DOCSLIB.ORG

Introductions to Expintegrale Introduction to the Exponential Integrals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introductions to Expintegrale Introduction to the Exponential Integrals Introductions to ExpIntegralE Introduction to the exponential integrals General The exponential-type integrals have a long history. After the early developments of differential calculus, mathemati- cians tried to evaluate integrals containing simple elementary functions, especially integrals that often appeared during investigations of physical problems. Despite the relatively simple form of the integrands, some of these integrals could not be evaluated through known functions. Examples of integrals that could not be evaluated in known functions are: 1 ât log t sin t à ât tH L cosH tL à ât t ãt H L à ât. t L. Euler (1768) introduced the first integral shown in the preceding list. Later L. Mascheroni (1790, 1819) used it à and introduced the second and third integrals, and A. M. Legendre (1811) introduced the last integral shown. T. Caluso (1805) used the first integral in an article and J. von Soldner (1809) introduced its notation through symbol li. F. W. Bessel (1812) used the second and third integrals. C. A. Bretschneider (1843) not only used the second and third integrals, but also introduced similar integrals for the hyperbolic functions: sinh t cosh t ât ât. t t O. SchlöH L milch (1846)H L and F. Arndt (1847) widely used such integrals containing the exponential and trigonometric à à functions. For the exponential, sine, and cosine integrals, J. W. L. Glaisher (1870) introduced the notations Ei, Si, and Ci. H. Amstein (1895) introduced the branch cut for the logarithmic integral with a complex argument. N. Nielsen (1904) used the notations Si and Ci for corresponding integrals. Different notations are used for the previous definite integrals by various authors when they are integrated from 0 to z or from z to ¥. Definitions of exponential integrals The exponential integral En z , exponential integral Ei z , logarithmic integral li z , sine integral Si z , hyperbolic sine integral Shi z , cosine integral Ci z , and hyperbolic cosine integral Chi z are defined as the following definite integrals, including the Euler gamma constant g 0.577216¼: H L H L H L H L H L H L H L http://functions.wolfram.com 2 ¥ ã-z t En z ât ; Re z > 0 n 1 t z ãt - 1 1 1 EiHzL à ât+ HlogL z - log + ý 0 t 2 z z 1 li zH L à ât H L 0 log t z sin t SiH zL à ât 0 tH L z sinhH L t ShiH Lz à ât 0 t z cos t H-L 1 Ci zH L à ât + log z + ý 0 t z coshH L t - 1 ChiH Lz à ât + logH L z + ý. 0 t The previous integralsH L are all interrelated and are called exponential integrals. H L à H L Instead of the above classical definitions through definite integrals, equivalent definitions through infinite series can be used, for example, the exponential integral Ei z can be defined by the following formula (see the following sections for the corresponding series for the other integrals): 1 1 ¥ zk H L Ei z log z - log + + ý. 2 z k=1 k k! A quickH L lookH Lat the exponentialâ integrals Here is a quick look at the graphics for the exponential integrals along the real axis. 4 2 E-3 x f 0 E-1 x H L -2 E1 x H L -4 E3 x H L -4 -2 0 2 4 E x x 5 H L H L http://functions.wolfram.com 3 4 2 Si x f 0 Ci x H L -2 Shi x H L -4 Chi x H L -7.5 -5 -2.5 0 2.5 5 7.5 x H L 4 2 Ei x f 0 li x H L -2 H L -4 -7.5 -5 -2.5 0 2.5 5 7.5 x Connections within the group of exponential integrals and with other function groups