Selberg s integral and linear forms in zeta values
|
|
- Edgar Hopkins
- 6 years ago
- Views:
Transcription
1 Selberg s integrl nd liner forms in zet vlues Tnguy Rivol Abstrct Using Selberg s integrl, we present some new Euler-type integrl representtions of certin nerly-poised hypergeometric series. These integrls re lso shown to produce liner forms in odd nd/or even zet vlues tht generlize previous work of the uthor. 1 Selberg s integrl nd nerly-poised series Much work hs been devoted to evluting multiple hypergeometric integrls fter Beukers proof of Apéry s theorem ζ(3 is irrtionl, in which he used the following integrls equtions [Be]: nd x n (1 x n y n (1 y n (1 (1 xy n+1 dxdy = n ζ(2 + b n x n (1 x n y n (1 y n z n (1 z n (1 (1 (1 xyz n+1 dxdydz = A n ζ(3 + B n for some (explicitly computble rtionl numbers n b n, A n nd B n. See in prticulr the work of Ht [H], Rhin nd Viol [RV1, RV2], Vsilyev [V], Sorokin [So], Zudilin [Zu]. Similrly, in order to prove tht infinitely mny odd zet vlues re irrtionl, the following multiple integrl ws used in [Ri] nd [BR] : J r,n := xrn j (1 x j n [,1] (1 x +1 x dx (2r+1n+2 dx, (1 where n,, r 1 re integers such tht ( + 1n > (2r + 1n + 2. This is interesting principlly becuse there exist explicitly computble rtionl numbers (for j =,..., such tht p r,n j, J r,n = p r,n, + 1 j=2 p r,n j, ζ(j
2 nd for j 2, p r,n j, = if (n j is odd (the prity phenomenon. In [Zu], the following integrl, which generlizes Beukers integrls bove, j=1 xn j (1 x j n Q (x 1, x 2,..., x dx n+1 1 dx, [,1] with Q (x 1, x 2,..., x = 1 (1 (1 x 3 x 2 x 1, is proved to produce liner forms in odd or even zet vlues. Zudilin gives n identity between such integrls nd slight modifiction of (1, which is not t ll obvious. In this note, we construct nother kind of hypergeometric integrl, bsed on Selberg s integrl, tht lso generlizes (1 nd produces liner forms in odd or even zet vlues (Theorem 1 in 2. We will lso show how it is relted to more usul Euler-type integrls (Propositions 1 nd 2 below. We first remind the reder of the definition of hypergeometric series p F q (z : [ ] α1, α 2,..., α p pf q ; z := β 1, β 2,..., β q k= (α 1 k (α 2 k (α p k (1 k (β 1 k (β q k z k. Here, p nd q re positive integers, α j nd β j re complex numbers such tht β j N, nd (x n := x(x + 1 (x + n 1 is the Pochhmmer symbol. We lso recll well-known result of Selberg, whose proof cn be found in [Se]: if the complex numbers α, β nd γ stisfy ( 1 Re(α > 1, Re(β > 1, Re(γ > min + 1, Re(α + 1, Re(β + 1, we hve tht Sel α,β,γ := [,1] +1 x α j (1 x j β = j<l x j x l 2γ dx dx Γ(γ + jγ + 1Γ(α + jγ + 1Γ(β + jγ + 1 Γ(γ + 1Γ(α + β + ( + jγ + 2. (2 Let us now define the integrl I α,β,γ,δ (z := j (1 x j β j<l x j x l 2γ [,1] (1 zx +1 x δ+1 dx dx, (3 where z is complex number, z 1 nd α, β, γ, δ, 1 re integers ( 1 such tht ( + 1β > δ (which ensures convergence on the circle z = 1. Then the following holds. 1 More generlly, α, β nd γ could be tken to be suitble complex numbers. 2
3 Proposition 1. Under the bove conditions, I α,β,γ,δ (z = Sel α,β,γ +2F +1 [ δ + 1, α + 1, α + γ + 1, α + 2γ + 1,..., α + γ + 1 α + β + 2γ + 2, α + β + (2 1γ + 2,..., α + β + γ + 2 ; z ]. (4 The hypergeometric function +2 F +1 on the RHS of (4 is sid to be nerlypoised becuse the sum of n upper prmeter (other thn δ + 1 nd suitble lower prmeter is invrint : for ll j =,...,, (α + jγ (α + β + (2 jγ + 2 = 2α + β + 2γ + 3. Proof. This cn be cn thought of s generting function rewriting of Selberg s identity. To simplify, we write c α,β,γ,δ := Then, using the expnsion Γ(γ + jγ + 1Γ(β + jγ + 1 Γ(δ + 1Γ(γ (1 t = δ+1 k= ( δ + k nd interverting the signs in (3, we get I α,β,γ,δ = (z ( δ + k z k k= = c α,β,γ,δ k k= [,1] +1 Γ(k + δ + 1 Γ(k + 1 k t k x α+k j (1 x j β ( j<l Γ(k + α + jγ + 1 Γ(k + α + β + ( + jγ + 2 x j x l 2γ dx dx z k, (5 where we hve used Selberg s identity (2 with α + k replcing α in the lst step. We now note tht, thnks to the identity (x n = Γ(x + n/γ(x, the eqution (5 is simply the RHS of (4. An interesting consequence of Proposition 1 is the construction of new integrl representtions for I α,β,γ,δ (z in which the discriminnt j<l x j x l does not pper. Proposition 2. Let σ be ny permuttion of the set {,..., }. Then I α,β,γ,δ (z = ( Γ(γ + jγ + 1Γ(β + jγ + 1 Γ(γ + 1Γ(β + ( j + σ(jγ + 1 [,1] +1 xα+jγ j (1 x j β+( j+σ(jγ (1 zx x δ+1 dx dx. (6 3
