EulerMaclaurin Summation Formula 1


 Clemence Craig
 1 years ago
 Views:
Transcription
1 Jnury 9, EulerMclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z, then <n b (f(x) + ψ(x)f (x)) dx + (f(b) f()). Proof: The proof proceeds long the lines of the Abel prtil summtion formul. (b )f(b) + f() (n )(f(n + ) f(n)) <n b n b (b )f(b) + f() (b )f(b) + f() (n ) n b n+ n b n (b )f(b) + f() + ( + ) bf(b) f() f(t)dt + f(t)dt + (t {t})f (t)dt {t}f (t)dt n+ n f (t)dt ([t] )f (t)dt f (t)dt ( {t} ) f (t)dt + (f(b) f()) (f(t) + ψ(t)f (t)) dt + (f(b) f()). [t]f (t)dt We define Bernoulli polynomils by the following three properties To determine the polynomils we introduce generting function F(t, x) B k (x) tk k!. B (x), () B k(x) kb k (x), k,,..., () B k (x)dx, k,,.... (3) These notes follow Anlytic Number Theory by H. Iwniec & E. Kowlski. Property () defines B k, k recursively up to constnt. The constnt is fixed by property (3). k
2 Observe F(t, x) x t k B k(x) tk k! kb k (x) tk k! t B k (x) tf(t, x). (k )! k k k Thus F(t, x) C(t)e tx (4) where C is some function of t. Now by the third defining property of B k (x), F(t, x)dx. Integrting both sides of (4) with respect to x over the intervl [, ] then implies nd hence; k C(t) t e t, B k (x) tk k! tetx e t. Using this generting function we cn find the first few Bernoulli polynomils: The Bernoulli numbers re defined by B (x), B (x) x, B (x) x x + 6, B 3 (x) x 3 3 x + x, B 4 (x) x 4 x 3 + x 3, B 5 (x) x 5 5 x x3 6 x, B 6 (x) x 6 3x x4 x + 4. B n B n (), tht is, the vlue of the Bernoulli polynomil t x. The generting function for Bernoulli numbers is clerly F(t) : t k B k k! t e t. n An esy clcultion shows F( t) F(t) + t; nd hence, F( t) F(t) t. This lst equlity implies tht B k+ for k,,.... Define ψ k (x) B k ({x}) (5) where {x} is the frctionl prt of x. Observe tht ψ(x) ψ (x) {x} which ppers in Lemm. Since {x} is periodic with period, so too re the functions ψ k (x) nd they hve generting function k ψ k (x) tk k! tet {x} e t
3 We now ssume tht f is twice continuously differentible in [, b]. We integrte by prts the term involving ψ in Lemm. First look t + f (x)ψ (x)dx f (t + )ψ (t + )dt since ψ is periodic. In the intervl [, ], ψ (t) B (t). Thus x ψ (t)dt (ψ (x) B ) f (t + )ψ (t)dt (6) where we used defining property () of the Bernoulli polynomils. Note tht ψ (x)dx since ψ is periodic. Integrting the lst integrl in (6) by prts gives + f (x)ψ (x)dx f (y + ) y ψ (t)dt y y f (y + ) (ψ (y) B ) + f (y + ) y ψ (t)dt f (y + )ψ (y)dy + B (f ( + ) f ()) f (x)ψ (x)dx + B (f ( + ) f ()) since ψ is periodic. One notes tht the bove formul is vlid over ny intervl [ + n, + n]; nmely, +n f (x)ψ (x)dx +n f (x)ψ (x)dx+ +n +n B (f ( + n) f ( + n )), n,,..., b. Summing over n we obtin We hve proved f (x)ψ (x)dx f (x)ψ (x)dx + B (f (b) f ()). Lemm : Let f be twice continuously differentible on [, b] where < b nd, b Z. Then <n b { f(x) } ψ (x)f (x) dx + ( ) l ( f l (b) f l () ) B l. l (Recll B.) If one continues to integrte by prts one obtins the Theorem (EulerMclurin formul): Suppose f is ktimes continuously differentible on the intervl [, b] with < b,, b Z. Then <n b } {f(x) ( )k ψ k (x)f (k) (x) dx + k! k ( ) l l ( ) f (l ) (b) f (l ) () B l. 3
4 Suppose f nd ll its derivtives go to zero s x. Then we obtin by letting b (nd dding f() to both sides) n f(x)dx + k f() ( ) l l f (l ) ()B l ( )k k! f (k) (x)ψ k (x)dx (7) Appliction of summtion formul to the Riemnn zetfunction Let s σ + it where σ is the rel prt of s nd t is the imginry prt of s. Let σ > nd define the Riemnn zetfunction ζ(s) ns, R(s) >. (8) n The series converges bsolutely nd uniformly in the hlfplne σ R(s) + ε: First observe tht Now pply the Weiestrss Mtest to the series n s n σ it n σ n ε n n +ε which is convergent for ll ε >. The series (8) clerly diverges t s. We now pply (7) with k to (8). Choose f(x) /x s. For R(s) >, The summtion formul then becomes ζ(s) s + s dx x s s. x s+ ψ (x)dx. This is derived under the ssumption tht σ >. Observe tht if we write ζ(s) s s x s+ ψ (x)dx then the righthnd side of the bove eqution defines holomorphic function for σ > since the integrl x s+ ψ (x)dx dx < (9) x+σ (We used ψ (x) /.) We now use the righthnd side (9) to define the lefthnd side of (9) for < σ. The two side gree for σ >. This is n exmple of nlytic continution. We hve mde sense out of the Riemnn zetfunction for < R(s). We see tht it hs simple pole t s nd is holomorphic for ll other points R(s) >. If we pply (7) to (8) for rbitrry positive integer k we obtin fter some elementry computtions ζ(s) s + + k l B l ( )k s(s + ) (s + l ) k! 4 s(s + ) (s + k )x s k ψ k (x)dx
