POCKETBOOKOF MATHEMATICAL FUNCTIONS Abridged Edition of Handbook of Mathematical Functions Milton Abramowitz and Irene A
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Integration in Terms of Exponential Integrals and Incomplete Gamma Functions
Integration in terms of exponential integrals and incomplete gamma functions Waldemar Hebisch Wrocªaw University [email protected] 13 July 2016 Introduction What is FriCAS? I FriCAS is an advanced computer algebra system I forked from Axiom in 2007 I about 30% of mathematical code is new compared to Axiom I about 200000 lines of mathematical code I math code written in Spad (high level strongly typed language very similar to FriCAS interactive language) I runtime system currently based on Lisp Some functionality added to FriCAS after fork: I several improvements to integrator I limits via Gruntz algorithm I knows about most classical special functions I guessing package I package for computations in quantum probability I noncommutative Groebner bases I asymptotically fast arbitrary precision computation of elliptic functions and elliptic integrals I new user interfaces (Emacs mode and Texmacs interface) FriCAS inherited from Axiom good (probably the best at that time) implementation of Risch algorithm (Bronstein, . ) I strong for elementary integration I when integral needed special functions used pattern matching Part of motivation for current work came from Rubi testsuite: Rubi showed that there is a lot of functions which are integrable in terms of relatively few special functions. I Adapting Rubi looked dicult I Previous work on adding special functions to Risch integrator was widely considered impractical My conclusion: Need new algorithmic approach. Ex post I claim that for considered here class of functions extension to Risch algorithm is much more powerful than pattern matching. Informally Ei(u)0 = u0 exp(u)=u Γ(a; u)0 = u0um=k exp(−u) for a = (m + k)=k and −k < m < 0. -
The Digamma Function and Explicit Permutations of the Alternating Harmonic Series
The digamma function and explicit permutations of the alternating harmonic series. Maxim Gilula February 20, 2015 Abstract The main goal is to present a countable family of permutations of the natural numbers that provide explicit rearrangements of the alternating harmonic series and that we can easily define by some closed expression. The digamma function presents its ubiquity in mathematics once more by being the key tool in computing explicitly the simple rearrangements presented in this paper. The permutations are simple in the sense that composing one with itself will give the identity. We show that the count- able set of rearrangements presented are dense in the reals. Then, slight generalizations are presented. Finally, we reprove a result given originally by J.H. Smith in 1975 that for any conditionally convergent real series guarantees permutations of infinite cycle type give all rearrangements of the series [4]. This result provides a refinement of the well known theorem by Riemann (see e.g. Rudin [3] Theorem 3.54). 1 Introduction A permutation of order n of a conditionally convergent series is a bijection φ of the positive integers N with the property that φn = φ ◦ · · · ◦ φ is the identity on N and n is the least such. Given a conditionally convergent series, a nat- ural question to ask is whether for any real number L there is a permutation of order 2 (or n > 1) such that the rearrangement induced by the permutation equals L. This turns out to be an easy corollary of [4], and is reproved below with elementary methods. Other \simple"rearrangements have been considered elsewhere, such as in Stout [5] and the comprehensive references therein. -
The Riemann and Hurwitz Zeta Functions, Apery's Constant and New
The Riemann and Hurwitz zeta functions, Apery’s constant and new rational series representations involving ζ(2k) Cezar Lupu1 1Department of Mathematics University of Pittsburgh Pittsburgh, PA, USA Algebra, Combinatorics and Geometry Graduate Student Research Seminar, February 2, 2017, Pittsburgh, PA A quick overview of the Riemann zeta function. The Riemann zeta function is defined by 1 X 1 ζ(s) = ; Re s > 1: ns n=1 Originally, Riemann zeta function was defined for real arguments. Also, Euler found another formula which relates the Riemann zeta function with prime numbrs, namely Y 1 ζ(s) = ; 1 p 1 − ps where p runs through all primes p = 2; 3; 5;:::. A quick overview of the Riemann zeta function. Moreover, Riemann proved that the following ζ(s) satisfies the following integral representation formula: 1 Z 1 us−1 ζ(s) = u du; Re s > 1; Γ(s) 0 e − 1 Z 1 where Γ(s) = ts−1e−t dt, Re s > 0 is the Euler gamma 0 function. Also, another important fact is that one can extend ζ(s) from Re s > 1 to Re s > 0. By an easy computation one has 1 X 1 (1 − 21−s )ζ(s) = (−1)n−1 ; ns n=1 and therefore we have A quick overview of the Riemann function. 1 1 X 1 ζ(s) = (−1)n−1 ; Re s > 0; s 6= 1: 1 − 21−s ns n=1 It is well-known that ζ is analytic and it has an analytic continuation at s = 1. At s = 1 it has a simple pole with residue 1. -