4 Proof. We first note tht in (4 n upper prmeter (other thn δ + 1 is lwys less thn lower prmeter, tht is to sy, for ny j, k, α + jγ + 1 < α + β + ( + kγ + 2. This inequlity cn be reformulted s follows : for ny permuttion σ of the set {,..., } nd for ny j, the inequlity α + jγ + 1 < α + β + ( + σ(jγ + 2 holds. Hence, we cn pply clssicl identity of Euler which expresses hypergeometric series s multiple integrl (cf. [Sl], p. 18 I α,β,γ,δ (z = ( Γ(γ + jγ + 1Γ(β + jγ + 1Γ(α + β + ( + σ(jγ + 2 Γ(γ + 1Γ(β + ( j + σ(jγ + 1Γ(α + β + ( + jγ + 2 j (1 x j β+( j+σ(jγ [,1] +1 (1 zx x δ+1 dx dx. (7 To conclude, we note tht, for ny permuttion σ, Γ(α + β + ( + σ(jγ + 2 = Γ(α + β + ( + jγ + 2, which simplifies the Gmm quotient in (7 nd proves the identity (6. 2 The well-poised cse Specil ttention should be pid to prticulr cse of Proposition 1 : when δ = 2α + β + 2γ + 1, then the hypergeometric function on the RHS of (4 is well-poised. In this cse, the corresponding integrl cn be evluted t z = 1 in term of odd (resp. even zet vlues, ccording to the vlues of the prmeters. (In the nerly-poised cse, such decomposition involves both the even nd odd zet vlues. Theorem 1. For integers α, β, γ nd 1 such tht β > 2α + 2γ + 2, the integrl J α,β,γ := xα j (1 x j β j<l x j x l 2γ dx [,1] (1 x +1 x 2α+β+2γ+2 dx (8 is convergent nd there exist explicitly computble rtionl numbers P α,β,γ j, j =,..., such tht (for J α,β,γ = P α,β,γ, + j=2 P α,β,γ j, ζ(j. (9 Furthermore, for j 2, P α,β,γ j,n = if (β j is odd. 4
5 Clerly, when γ =, the integrl J α,β,γ in the introduction s prticulr cse. includes the integrl J r,n mentioned Proof. It is now well-estblished tht the prity phenomenon is consequence of the well-poisedness of the underlying hypergeometric function (see [RZ] for more detils of this ide. We will therefore simply sketch the resoning nd give references t ech step. With the nottion used in the proof of Proposition 1 nd with δ = 2α + β + 2γ + 1, trivil trnsformtion of the Pochhmmer symbols of the hypergeometric series (4 gives the following identity : J α,β,γ = c α,β,γ,δ k= (k + 1 δ (k + α + jγ + 1. (1 β+γ+1 The expnsion in prtil frctions of the rtionl function R(X := (X + 1 δ (X + α + jγ + 1 β+γ+1 immeditely implies (9 (see [Ri], [BR]. Furthermore, pplying the trivil identity (x n = ( 1 n ( x n+1 n to R(X, we obtin the crucil symmetry reltion R( X δ 1 = ( 1 (β+1 R(X. (11 From (11 (essentilly by the uniqueness of the decomposition in prtil frctions, we then deduce tht P α,β,γ j, = if j 2 nd j + (β + 1 is odd, which completes the proof (see [Fi], [Co], or [Ri], [BR] for the slightly different originl rgument. An interesting problem would be to prove the identity (9 nd the prity phenomenom without expnding the integrl (8 s the series (1 (the method used in [RV1, RV2] could be relevnt here. Furthermore, hopefully we will find new diophntine pplictions of Theorem 1, other thn those lredy known for γ =. Acknowledgment. I thnk F. Amoroso for suggesting the ide of using Selberg s integrl to produce new rtionl liner forms in zet vlues. References [Be] F. Beukers, A note on the irrtionlity of ζ(2 nd ζ(3, Bull. London Mth. Soc. 11 (1979, no. 3, [BR] K. Bll, T. Rivol, Irrtionlité d une infinité de vleurs de l fonction zêt ux entiers impirs, Invent. Mth. 146 (21, no. 1,
6 [Co] P. Colmez, Arithmétique de l fonction zêt, Actes des journées X-UPS 22 L fonction zêt. [Fi] S. Fischler, Irrtionlité de vleurs de zêt (d près Apéry, Rivol,..., Séminire Bourbki 22-23, exposé numéro 91, novembre 22. [H] M. Ht, A new irrtionlity mesure for ζ(3, Act Arith. 92 (2, no. 1, [RV1] G. Rhin, C. Viol, On permuttion group relted to ζ(2, Act Arith. 77 (1996, no. 1, [RV2] G. Rhin, C. Viol, The group structure for ζ(3, Act Arith. 97 (21, no. 3, [Ri] T. Rivol, L fonction zêt de Riemnn prend une infinité de vleurs irrtionnelles ux entiers impirs, C. R. Acd. Sci. Pris Sér. I Mth. 331 (2, no. 4, [RZ] T. Rivol, W. Zudilin, Diophntine properties of numbers relted to Ctln s constnt, to pper in Mth. Annlen (23. [Se] A. Selberg, Remrks on multiple integrl (Norwegin, Norsk Mt. Tidsskr. 