5 Since ψ k is periodic nd equl to the polynomil B k (x) on [, ), ψ k (x) is bounded function on ll of R. Thus the integrl on the righthnd side is convergent for ll σ+k > nd thus defines holomorphic function for σ > k. By repeting the bove rgument we see tht we hve nlyticlly continued the Riemnn zetfunction to the righthlf plne σ > k, for ll k,, 3,.... For exmple, it now mkes sense to sk for the vlue ζ ( ). 3 We summrize our findings in Theorem: The Riemnn zetfunction ζ(s) defined by (8) for R(s) > cn be nlyticlly continued to C {} where it is holomorphic nd t s, ζ(s) hs simple pole. 3 Before we nlyticlly continued ζ(s) it clerly mkes no sense in (8) to sk for the derivtive t s since the series only converges for R(s) >. It is fct tht ζ ( )
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
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 informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationOrthogonal Polynomials and LeastSquares Approximations to Functions
Chpter Orthogonl Polynomils nd LestSqures Approximtions to Functions **4/5/3 ET. Discrete LestSqures Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More informationChapter 6. Riemann Integral
Introduction to Riemnn integrl Chpter 6. Riemnn Integrl WonKwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl
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 informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
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 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 informationMATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.
MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationMAT612REAL ANALYSIS RIEMANN STIELTJES INTEGRAL
MAT612REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
More informationIII. Lecture on Numerical Integration. File faclib/dattab/lecturenotes/numericalinter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecturenotes/numericalinter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the
More informationProperties of the Riemann Stieltjes Integral
Properties of the Riemnn Stieltjes Integrl Theorem (Linerity Properties) Let < c < d < b nd A,B IR nd f,g,α,β : [,b] IR. () If f,g R(α) on [,b], then Af +Bg R(α) on [,b] nd [ ] b Af +Bg dα A +B (b) If
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCKKURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When relvlued
More information1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.
Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the
More informationA unified generalization of perturbed midpoint and trapezoid inequalities and asymptotic expressions for its error term
An. Ştiinţ. Univ. Al. I. Cuz Işi. Mt. (N.S. Tomul LXIII, 07, f. A unified generliztion of perturbed midpoint nd trpezoid inequlities nd symptotic expressions for its error term Wenjun Liu Received: 7.XI.0
More informationMATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.
MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded
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 relvlues nd smooth The pproximtion of n integrl by numericl
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationNew Integral Inequalities for ntime Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353362 New Integrl Inequlities for ntime Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More informationNew Integral Inequalities of the Type of HermiteHadamard Through Quasi Convexity
Punjb University Journl of Mthemtics (ISSN 11656) Vol. 45 (13) pp. 3338 New Integrl Inequlities of the Type of HermiteHdmrd Through Qusi Convexity S. Hussin Deprtment of Mthemtics, College of Science,
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 informationWeek 7 Riemann Stieltjes Integration: Lectures 1921
Week 7 Riemnn Stieltjes Integrtion: Lectures 1921 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0
More informationS. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:
FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy
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, BbeşBolyi University Str. Koglnicenu
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 informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationWeek 10: Riemann integral and its properties
Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the
More informationON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More information(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35
7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know
More informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationMATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral.