Hydrology 510 Quantitative Methods in Hydrology
New Mexico Tech Hyd 510 Hydrology Program Quantitative Methods in Hydrology Hydrology 510 Quantitative Methods in Hydrology Motive Preview: Example of a function and its limits Consider the solute concentration, C [ML-3], in a confined aquifer downstream of a continuous source of solute with fixed concentration, C0, starting at time t=0. Assume that the solute advects in the aquifer while diffusing into the bounding aquitards, above and below (but which themselves have no significant flow). (A homology to this problem would be solute movement in a fracture aquitard (diffusion controlled) Flow aquifer Solute (advection controlled) aquitard (diffusion controlled) x with diffusion into the porous-matrix walls bounding the fracture: the so-called “matrix diffusion” problem. The difference with the original problem is one of spatial scale.) An approximate solution for this conceptual model, describing the space-time variation of concentration in the aquifer, is the function: 1/ 2 x x D C x D C(x,t) = C0 erfc or = erfc B vt − x vB C B v(vt − x) 0 where t is time [T] since the solute was first emitted x is the longitudinal distance [L] downstream, v is the longitudinal groundwater (seepage) velocity [L/T] in the aquifer, D is the effective molecular diffusion coefficient in the aquitard [L2/T], B is the aquifer thickness [L] and erfc is the complementary error function. Describe the behavior of this function. Pick some typical numbers for D/B2 (e.g., D ~ 10-9 m2 s-1, typical for many solutes, and B = 2m) and v (e.g., 0.1 m d-1), and graph the function vs. -
Multiple-Precision Exponential Integral and Related Functions
Multiple-Precision Exponential Integral and Related Functions David M. Smith Loyola Marymount University This article describes a collection of Fortran-95 routines for evaluating the exponential integral function, error function, sine and cosine integrals, Fresnel integrals, Bessel functions and related mathematical special functions using the FM multiple-precision arithmetic package. Categories and Subject Descriptors: G.1.0 [Numerical Analysis]: General { computer arith- metic, multiple precision arithmetic; G.1.2 [Numerical Analysis]: Approximation { special function approximation; G.4 [Mathematical Software]: { Algorithm design and analysis, efficiency, portability General Terms: Algorithms, exponential integral function, multiple precision Additional Key Words and Phrases: Accuracy, function evaluation, floating point, Fortran, math- ematical library, portable software 1. INTRODUCTION The functions described here are high-precision versions of those in the chapters on the ex- ponential integral function, error function, sine and cosine integrals, Fresnel integrals, and Bessel functions in a reference such as Abramowitz and Stegun [1965]. They are Fortran subroutines that use the FM package [Smith 1991; 1998; 2001] for multiple-precision arithmetic, constants, and elementary functions. The FM package supports high precision real, complex, and integer arithmetic and func- tions. All the standard elementary functions are included, as well as about 25 of the mathematical special functions. An interface module provides easy use of the package through three multiple- precision derived types, and also extends Fortran's array operations involving vectors and matrices to multiple-precision. The routines in this module can detect an attempt to use a multiple precision variable that has not been defined, which helps in debugging. There is great flexibility in choosing the numerical environment of the package. -
COMPLETE MONOTONICITY for a NEW RATIO of FINITE MANY GAMMA FUNCTIONS Feng Qi
COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITE MANY GAMMA FUNCTIONS Feng Qi To cite this version: Feng Qi. COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITE MANY GAMMA FUNCTIONS. 2020. hal-02511909 HAL Id: hal-02511909 https://hal.archives-ouvertes.fr/hal-02511909 Preprint submitted on 19 Mar 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITE MANY GAMMA FUNCTIONS FENG QI Dedicated to people facing and fighting COVID-19 Abstract. In the paper, by deriving an inequality involving the generating function of the Bernoulli numbers, the author introduces a new ratio of finite many gamma functions, finds complete monotonicity of the second logarithmic derivative of the ratio, and simply reviews complete monotonicity of several linear combinations of finite many digamma or trigamma functions. Contents 1. Preliminaries and motivations 1 2. A lemma 3 3. Complete monotonicity 4 4. A simple review 5 References 7 1. Preliminaries and motivations Let f(x) be an infinite differentiable function on (0; 1). If (−1)kf (k)(x) ≥ 0 for all k ≥ 0 and x 2 (0; 1), then we call f(x) a completely monotonic function on (0; 1). -
Euler-Maclaurin and Euler-Boole Formulas
Appendix A Euler-MacLaurin and Euler-Boole Formulas A.1 A Taylor Formula The classical Taylor formula Z Xm xk x .x t/m f .x/ D @kf .0/ C @mC1f .t/dt kŠ 0 mŠ kD0 xk can be generalized if we replace the polynomial kŠ by other polynomials (Viskov 1988; Bourbaki 1959). Definition If is a linear form on C0.R/ such that .1/ D 1,wedefinethe polynomials .Pn/ by: P0 D 1 @Pn D Pn1 , .Pn/ D 0 for n 1 P . / k The Generating Function k0 Pk x z We have formally X X X k k k @x. Pk.x/z / D Pk1.x/z D z. Pk.x/z / k0 k1 k0 © Springer International Publishing AG 2017 175 B. Candelpergher, Ramanujan Summation of Divergent Series, Lecture Notes in Mathematics 2185, DOI 10.1007/978-3-319-63630-6 176 A Euler-MacLaurin and Euler-Boole Formulas thus X k xz Pk.x/z D C.z/e k0 To evaluate C.z/ we use the notation x for and by definition of .Pn/ we can write X X k k x. Pk.x/z / D x.Pk.x//z D 1 k0 k0 X k xz xz x. Pk.x/z / D x.C.z/e / D C.z/x.e / k0 1 this gives C.z/ D xz . Thus the generating function of the sequence .Pn/ is x.e / X n xz Pn.x/z D e =M.z/ n xz where the function M is defined by M.z/ D x.e /: Examples P xn n xz . -
NM Temme 1. Introduction the Incomplete Gamma Functions Are Defined by the Integrals 7(A,*)
Methods and Applications of Analysis © 1996 International Press 3 (3) 1996, pp. 335-344 ISSN 1073-2772 UNIFORM ASYMPTOTICS FOR THE INCOMPLETE GAMMA FUNCTIONS STARTING FROM NEGATIVE VALUES OF THE PARAMETERS N. M. Temme ABSTRACT. We consider the asymptotic behavior of the incomplete gamma func- tions 7(—a, —z) and r(—a, —z) as a —► oo. Uniform expansions are needed to describe the transition area z ~ a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for 7(0, z) and V(a,z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given. 1. Introduction The incomplete gamma functions are defined by the integrals 7(a,*)= / T-Vcft, r(a,s)= / t^e^dt, (1.1) where a and z are complex parameters and ta takes its principal value. For 7(0,, z), we need the condition ^Ra > 0; for r(a, z), we assume that |arg2:| < TT. Analytic continuation can be based on these integrals or on series representations of 7(0,2). We have 7(0, z) + r(a, z) = T(a). Another important function is defined by 7>>*) = S7(a,s) = =^ fu^e—du. (1.2) This function is a single-valued entire function of both a and z and is real for pos- itive and negative values of a and z. For r(a,z), we have the additional integral representation e-z poo -zt j.-a r(a, z) = — r / dt, ^a < 1, ^z > 0, (1.3) L [i — a) JQ t ■+■ 1 which can be verified by differentiating the right-hand side with respect to z. -
On Some Series Representations of the Hurwitz Zeta Function Mark W
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 216 (2008) 297–305 www.elsevier.com/locate/cam On some series representations of the Hurwitz zeta function Mark W. Coffey Department of Physics, Colorado School of Mines, Golden, CO 80401, USA Received 21 November 2006; received in revised form 3 May 2007 Abstract A variety of infinite series representations for the Hurwitz zeta function are obtained. Particular cases recover known results, while others are new. Specialization of the series representations apply to the Riemann zeta function, leading to additional results. The method is briefly extended to the Lerch zeta function. Most of the series representations exhibit fast convergence, making them attractive for the computation of special functions and fundamental constants. © 2007 Elsevier B.V. All rights reserved. MSC: 11M06; 11M35; 33B15 Keywords: Hurwitz zeta function; Riemann zeta function; Polygamma function; Lerch zeta function; Series representation; Integral representation; Generalized harmonic numbers 1. Introduction (s, a)= ∞ (n+a)−s s> a> The Hurwitz zeta function, defined by n=0 for Re 1 and Re 0, extends to a meromorphic function in the entire complex s-plane. This analytic continuation to C has a simple pole of residue one. This is reflected in the Laurent expansion ∞ n 1 (−1) n (s, a) = + n(a)(s − 1) , (1) s − 1 n! n=0 (a) (a)=−(a) =/ wherein k are designated the Stieltjes constants [3,4,9,13,18,20] and 0 , where is the digamma a a= 1 function. -