26 (1944, [Sl] L.J. Slter, Generlized Hypergeometric Functions, Cmbridge University Press, [So] V.N. Sorokin, On Apéry theorem (Russin. Russin summry, Vestnik Moskov. Univ. Ser. I Mt. Mekh. 1998, no. 3, 48 53, 74; trnsltion in Moscow Univ. Mth. Bull. 53 (1998, no. 3, [V] D.V. Vsilyev, On smll liner forms for the vlues of the Riemnn zetfunction t odd points, Preprint no. 1 (558, Nt. Acd. Sci. Belrus, Institute Mth., Minsk (21. [Zu] W. Zudilin, Well-poised hypergeometric service for diophntine problems of zet vlues, Actes des 12èmes rencontres rithmétiques de Cen (June 29 3, 21, to pper in J. Théorie Nombres Bordeux (23. Tnguy Rivol Lbortoire de Mthémtiques Nicols Oresme CNRS UMR 6139 Université de Cen BP Cen cedex, Frnce rivol@mth.unicen.fr 6
Selberg s integral and linear forms in zeta values
Journl of Computtionl nd Applied Mthemtics 160 (2003 265 270 www.elsevier.com/locte/cm Selberg s integrl nd liner forms in zet vlues Tnguy Rivol Lbortoire de Mthemtiques Nicols Oresme, CNRS UMR 6139, Universite
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationFrobenius numbers of generalized Fibonacci semigroups
Frobenius numbers of generlized Fiboncci semigroups Gretchen L. Mtthews 1 Deprtment of Mthemticl Sciences, Clemson University, Clemson, SC 29634-0975, USA gmtthe@clemson.edu Received:, Accepted:, Published:
More informationarxiv: v2 [math.nt] 5 Aug 2018
HYPERGEOMETRY INSPIRED BY IRRATIONALITY QUESTIONS CHRISTIAN KRATTENTHALER AND WADIM ZUDILIN riv:80.08856v mth.nt 5 Aug 08 Abstrct. We report new hypergeometric constructions of rtionl pproximtions to Ctln
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationIntegral points on the rational curve
Integrl points on the rtionl curve y x bx c x ;, b, c integers. Konstntine Zeltor Mthemtics University of Wisconsin - Mrinette 750 W. Byshore Street Mrinette, WI 5443-453 Also: Konstntine Zeltor P.O. Box
More informationOn Error Sum Functions Formed by Convergents of Real Numbers
3 47 6 3 Journl of Integer Sequences, Vol. 4 (), Article.8.6 On Error Sum Functions Formed by Convergents of Rel Numbers Crsten Elsner nd Mrtin Stein Fchhochschule für die Wirtschft Hnnover Freundllee
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu
More informationUniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationIN GAUSSIAN INTEGERS X 3 + Y 3 = Z 3 HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A2 IN GAUSSIAN INTEGERS X + Y = Z HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH Elis Lmpkis Lmpropoulou (Term), Kiprissi, T.K: 24500,
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationWHEN IS A FUNCTION NOT FLAT? 1. Introduction. {e 1 0, x = 0. f(x) =
WHEN IS A FUNCTION NOT FLAT? YIFEI PAN AND MEI WANG Abstrct. In this pper we prove unique continution property for vector vlued functions of one vrible stisfying certin differentil inequlity. Key words:
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationDecomposition of terms in Lucas sequences
Journl of Logic & Anlysis 1:4 009 1 3 ISSN 1759-9008 1 Decomposition of terms in Lucs sequences ABDELMADJID BOUDAOUD Let P, Q be non-zero integers such tht D = P 4Q is different from zero. The sequences
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More information1. On some properties of definite integrals. We prove
This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More informationREGULARITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2
EGULAITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2 OVIDIU SAVIN AND ENICO VALDINOCI Abstrct. We show tht the only nonlocl s-miniml cones in 2 re the trivil ones for ll s 0, 1). As consequence we obtin tht
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationTHE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p
THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS PETE L CLARK Circ 1870, Zolotrev observed tht the Legendre symbol ( p ) cn be interpreted s the sign of multipliction by viewed s permuttion of the set Z/pZ
More informationSection 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
More informationProperties of the Riemann Integral
Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More informationMath Solutions to homework 1
Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationAnalytical Methods Exam: Preparatory Exercises
Anlyticl Methods Exm: Preprtory Exercises Question. Wht does it men tht (X, F, µ) is mesure spce? Show tht µ is monotone, tht is: if E F re mesurble sets then µ(e) µ(f). Question. Discuss if ech of the
More informationChapter 1. Basic Concepts
Socrtes Dilecticl Process: The Þrst step is the seprtion of subject into its elements. After this, by deþning nd discovering more bout its prts, one better comprehends the entire subject Socrtes (469-399)
More informationKRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationAnti-derivatives/Indefinite Integrals of Basic Functions
Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS
ON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS A. BAZSÓ Astrct. Depending on the prity of the positive integer n the lternting power sum T k n = k + k + + k...+ 1 n 1 n 1 + k. cn e extended to polynomil
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions
ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner non-repeted ctors. The cover-up rule my be used: 4 4 4
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More information#A11 INTEGERS 11 (2011) NEW SEQUENCES THAT CONVERGE TO A GENERALIZATION OF EULER S CONSTANT
#A INTEGERS (20) NEW SEQUENCES THAT CONVERGE TO A GENERALIZATION OF EULER S CONSTANT Alin Sîntămărin Deprtment of Mthemtics, Technicl University of Cluj-Npoc, Cluj-Npoc, Romni Alin.Sintmrin@mth.utcluj.ro
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationThe presentation of a new type of quantum calculus
DOI.55/tmj-27-22 The presenttion of new type of quntum clculus Abdolli Nemty nd Mehdi Tourni b Deprtment of Mthemtics, University of Mzndrn, Bbolsr, Irn E-mil: nmty@umz.c.ir, mehdi.tourni@gmil.com b Abstrct
More informationIf deg(num) deg(denom), then we should use long-division of polynomials to rewrite: p(x) = s(x) + r(x) q(x), q(x)
Mth 50 The method of prtil frction decomposition (PFD is used to integrte some rtionl functions of the form p(x, where p/q is in lowest terms nd deg(num < deg(denom. q(x If deg(num deg(denom, then we should
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationOn the Generalized Weighted Quasi-Arithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 2039-2048 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More information1 1D heat and wave equations on a finite interval
1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationCALCULUS WITHOUT LIMITS
CALCULUS WITHOUT LIMITS The current stndrd for the clculus curriculum is, in my opinion, filure in mny spects. We try to present it with the modern stndrd of mthemticl rigor nd comprehensiveness but of
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationREPRESENTATION THEORY OF PSL 2 (q)
REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book [1]. The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationNew Integral Inequalities for n-time Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationDYNAMICAL SYSTEMS SUPPLEMENT 2007 pp Natalija Sergejeva. Department of Mathematics and Natural Sciences Parades 1 LV-5400 Daugavpils, Latvia
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 920 926 ON THE UNUSUAL FUČÍK SPECTRUM Ntlij Sergejev Deprtment of Mthemtics nd Nturl Sciences Prdes 1 LV-5400
More informationDISCRETE MATHEMATICS HOMEWORK 3 SOLUTIONS
DISCRETE MATHEMATICS 21228 HOMEWORK 3 SOLUTIONS JC Due in clss Wednesdy September 17. You my collborte but must write up your solutions by yourself. Lte homework will not be ccepted. Homework must either
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationIndefinite Integral. Chapter Integration - reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.
More informationLECTURE. INTEGRATION AND ANTIDERIVATIVE.
ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationVyacheslav Telnin. Search for New Numbers.
Vycheslv Telnin Serch for New Numbers. 1 CHAPTER I 2 I.1 Introduction. In 1984, in the first issue for tht yer of the Science nd Life mgzine, I red the rticle "Non-Stndrd Anlysis" by V. Uspensky, in which
More information1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More information