The Dirac distribution


 Gwendolyn Fisher
 1 years ago
 Views:
Transcription
1 A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution is usully depicted by the rrow of unit length see Fig Figure : Grph of the Dirc distribution δx 2 The sifting property of the Dirc distribution my serve s nother possible definition: Let s suppose tht funciton fx is continnous over the intervl x, x 2 or tht it hs t most finite number of finite discontinuities over tht intervl Then fx δx x dx x x2 2 [ fx + fx + ], if x x, x 2, 2 fx+, if x x, 2 fx, if x x 2,, if x x, x 2 Of course, if the function is continuous, the first of the reltions 2 reduces to the form x2 x fx δx x dx fx, if x x, x 2, 2 which is the most frequently ppering form of the sifting property see Fig 2 2 Figure 2: Sifting property of the Dirc distribution 3 Very often the Dirc distribution is defined s the limit of the sequence of functions δ p x The function δ p x hve to stisfy two conditions: δx lim δ px
2 2 A DIRAC DISTRIBUTION δ p x lim δ p x dx nd lim 3 lim x δ p x In most cses the functions δ p x stisfy more severe conditions: δ p x dx nd lim δ px 3 A2 Exmples of functions δ p x Probbly the most obvious exmple of functions δ p x is δ p x p rect px see Fig 3 Evidently, these functions stisfy the conditions A3 Figure 3: Grph of the function δ p x p rect px b Another obvious exmple provide the functions δ p x p tri px 2 see Fig 4 Also these functions obviously stisfy the conditions A3 Figure 4: Grph of the function δ p x p tri px c An importnt exmple is the sequence of functions p δ p x π exp px2 3 see Fig 5 Let us show, tht lso these functions stisfy the conditions A3: p δ p x dx exp px 2 dx exp t 2 dt π π The integrl I exp t2 dt is evluted s follows: I 2 exp x 2 dx,
3 A DIRAC DISTRIBUTION 3 Figure 5: Grph of the function δ p x p/π exp px 2 so tht I π π/2 exp x 2 dx exp y 2 dy 4 exp r 2 r dϕ dr 2π exp s ds π exp r 2 r dr exp [ x 2 + y 2 ] dx dy Hence I π The second condition A3 is stisfied s well: lim δ px p lim π exppx 2 2x 2 π lim exp px 2 p x x d Also the functions see Fig 6 stisfy the conditions A3: δ p x dx π δ p x π p dx + p 2 x 2 π lim δ px π lim p + p 2 x 2 p + p 2 x 2 4 dt + t 2 π rctg t π lim x 2px 2 t t x e In clcultions nd proofs of theorems bout the Fourier trnsform we often meet formlly different expressions of the function δ p x 2π p p exp±itx dt π p see Fig 7 The first of the conditions A3 is stisfied: cos tx dt π x p π px 5
4 4 A DIRAC DISTRIBUTION Figure 6: Grph of the function δ p x π p +p 2 x 2 Figure 7: Grph of the function δ p x πx δ p x dx π 2 π x sin y y dx π dy 2 π π 2 cf eg [2], 372, [3], 5225 The second condition A3 is however, not stisfied becuse the corresponding limit does not exist: If x, the function δ p x πx tkes the vlues from the intervl πx, πx, which does not depend on p The condition A3 is, of course, stisfied becuse lim x δ p x p π, nd hence δ p x lim lim x δ p x lim π lim πx p px x x A3 Properties of the Dirc distribution sin y y dy Let us denote by x n the roots of the eqution fx nd suppose tht f x n Then
5 A DIRAC DISTRIBUTION 5 δfx n δx x n f x n Proof: Let us choose the numbers n, b n, in neighbourhood of ech root x n in such wy tht n < x n < b n nd the function fx is monotonous in the intervl < n, b n > Then where gxδfx dx n I n, 2 I n b n n gxδfx dx By the substitution x x n f x n t we get I n f x n b n b n x nf x n n x nf x n n gxδx x n f x n dx 3 g t f x n + x n δt dt 4 If f x n <, the upper limit of integrtion is greter thn the lower one nd it is I n f x n n x nf x n b n x nf x n g t f x n + x n δt dt gx n f x n From 2 nd 3 nd from the sifting property of the Dirc distribution A2 it follows gxδfx dx n gx n f x n n b n f gxδx x n dx 5 x n n If f x n >, the reltion 5 follows imeditely from 2, 4 nd A2 Thus, the eqution is proved Importnt consequences of eqution re: δ x δx, 6 b It is ie where δx x δ δ sin π x π δx 2 2 δx 2 m x x, 7 δx m, 8 δx + δx d x, dx x δx dhx dx, Hx 2 + x x 2
