Expint: Exponential Integral and Incomplete Gamma Function
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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. -
Math/Library Special Functions Quick Tips on How to Use This Online Manual
Fortran Subroutines for Mathematical Applications Math/Library Special Functions Quick Tips on How to Use this Online Manual Click to display only the page. Click to go back to the previous page from which you jumped. Click to display both bookmark and the page. Click to go to the next page. Double-click to jump to a topic Click to go to the last page. when the bookmarks are displayed. Click to jump to a topic when the Click to go back to the previous view and bookmarks are displayed. page from which you jumped. Click to display both thumbnails Click to return to the next view. and the page. Click and use to drag the page in vertical Click to view the page at 100% zoom. direction and to select items on the page. Click and drag to page to magnify Click to fit the entire page within the the view. window. Click and drag to page to reduce the view. Click to fit the page width inside the window. Click and drag to the page to select text. Click to find part of a word, a complete word, or multiple words in a active document. Click to go to the first page. Printing an online file: Select Print from the File menu to print an online file. The dialog box that opens allows you to print full text, range of pages, or selection. Important Note: The last blank page of each chapter (appearing in the hard copy documentation) has been deleted from the on-line documentation causing a skip in page numbering before the first page of the next chapter, for instance, Chapter 1 in the on-line documentation ends on page 317 and Chapter 2 begins on page 319. -
Introduction to Analytic Number Theory the Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
Math 229: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler’s identity: ∞ ∞ X Y X Y 1 [ζ(s) :=] n−s = p−cps = . (1) 1 − p−s n=1 p prime cp=0 p prime We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. Riemann’s insight was to consider (1) as an identity between functions of a complex variable s. We follow the curious but nearly universal convention of writing the real and imaginary parts of s as σ and t, so s = σ + it. We already observed that for all real n > 0 we have |n−s| = n−σ, because n−s = exp(−s log n) = n−σe−it log n and e−it log n has absolute value 1; and that both sides of (1) converge absolutely in the half-plane σ > 1, and are equal there either by analytic continuation from the real ray t = 0 or by the same proof we used for the real case. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the “Riemann zeta function”), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula ∞ ∞ 1 X Z n+1 X Z n+1 ζ(s) − = n−s − x−s dx = (n−s − x−s) dx, s − 1 n=1 n n=1 n since for x ∈ [n, n + 1] (n ≥ 1) and σ > 0 we have Z x −s −s −1−s −1−σ |n − x | = s y dy ≤ |s|n n so the formula for ζ(s) − (1/(s − 1)) is a sum of analytic functions converging absolutely in compact subsets of {σ + it : σ > 0} and thus gives an analytic function there. -
IMSL Fortran Library User's Guide MATH/LIBRARY Special Functions
IMSL Fortran Library User’s Guide MATH/LIBRARY Special Functions Mathematical Functions in Fortran Trusted For Over 30 Years IMSL MATH/LIBRARY User’s Guide Special Functions Mathematical Functions in Fortran P/N 7681 [ www.vni.com ] Visual Numerics, Inc. – United States Visual Numerics International Ltd. Visual Numerics SARL Corporate Headquarters Centennial Court Immeuble le Wilson 1 2000 Crow Canyon Place, Suite 270 Easthampstead Road 70, avenue due General de Gaulle San Ramon, CA 94583 Bracknell, Berkshire RG12 1YQ F-92058 PARIS LA DEFENSE, Cedex PHONE: 925-807-0138 UNITED KINGDOM FRANCE FAX: 925-807-0145 e-mail: [email protected] PHONE: +44 (0) 1344-458700 PHONE: +33-1-46-93-94-20 Westminster, CO FAX: +44 (0) 1344-458748 FAX: +33-1-46-93-94-39 PHONE: 303-379-3040 e-mail: [email protected] e-mail: [email protected] Houston, TX PHONE: 713-784-3131 Visual Numerics S. A. de C. V. Visual Numerics International GmbH Visual Numerics Japan, Inc. Florencia 57 Piso 10-01 Zettachring 10 GOBANCHO HIKARI BLDG. 4TH Floor Col. Juarez D-70567Stuttgart 14 GOBAN-CHO CHIYODA-KU Mexico D. F. C. P. 06600 GERMANY TOKYO, JAPAN 102 MEXICO PHONE: +49-711-13287-0 PHONE: +81-3-5211-7760 PHONE: +52-55-514-9730 or 9628 FAX: +49-711-13287-99 FAX: +81-3-5211-7769 FAX: +52-55-514-4873 e-mail: [email protected] e-mail: [email protected] Visual Numerics, Inc. Visual Numerics Korea, Inc. World Wide Web site: http://www.vni.com 7/F, #510, Sect. 5 HANSHIN BLDG. -