MATH 409 Advnced Clculus I Lecture 18: Drboux sums. The Riemnn integrl. Prtitions of n intervl Definition. A prtition of closed bounded intervl [, b] is finite subset P [,b] tht includes the endpoints
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 informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. LmiAthens Lmi 3500 Greece Abstrct Using
More informationChapter 4. Lebesgue Integration
4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationFor a continuous function f : [a; b]! R we wish to define the Riemann integral
Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationRAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IITJEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties
M.Sc. (Mths), B.Ed, M.Phil (Mths) MATHEMATICS Mob. : 947084408 9546359990 M.Sc. (Mths), B.Ed, M.Phil (Mths) RAM RAJYA MORE, SIWAN XI th, XII th, TARGET IITJEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII
More informationA Convergence Theorem for the Improper Riemann Integral of Banach Spacevalued Functions
Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 24512460 HIKARI Ltd, www.mhikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch
More informationThe Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2
The Bernoulli Numbers John C. Bez, December 23, 2003 The numbers re defined by the eqution e 1 n 0 k. They re clled the Bernoulli numbers becuse they were first studied by Johnn Fulhber in book published
More informationCalculus MATH 172Fall 2017 Lecture Notes
Clculus MATH 172Fll 2017 Lecture Notes These notes re concise summry of wht hs been covered so fr during the lectures. All the definitions must be memorized nd understood. Sttements of importnt theorems
More informationMidpoint Approximation
Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do
More informationNOTES AND PROBLEMS: INTEGRATION THEORY
NOTES AND PROBLEMS: INTEGRATION THEORY SAMEER CHAVAN Abstrct. These re the lecture notes prepred for prticipnts of AFSI to be conducted t Kumun University, Almor from 1st to 27th December, 2014. Contents
More informationMath 324 Course Notes: Brief description
Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd
More informationMTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008
MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul
More informationON BERNOULLI BOUNDARY VALUE PROBLEM
LE MATEMATICHE Vol. LXII (2007) Fsc. II, pp. 163 173 ON BERNOULLI BOUNDARY VALUE PROBLEM FRANCESCO A. COSTABILE  ANNAROSA SERPE We consider the boundry vlue problem: x (m) (t) = f (t,x(t)), t b, m > 1
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 informationThe RiemannStieltjes Integral
Chpter 6 The RiemnnStieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More informationKOÇ UNIVERSITY MATH 106 FINAL EXAM JANUARY 6, 2013
KOÇ UNIVERSITY MATH 6 FINAL EXAM JANUARY 6, 23 Durtion of Exm: 2 minutes INSTRUCTIONS: No clcultors nd no cell phones my be used on the test. No questions, nd tlking llowed. You must lwys explin your nswers
More informationFunctions of bounded variation
Division for Mthemtics Mrtin Lind Functions of bounded vrition Mthemtics Clevel thesis Dte: 20060130 Supervisor: Viktor Kold Exminer: Thoms Mrtinsson Krlstds universitet 651 88 Krlstd Tfn 054700 10
More informationMATH Summary of Chapter 13
MATH 21259 ummry of hpter 13 1. Vector Fields re vector functions of two or three vribles. Typiclly, vector field is denoted by F(x, y, z) = P (x, y, z)i+q(x, y, z)j+r(x, y, z)k where P, Q, R re clled
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI Question No: 1 { Mrks: 1 )  Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...
More informationPower Series, Taylor Series
CHAPTER 5 Power Series, Tylor Series In Chpter 4, we evluted complex integrls directly by using Cuchy s integrl formul, which ws derived from the fmous Cuchy integrl theorem. We now shift from the pproch
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationSummary of Elementary Calculus
Summry of Elementry Clculus Notes by Wlter Noll (1971) 1 The rel numbers The set of rel numbers is denoted by R. The set R is often visulized geometriclly s numberline nd its elements re often referred
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A 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 informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
More informationNumerical quadrature based on interpolating functions: A MATLAB implementation
SEMINAR REPORT Numericl qudrture bsed on interpolting functions: A MATLAB implementtion by Venkt Ayylsomyjul A seminr report submitted in prtil fulfillment for the degree of Mster of Science (M.Sc) in
More informationCalculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties
Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwthchen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:
More informationNecessary and Sufficient Conditions for Differentiating Under the Integral Sign
Necessry nd Sufficient Conditions for Differentiting Under the Integrl Sign Erik Tlvil 1. INTRODUCTION. When we hve n integrl tht depends on prmeter, sy F(x f (x, y dy, it is often importnt to know when
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationTest 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).
Test 3 Review Jiwen He Test 3 Test 3: Dec. 46 in CASA Mteril  Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 1417 in CASA You Might Be Interested
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationSummer MTH142 College Calculus 2. Section J. Lecture Notes. Yin Su University at Buffalo
Summer 6 MTH4 College Clculus Section J Lecture Notes Yin Su University t Bufflo yinsu@bufflo.edu Contents Bsic techniques of integrtion 3. Antiderivtive nd indefinite integrls..............................................
More informationf (x)dx = f(b) f(a). a b f (x)dx is the limit of sums
Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationTHIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions
THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, HuiHsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No.
More information4402 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam
4402 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP
More informationMath RE  Calculus II Area Page 1 of 12
Mth RE  Clculus II re Pge of re nd the Riemnn Sum Let f) be continuous function nd = f) f) > on closed intervl,b] s shown on the grph. The Riemnn Sum theor shows tht the re of R the region R hs re=
More information4.1. Probability Density Functions
STT 1 4.14. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile  vers  discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More information