Representations through more general functions The exponential integrals En z , Ei z , li z , Si z , Shi z , Ci z , and Chi z are the particular cases of the more general hypergeometric and Meijer G functions. For example, they can be representedH L H L throughH L hypergeometricH L H L H functionsL pHFqL or the Tricomi confluent hypergeomet- ric function U: 1 n-1 En z z G 1 - n - 1F1 1 - n; 2 - n; -z 1 - n E z zn-1 ã-z U n, n, z nH L H L H L 1 1 Ei z z 2F2 1, 1; 2, 2; z + log z - log + ý H L H L 2 z 1 1 EiHzL -ãz UH 1, 1, -z -Llog -z +H L log z - log 2 z 1 1 li zH L log z 2HF2 1, 1; L2, 2; logH zL + logH logL z - log + ý 2 log z 1 3 3 z2 H L H L H H LL H H LL Si z z 1F2 ; , ; - 2 2 2 4 H L 2 H L 1 3 3 z Shi z z 1F2 ; , ; 2 2 2 4 2 2 H L z 3 z Ci z - 2F3 1, 1; 2, 2, ; - + log z + ý 4 2 4 H L H L http://functions.wolfram.com 4 z2 3 z2 Chi z 2F3 1, 1; 2, 2, ; + log z + ý. 4 2 4 RepresentationsH L of the exponential integralsH L En z and Ei z , the sine and cosine integrals Si z and Ci z , and the hyperbolic sine and cosine integrals Shi z and Chi z through classical Meijer G functions are rather simple: 2,0 n H L H L H L H L En z G1,2 z n - 1, 0 H L H L 1,2 1, 1 1 1 EiHzL -G2,3 -z + ý - log - log z 1, 0, 0 2 z 2 1 H L p 1,1 z 2 H L Si z z G1,3 4 4 0, - 1 , - 1 2 2 H L p z2 1 1 Ci z - G2,0 - log z2 - 2 log z 1,3 0, 0, 1 2 4 2 2 2 1 H L p 1,1 z 2 I I M H LM Shi z z G1,3 - 4 4 0, - 1 , - 1 2 2 H L p z2 1 1 Chi z - G2,0 - - log -z2 - 2 log z . 1,3 0, 0, 1 2 4 2 2 HereH Lý is the Euler gamma constant ýI 0.577216I M ¼H andLM the complicated-looking expression containing the two logarithm simplifies piecewise: 1 1 - log - log z log z ; z Ï -¥, 0 2 z 1 1 - log - logHzL logHzL - p ä ;8z Î -¥< , 0 . 2 z But the last four formulas that contain the Meijer G function can be simplified further by changing the classical H L H L 8 < Meijer G functions to the generalized one. These formulas do not include factors z and terms 1 log z2 - 2 log z : 2 p z 1 1 Si z G1,1 , 1,3 1 , 0, 0 I I M H LM 2 2 2 2 H L p z 1 1 Ci z - G2,0 , 1,3 0, 0, 1 2 2 2 2 H L ä p ä z 1 1 Shi z - G1,1 , 1,3 1 , 0, 0 2 2 2 2 1 H L 1 z 1 , 1 3 2 2,0 2 Chi z - p G2,4 , . 2 2 2 0, 0, 1 , 1 2 2 H L http://functions.wolfram.com 5 The corresponding representations of the logarithmic integral li z through the classical Meijer G function is more complicated and includes composition of the G function and a logarithmic function: 1 1 1,2 1, 1 H L li z ý - log - log log z - G2,3 -log z . 2 log z 1, 0, 0 All six exponential integrals of one variable are the particular cases of the incomplete gamma function: H L H H LL H L H L 1 1 Ei z -G 0, -z + log z - log - log -z 2 z ä SiHzL GH 0, -Lä z - G 0,HäLz + log -ä z - logH äLz 2 1 ShiH Lz H GH 0, z -L G 0H, -z -L log -H z +Llog z H LL 2 1 Ci zH L logHz H- L G 0H, -ä z L+ G 0,Hä zL + logH-LäLz + log ä z 2 1 ChiH Lz - H L G 0, H-zH + G 0L, z +Hlog -L z - logH z L H LL 2 1 1 li z HL -G 0, H-logH z L+ H logLlog z H - logL H LL - log -log z . 2 log z Representations through related equivalent functions H L H H LL H H LL H H LL H L The exponential integral En z can be represented through the incomplete gamma function or the regularized incomplete gamma function: n-1 En z z G 1 - n, z H L n-1 En z z G 1 - n Q 1 - n, z . H L H L Relations to inverse functions H L H L H L The exponential integral En z is connected with the inverse of the regularized incomplete gamma function Q-1 a, z by the following formula: -1 -1 H nL-1 En Q 1 - n, z Q 1 - n, z G 1 - n z. H L Representations through