Fortran Math Special Functions Library
IMSL® Fortran Math Special Functions Library Version 2021.0 Copyright 1970-2021 Rogue Wave Software, Inc., a Perforce company. Visual Numerics, IMSL, and PV-WAVE are registered trademarks of Rogue Wave Software, Inc., a Perforce company. IMPORTANT NOTICE: Information contained in this documentation is subject to change without notice. Use of this docu- ment is subject to the terms and conditions of a Rogue Wave Software License Agreement, including, without limitation, the Limited Warranty and Limitation of Liability. ACKNOWLEDGMENTS Use of the Documentation and implementation of any of its processes or techniques are the sole responsibility of the client, and Perforce Soft- ware, Inc., assumes no responsibility and will not be liable for any errors, omissions, damage, or loss that might result from any use or misuse of the Documentation PERFORCE SOFTWARE, INC. MAKES NO REPRESENTATION ABOUT THE SUITABILITY OF THE DOCUMENTATION. THE DOCU- MENTATION IS PROVIDED "AS IS" WITHOUT WARRANTY OF ANY KIND. PERFORCE SOFTWARE, INC. HEREBY DISCLAIMS ALL WARRANTIES AND CONDITIONS WITH REGARD TO THE DOCUMENTATION, WHETHER EXPRESS, IMPLIED, STATUTORY, OR OTHERWISE, INCLUDING WITHOUT LIMITATION ANY IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PAR- TICULAR PURPOSE, OR NONINFRINGEMENT. IN NO EVENT SHALL PERFORCE SOFTWARE, INC. BE LIABLE, WHETHER IN CONTRACT, TORT, OR OTHERWISE, FOR ANY SPECIAL, CONSEQUENTIAL, INDIRECT, PUNITIVE, OR EXEMPLARY DAMAGES IN CONNECTION WITH THE USE OF THE DOCUMENTATION. The Documentation is subject to change at any time without notice. IMSL https://www.imsl.com/ Contents Introduction The IMSL Fortran Numerical Libraries . 1 Getting Started . 2 Finding the Right Routine . 3 Organization of the Documentation . 4 Naming Conventions . -
INTEGRALS of POWERS of LOGGAMMA 1. Introduction The
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 2, February 2011, Pages 535–545 S 0002-9939(2010)10589-0 Article electronically published on August 18, 2010 INTEGRALS OF POWERS OF LOGGAMMA TEWODROS AMDEBERHAN, MARK W. COFFEY, OLIVIER ESPINOSA, CHRISTOPH KOUTSCHAN, DANTE V. MANNA, AND VICTOR H. MOLL (Communicated by Ken Ono) Abstract. Properties of the integral of powers of log Γ(x) from 0 to 1 are con- sidered. Analytic evaluations for the first two powers are presented. Empirical evidence for the cubic case is discussed. 1. Introduction The evaluation of definite integrals is a subject full of interconnections of many parts of mathematics. Since the beginning of Integral Calculus, scientists have developed a large variety of techniques to produce magnificent formulae. A partic- ularly beautiful formula due to J. L. Raabe [12] is 1 Γ(x + t) (1.1) log √ dx = t log t − t, for t ≥ 0, 0 2π which includes the special case 1 √ (1.2) L1 := log Γ(x) dx =log 2π. 0 Here Γ(x)isthegamma function defined by the integral representation ∞ (1.3) Γ(x)= ux−1e−udu, 0 for Re x>0. Raabe’s formula can be obtained from the Hurwitz zeta function ∞ 1 (1.4) ζ(s, q)= (n + q)s n=0 via the integral formula 1 t1−s (1.5) ζ(s, q + t) dq = − 0 s 1 coupled with Lerch’s formula ∂ Γ(q) (1.6) ζ(s, q) =log √ . ∂s s=0 2π An interesting extension of these formulas to the p-adic gamma function has recently appeared in [3]. -
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. -
32 FA15 Abstracts
32 FA15 Abstracts IP1 metric simple exclusion process and the KPZ equation. In Vector-Valued Nonsymmetric and Symmetric Jack addition, the experiments of Takeuchi and Sano will be and Macdonald Polynomials briefly discussed. For each partition τ of N there are irreducible representa- Craig A. Tracy tions of the symmetric group SN and the associated Hecke University of California, Davis algebra HN (q) on a real vector space Vτ whose basis is [email protected] indexed by the set of reverse standard Young tableaux of shape τ. The talk concerns orthogonal bases of Vτ -valued polynomials of x ∈ RN . The bases consist of polyno- IP6 mials which are simultaneous eigenfunctions of commuta- Limits of Orthogonal Polynomials