of evaluating the exponential integral E 1 ( z ) which occurs widely in applications, most notably in quantum-mechanical electronic structure calculations. The object of the present sequel to the work by Gautschi et al. [Expansions of the exponential integral in incomplete Gamma functions, Appl. Math. Lett. 16 (2003) 1095–1099] is to give a rather elementary demonstration of the aforementioned expansion formula and to show how easily it can be put in a much more general setting. Some analogous expansion formulas in series of the complementary incomplete Gamma function Γ ( α , z ) are also considered.

Expansions of the exponential integral in incomplete gamma functions

An apparently new expansion of the exponential integral E 1 in incomplete gamma functions is presented and shown to be a limiting case of a more general expansion given by Tricomi in 1950 without proof. This latter expansion is proved here by interpreting it as a “multiplication theorem”. A companion result, not mentioned by THcomi, holds for the complementary incomplete gamma function and can be applied to yield an expansion connecting E 1 of different arguments. A general method is described for converting a power series into an expansion in incomplete gamma functions. In a special case, this provides an alternative derivation of Tricomi's expansion. Numerical properties of the new expansion for E 1 are discussed.

Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function

Uniform asymptotic expansions are derived for solutions of the differential equation d 2 W/dζ 2 = ( u 2 ζ 2 + βu + ψ( u , ζ))W, which are uniformly valid for u real and large, β bounded (real or complex), and £ lying in a well-defined bounded or unbounded complex domain, which contains the origin. The function ψ(u, ζ) is assumed to be holomorphic in this domain, and is o ( u / ln( u )) uniformly as u —> oo. The approximations involve parabolic cylinder functions, and include explicit and realistic error bounds. The new theory is then applied to the complementary incomplete gamma function Γ( α , α x ), furnishing an asymptotic approximation which is uniformly valid for x lying in a complex domain which properly contains all x satisfying |arg(x)| oo.

LADAR detection statistics in the presence of pointing errors

The probability of detection of optically rough targets with pulsed LADAR systems that use direct detection is considered. It is assumed that the LADAR operates under conditions of both unintentional pointing offset bias (i.e., bore-sight error) andjitter. Under these conditions the probabilities of detection of targets in both the near field and the far field of the collecting aperture (i.e., for resolved, partially resolved, and unresolved targets) and for both large and small photoelectron counts are derived, and in many cases of practical interest accurate, elementary analytic approximations that are useful for parametric system studies are obtained. A number of technical references are appended, in which some of the key results are derived. In particular, an interesting new mathematical result involving the complementary incomplete gamma function and an analytic expression for the probability distribution function of a signal photoelectron count obeying BoseEinstein statistics (such as that arising from unresolved targets) immersed in Poisson noise is derived.

Numerical calculation of incomplete gamma functions by the trapezoidal rule

The trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function Г (a,x) in the regiona 0. Since this application of the rule is not standard, a careful investigation of the remainder terms using the Euler-Maclaurin formula is carried out. The outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with surprising accuracy in the stated region.

Several expansion formulae of the complementary incomplete gamma function and possible applications

The complementary incomplete Gamma function, defined by Euler's integral ▪ d t, appears in many physicochemical problems. In this paper, several expansion formulae of the above integral are presented. Some properties of these asymptotic series are discussed from the point of view of the possibilities of their application to kinetics problems.

A Computational Procedure for Incomplete Gamma Functions

We develop a computational procedure, based on Taylor's series and continued fractions, for evaluating Tncomi's incomplete gamma functmn 7*(a, x) = (x-/F(a))S~ e-~t'-ldt and the complementary incomplete gamma function F(a, x) = $7 e-tt -1 dt, suitably normalized, m the region x >_. 0, -oo < a < oo.