A New Entire Factorial Function
A new entire factorial function Matthew D. Klimek Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY, 14853, USA Department of Physics, Korea University, Seoul 02841, Republic of Korea Abstract We introduce a new factorial function which agrees with the usual Euler gamma function at both the positive integers and at all half-integers, but which is also entire. We describe the basic features of this function. The canonical extension of the factorial to the complex plane is given by Euler's gamma function Γ(z). It is the only such extension that satisfies the recurrence relation zΓ(z) = Γ(z + 1) and is logarithmically convex for all positive real z [1, 2]. (For an alternative function theoretic characterization of Γ(z), see [3].) As a consequence of the recurrence relation, Γ(z) is a meromorphic function when continued onto − the complex plane. It has poles at all non-positive integer values of z 2 Z0 . However, dropping the two conditions above, there are infinitely many other extensions of the factorial that could be constructed. Perhaps the second best known factorial function after Euler's is that of Hadamard [4] which can be expressed as sin πz z z + 1 H(z) = Γ(z) 1 + − ≡ Γ(z)[1 + Q(z)] (1) 2π 2 2 where (z) is the digamma function d Γ0(z) (z) = log Γ(z) = : (2) dz Γ(z) The second term in brackets is given the name Q(z) for later convenience. We see that the Hadamard gamma is a certain multiplicative modification to the Euler gamma. -
Exponential Asymptotics with Coalescing Singularities
Exponential asymptotics with coalescing singularities Philippe H. Trinh and S. Jonathan Chapman Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford E-mail: [email protected], [email protected] Abstract. Problems in exponential asymptotics are typically characterized by divergence of the associated asymptotic expansion in the form of a factorial divided by a power. In this paper, we demonstrate that in certain classes of problems that involve coalescing singularities, a more general type of exponential-over-power divergence must be used. As a model example, we study the water waves produced by flow past an obstruction such as a surface-piercing ship. In the low speed or low Froude limit, the resultant water waves are exponentially small, and their formation is attributed to the singularities in the geometry of the obstruction. However, in cases where the singularities are closely spaced, the usual asymptotic theory fails. We present both a general asymptotic framework for handling such problems of coalescing singularities, and provide numerical and asymptotic results for particular examples. 1. Introduction Many problems in exponential asymptotics involve the analysis of singularly perturbed differential equations where the associated solution is expressed as a divergent asymptotic expansion. It has been noted by authors such as Dingle [14] and Berry [3] that in many cases, the divergence of the sequence occurs in the form of a factorial divided by a arXiv:1403.7182v2 [math.CA] 4 Oct 2014 power. In this paper, we use a model problem from the theory of water waves and ship hydrodynamics to demonstrate how in certain classes of problems, a more general form of divergence must be used in order to perform the exponential asymptotic analysis. -
Transcendence of Various Infinite Series Applications of Baker's
Transcendence of Various Infinite Series and Applications of Baker's Theorem by Chester Jay Weatherby A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Doctor of Philosophy Queen's University Kingston, Ontario, Canada April 2009 Copyright c Chester Jay Weatherby, 2009 Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l’édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-65340-1 Our file Notre référence ISBN: 978-0-494-65340-1 NOTICE: AVIS: The author has granted a non- L’auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l’Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distribute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L’auteur conserve la propriété du droit d’auteur ownership and moral rights in this et des droits moraux qui protège cette thèse. Ni thesis. Neither the thesis nor la thèse ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent être imprimés ou autrement printed or otherwise reproduced reproduits sans son autorisation.