and Contrac- tive algebras of differential-difference operators, which are tions of Lie Algebras parametrized by κ and (q, t) respectively. These polynomi- als reduce to the ordinary Jack and Macdonald polynomials In this talk, I will discuss the connection between superin- when the partition has just one part (N). The polynomi- tegrable systems and classical systems of orthogonal poly- als are constructed by means of the Yang-Baxter graph. nomials in particular in the expansion coefficients between There is a natural bilinear form, which is positive-definite separable coordinate systems, related to representations of for certain ranges of parameter values depending on τ,and the (quadratic) symmetry algebras. This connection al- there are integral kernels related to the bilinear form for lows us to extend the Askey scheme of classical orthogonal the group case, of Gaussian and of torus type. The mate- polynomials and the limiting processes within the scheme. -
Sums of Powers and the Bernoulli Numbers Laura Elizabeth S
Eastern Illinois University The Keep Masters Theses Student Theses & Publications 1996 Sums of Powers and the Bernoulli Numbers Laura Elizabeth S. Coen Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). Masters Theses. 1896. https://thekeep.eiu.edu/theses/1896 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. THESIS REPRODUCTION CERTIFICATE TO: Graduate Degree Candidates (who have written formal theses) SUBJECT: Permission to Reproduce Theses The University Library is rece1v1ng a number of requests from other institutions asking permission to reproduce dissertations for inclusion in their library holdings. Although no copyright laws are involved, we feel that professional courtesy demands that permission be obtained from the author before we allow theses to be copied. PLEASE SIGN ONE OF THE FOLLOWING STATEMENTS: Booth Library of Eastern Illinois University has my permission to lend my thesis to a reputable college or university for the purpose of copying it for inclusion in that institution's library or research holdings. u Author uate I respectfully request Booth Library of Eastern Illinois University not allow my thesis -
Notes on Riemann's Zeta Function
NOTES ON RIEMANN’S ZETA FUNCTION DRAGAN MILICIˇ C´ 1. Gamma function 1.1. Definition of the Gamma function. The integral ∞ Γ(z)= tz−1e−tdt Z0 is well-defined and defines a holomorphic function in the right half-plane {z ∈ C | Re z > 0}. This function is Euler’s Gamma function. First, by integration by parts ∞ ∞ ∞ Γ(z +1)= tze−tdt = −tze−t + z tz−1e−t dt = zΓ(z) Z0 0 Z0 for any z in the right half-plane. In particular, for any positive integer n, we have Γ(n) = (n − 1)Γ(n − 1)=(n − 1)!Γ(1). On the other hand, ∞ ∞ Γ(1) = e−tdt = −e−t = 1; Z0 0 and we have the following result. 1.1.1. Lemma. Γ(n) = (n − 1)! for any n ∈ Z. Therefore, we can view the Gamma function as a extension of the factorial. 1.2. Meromorphic continuation. Now we want to show that Γ extends to a meromorphic function in C. We start with a technical lemma. Z ∞ 1.2.1. Lemma. Let cn, n ∈ +, be complex numbers such such that n=0 |cn| converges. Let P S = {−n | n ∈ Z+ and cn 6=0}. Then ∞ c f(z)= n z + n n=0 X converges absolutely for z ∈ C − S and uniformly on bounded subsets of C − S. The function f is a meromorphic function on C with simple poles at the points in S and Res(f, −n)= cn for any −n ∈ S. 1 2 D. MILICIˇ C´ Proof. Clearly, if |z| < R, we have |z + n| ≥ |n − R| for all n ≥ R. -
Special Values of the Gamma Function at CM Points
Special values of the Gamma function at CM points M. Ram Murty & Chester Weatherby The Ramanujan Journal An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan ISSN 1382-4090 Ramanujan J DOI 10.1007/s11139-013-9531-x 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media New York. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Ramanujan J DOI 10.1007/s11139-013-9531-x Special values of the Gamma function at CM points M. Ram Murty · Chester Weatherby Received: 10 November 2011 / Accepted: 15 October 2013 © Springer Science+Business Media New York 2014 Abstract Little is known about the transcendence of certain values of the Gamma function, Γ(z). In this article, we study values of Γ(z)when Q(z) is an imaginary quadratic field. We also study special values of the digamma function, ψ(z), and the polygamma functions, ψt (z).