other exponential integrals I H LM H L H L The exponential integrals En z , Ei z , li z , Si z , Shi z , Ci z , and Chi z are interconnected through the follow- ing formulas: 1 H L 1 H L H L H L H L H L H L Ei z -E1 -z + log z - log - log -z 2 z Ei z li ãz ; -p < Im z £ p H L H L H L H L Ei log z li z H L H L H L H H LL H L http://functions.wolfram.com 6 1 1 Ei z Ci ä z - ä Si ä z - log - log z - log ä z 2 z 1 1 EiHzL ChiH zL + ShiHz -L log + log zH L H L 2 z 1 1 li zH L -E1 H-Llog z H+L log log z - logH L - log -log z 2 log z li z Ei log z H L H H LL H H LL H H LL H L 1 1 li z Ci ä log z - ä Si ä log z - log - log log z - log ä log z H L H H LL 2 log z 1 1 liHzL ChiH logHzLL + ShiHlog zH L-L log + log logH zH LL H H LL 2 log zH L ä SiH zL HE1 -H äLLz - E1Hä z H+LLlog -ä z - log ä z H H LL 2 H L ä ä ä SiHzL H2 HEi -Lä z - EiH äLz + logH L - logH -LL - log -ä z + log ä z 4 z z 1 p SiHzL liH ãäHz - Lli ã-ä Hz -LL sgn Re z ; Re z < pH L H L 2 ä 2 Si z -ä Shi ä z H L I I M I MM H H LL H L¤ 1 Shi z E1 z - E1 -z - log -z + log z H L 2 H L 1 1 1 ShiHzL H2 HEiL z - EiH -Lz + logH L - logH LL- + log -z - log z 4 z z 1 ä p ShiHzL li Hãz H-Lli ã-Hz -LL sgn Im z ; Im z < pH L H L 2 2 Shi z -ä Si ä z H L H H L H LL H H LL H L¤ 1 Ci z - E1 -ä z + E1 ä z + log -ä z + log ä z + log z H L 2 H L 1 ä ä CiHzL 2H EiH -ä zL + EiHä zL + logH L+ log H- LL - logH-Lä z - log ä z + log z 4 z z 1 p ä CiHzL li Hã-äHz + liL ãä z H +LL sgn Im z 1 - sgn Re zH ;L Re z H <Lp H L 2 2 Ci z Chi ä z - log ä z + log z H L I I M I MM H H LL H H H LLL H L¤ 1 Chi z - E1 -z + E1 z + log -z - log z H L 2H L H L H L 1 1 1 ChiHzL 2H EiH -zL + EiHzL + logH L + logH -LL - log -z + 3 log z 4 z z H L H H L H LL H L H L http://functions.wolfram.com 7 1 p ä 1 1 Chi z li ã-z + li ãz + sgn Im z + log + log z ; Im z < p 2 2 2 z Chi z Ci ä z + log z - log ä z .
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Expected Logarithm and Negative Integer Moments of a Noncentral 2-Distributed Random Variable
    entropy Article Expected Logarithm and Negative Integer Moments of a Noncentral c2-Distributed Random Variable † Stefan M. Moser 1,2 1 Signal and Information Processing Lab, ETH Zürich, 8092 Zürich, Switzerland; [email protected]; Tel.: +41-44-632-3624 2 Institute of Communications Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan † Part of this work has been presented at TENCON 2007 in Taipei, Taiwan, and at ISITA 2008 in Auckland, New Zealand. Received: 21 July 2020; Accepted: 15 September 2020; Published: 19 September 2020 Abstract: Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral c2-distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed. Keywords: central c2 distribution; chi-square distribution; expected logarithm; exponential distribution; negative integer moments; noncentral c2 distribution; squared Rayleigh distribution; squared Rice distribution 1. Introduction The noncentral c2 distribution is a family of probability distributions of wide interest. They appear in situations where one or several independent Gaussian random variables (RVs) of equal variance (but potentially different means) are squared and summed together. The noncentral c2 distribution contains as special cases among others the central c2 distribution, the exponential distribution (which is equivalent to a squared Rayleigh distribution), and the squared Rice distribution. In this paper, we present closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral c2-distributed RV with even or odd degrees of freedom.
    [Show full text]
  • POCKETBOOKOF MATHEMATICAL FUNCTIONS Abridged Edition of Handbook of Mathematical Functions Milton Abramowitz and Irene A
    POCKETBOOKOF MATHEMATICAL FUNCTIONS Abridged edition of Handbook of Mathematical Functions Milton Abramowitz and Irene A. Stegun (eds.) Material selected by Michael Danos and Johann Rafelski 1984 Verlag Harri Deutsch - Thun - Frankfurt/Main CONTENTS Forewordtothe Original NBS Handbook 5 Pref ace 6 2. PHYSICAL CONSTANTS AND CONVERSION FACTORS 17 A.G. McNish, revised by the editors Table 2.1. Common Units and Conversion Factors 17 Table 2.2. Names and Conversion Factors for Electric and Magnetic Units 17 Table 2.3. AdjustedValuesof Constants 18 Table 2.4. Miscellaneous Conversion Factors 19 Table 2.5. FactorsforConvertingCustomaryU.S. UnitstoSIUnits 19 Table 2.6. Geodetic Constants 19 Table 2.7. Physical andNumericalConstants 20 Table 2.8. Periodic Table of the Elements 21 Table 2.9. Electromagnetic Relations 22 Table 2.10. Radioactivity and Radiation Protection 22 3. ELEMENTARY ANALYTICAL METHODS 23 Milton Abramowitz 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometrie Progressions; Arithmetic, Geometrie, Harmonie and Generalized Means 23 3.2. Inequalities 23 3.3. Rules for Differentiation and Integration 24 3.4. Limits, Maxima and Minima 26 3.5. Absolute and Relative Errors 27 3.6. Infinite Series 27 3.7. Complex Numbers and Functions 29 3.8. Algebraic Equations 30 3.9. Successive Approximation Methods 31 3.10. TheoremsonContinuedFractions 32 4. ELEMENTARY TRANSCENDENTAL FUNCTIONS 33 Logarithmic, Exponential, Circular and Hyperbolic Functions Ruth Zucker 4.1. Logarithmic Function 33 4.2. Exponential Function 35 4.3. Circular Functions 37 4.4. Inverse Circular Functions 45 4.5. Hyperbolic Functions 49 4.6. InverseHyberbolicFunctions 52 5. EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS .... 56 Walter Gautschi and William F.
    [Show full text]
  • A Table of Integrals of the Error Functions*
    JOURNAL OF RESEAR CH of the Natiana l Bureau of Standards - B. Mathematical Sciences Vol. 73B, No. 1, January-March 1969 A Table of Integrals of the Error Functions* Edward W. Ng** and Murray Geller** (October 23, 1968) This is a compendium of indefinite and definite integrals of products of the Error function with elementary or transcendental functions. A s'ubstantial portion of the results are ne w. Key Words: Astrophysics; atomic physics; Error functions; indefi nite integrals; special functions; statistical analysis. 1. Introduction Integrals of the error function occur in a great variety of applicati ons, us ualJy in problems involvin g multiple integration where the integrand contains expone ntials of the squares of the argu­ me nts. Examples of applications can be cited from atomic physics [16),1 astrophysics [13] , and statistical analysis [15]. This paper is an attempt to give a n up-to-date exhaustive tabulation of s uch integrals. All formulas for indefi nite integrals in sections 4.1, 4.2, 4.5, a nd 4.6 below were derived from integration by parts and c hecked by differe ntiation of the resulting expressions_ Section 4.3 and the second hali' of 4.5 cover all formulas give n in [7] , with omission of trivial duplications and with a number of additions; section 4.4 covers essentially formulas give n in [4], Vol. I, pp. 233-235. All these formulas have been re-derived and checked, eithe r from the integral representation or from the hype rgeometric series of the error function. Sections 4.7,4_8 and 4_9 ori ginated in a more varied way.
    [Show full text]
  • Error Analysis in the Series Evaluation of the Exponential Type Integral Ezel(Z)
    An International Joumal computers & mathematics with applications PERGAMON Computers and Mathematics with Applications 43 (2002) 207-266 www.elsevier.com/locate/camwa Error Analysis in the Series Evaluation of the Exponential Type Integral eZEl(z) W. C. HASSENPFLUG Department of Mechanical Engineering University of Stellenbosch, Stellenbosch, South Africa (Received July 1999; accepted August 1999) Abstract--The method of convolution algebra is used to compute values of the exponential type integral eZ E1 (z), y(z) = f0 ~ se-S + z ds, by expansion of the integrand in a string of Taylor series' along the real s-axis for any complex parameter z, accurate within 4-1 of the last digit of seven-digit computation. Accuracy is verified by comparison with existing tables of E1 and related integrals. This method is used to assess the accuracy of the error estimates of all subsequent computations. Three errors of a Taylor series are identified. These consist of a Taylor series truncation error, a digital truncation error, and a stability error. Methods are developed to estimate the error. By iteration a numerical radius of convergence for a given accuracy is determined. The z-plane is divided into three regions in which three different types of series are used to expand the function f(z) directly in z. Around the center the well-known Frobenius series is used. In the outer region the well-known asymptotic approximation is used. Their accuracy boundaries are determined. In the near-annular region in between, a set of Taylor series is introduced. As the result, the function f(z) can be computed fast with the appropriate series for any complex argument z, to an accuracy within less than relative error of 5 x 10 -7.
    [Show full text]
  • Expint: Exponential Integral and Incomplete Gamma Function
    expint: Exponential integral and incomplete gamma function Vincent Goulet Université Laval 1 Introduction The exponential integral Z ¥ e−t E1(x) = dt, x 2 R x t and the incomplete gamma function Z ¥ G(a, x) = ta−1e−t dt, x > 0, a 2 R x are two closely related functions that arise in various fields of mathematics. expint is a small package that intends to fill a gap in R’s support for mathematical functions by providing facilities to compute the exponential integral and the incomplete gamma function. Furthermore, and perhaps most conveniently for R package developers, the package also gives easy access to the underlying C workhorses through an API. The C routines are derived from the GNU Scientific Library (GSL; Galassi et al., 2009). Package expint started its life in version 2.0-0 of package actuar (Dutang et al., 2008) where we extended the range of admissible values in the com- putation of limited expected value functions. This required an incomplete gamma function that accepts negative values of argument a, as explained at the beginning of Appendix A of Klugman et al.(2012). 2 Exponential integral Abramowitz and Stegun(1972, Section 5.1) first define the exponential integral as Z ¥ e−t E1(x) = dt. (1) x t 1 An alternative definition (to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero) is Z ¥ e−t Z x et Ei(x) = − dt = dt, x > 0. −x t −¥ t The above two definitions are related as follows: E1(−x) = − Ei(x), x > 0.
    [Show full text]
  • Chapter S – Approximations of Special Functions
    s – Approximations of Special Functions Introduction – s Chapter s – Approximations of Special Functions 1. Scope of the Chapter This chapter provides functions for computing some of the most commonly required special functions of mathematics. See Chapter g01 for probability distribution functions and their inverses. 2. Accuracy The majority of the functions in this chapter evaluate real-valued functions of a single real variable x. The accuracy of the computed value is discussed in each function document, along the following lines. Let ∆ be the absolute error in the argument x,andδ be the relative error in x. If we ignore errors that arise in the argument by propagation of data errors etc. and consider only those errors that result from the fact that a real number is being represented in the computer in floating-point form with finite precision, then δ is bounded and this bound is independent of the magnitude of x; for example, on an n-digit machine |δ|≤10−n. (This of course implies that the absolute error ∆ = xδ is also bounded but the bound is now dependent on x.) Let E be the absolute error in the computed function value f(x), and be the relative error. Then E |f (x)|∆ E |xf (x)|δ |xf (x)/f(x)|δ. If possible, the function documents discuss the last of these relations, that is the propagation of relative error, in terms of the error amplification factor |xf (x)/f(x)|. But in some cases, such as near zeros of the function which cannot be extracted explicitly, absolute error in the result is the quantity of significance and here the factor |xf (x)| is described.
    [Show full text]
  • 1 Polyexponentials Khristo N. Boyadzhiev Ohio Northern University Departnment of Mathematics Ada, OH 45810 [email protected]
    Polyexponentials Khristo N. Boyadzhiev Ohio Northern University Departnment of Mathematics Ada, OH 45810 [email protected] 1. Introduction. The polylogarithmic function [15] (1.1) and the more general Lerch Transcendent (or Lerch zeta function) [4], [14] (1.2) have established themselves as very useful special functions in mathematics. Here and further we assume that . In this article we shall discuss the function (1.3) and survey some of its properties. Our intention is to show that this function can also be useful in a number of situations. It extends the natural exponential function and also the exponential integral (see Section 2). As we shall see, the function is relevant also to the theory of the Lerch Transcendent (1.2 ) and the Hurwitz zeta function , (1.4) About a century ago, in 1905, Hardy [6] published a paper on the function 1 , (1.5) where he focused on the variable and studied the zeroes of and its asymptotic behavior. We shall use a different notation here in order to emphasize the relation of to the natural exponential function. Hardy wrote about : “All the functions which have been considered so far are in many ways analogous to the ordinary exponential function” [6, p. 428]. For this reason we call polyexponential (or polyexponentials, if is considered a parameter). Our aim is to study some of its properties (complimentary to those in [6]} and point out some applications. In Section 2 we collect some basic properties of the polyexponential function. Next, in Section 3, we discuss exponential polynomials and the asymptotic expansion of in the variable .
    [Show full text]