6 6 A DIRAC DISTRIBUTION is the Heviside function Proof: 2 d x dx x 2 lim d 2 dx π rctg px π lim p + p 2 δx 3 x2 cf A24 c The following properties of the Dirc distribution re frequently used eg while evluting convolutions nd cross coreltions: fx δx f δx, 4 d c δx δx b dx δ b, c < min, b, d > mx, b 5 A4 The Dirc distribution obtined from complete system of orthonorml functions Interesting nd often useful expressions of the Dirc distribution cn be obtined from complete systems of orthogonl functions Let functions ψ n x, n being integers, form complete orthonorml system of functions on n intervl x, x + nd let x nd x be inner points of tht intervl Then ψnx ψ n x δx x, n where the summtion goes over ll n for which the orthonorml system {ψ n x} is complete Proof: To prove we shll demonstrte tht the left hnd side of eqution hs the sifting property of the Dirc distribution ie tht x+ x fx δx x dx fx, x+ x fx n ψ nx ψ n x dx fx 2 To prove 2 we expnd function fx into the system of orthonorml functions {ψ n x}, ie fx m c m ψ m x, 3 where c m x+ x fx ψ mx dx Now we insert the series 3 into the left hnd side of eqution 2, exchnge the order of integrtion nd ddition nd mke use of the condition of orthonormlity x+ x ψ nx ψ m x dx δ m,n : x+ x m c m ψ m x n x+ ψnx ψ n x dx c m ψ n x ψnx ψ m x dx m n x c m ψ n x δ m,n m n c m ψ m x fx m
7 A DIRAC DISTRIBUTION 7 Thus, we hve got the right hnd side of 2 nd the sttement is proved The functions ψ n x exp in2π x, n, ±, ±2, form the complete orthonorml system on ny intervl of the length nd hence lso on the intervl /2, /2 Therefore, ccording to it is n exp in2π x x δx x, x, x 2, 2 Every summnd of the infinite geometric series on the left hnd side of the foregoing reltion is periodic function with the period Consequently the sum of the series hs the sme period nd for ll x, x it holds n exp in2π x x m δx x m 4 Reltion 4 is importnt for the proof of the fct, tht the Fourier trnsform of the lttice function is proportionl to the lttice function chrcterizing the reciprocl lttice cf section 43 This is true for the lttices of ny dimensions N, N being integer N To be prepred for the proof in the spce of the dimension N 2 we denote the length of the intervl, so tht in eqution 4 my be both positive nd negtive The series t the left hnd side of 4 my be rewritten in vrious forms For exmple + 2 n cos n2π x x m δx x m 5 The series t the left hnd side of 4 is geometric series of the rtio exp i2π x x We my replce it by the limit n exp in2π x x lim By summing 2p + terms of the limit we get p n p exp in2π x x lim p n p exp in2π x x lim lim { { exp exp ip2π x x [ ]} exp i2p + 2π x x exp i2π x x ip2π x x [ exp i2p + π x x ] exp iπ x x ] [ ]} exp i2p + π x x exp [ i2p + π x x lim exp iπ x x ] sin [ 2p + π x x sin π x x exp iπ x x Hence lim sin [ ] 2p + π x x sin π x x m δ x x m 6
8 8 A DIRAC DISTRIBUTION A5 The Dirc distribution in E N In Crtesin coordintes From the fct tht D f xδ x x d N x f x, if x D D fx, x 2,, x N δx x δx 2 x 2 δx N x N dx dx 2 dx N fx, x 2,, x N, it follows tht in Crtesin coordintes Obviously N δ x x δx x δx 2 x 2 δx N x N δx k x k 2 k δ x δ x 3 N b Generl coordintes Let the Crtesin coordintes x, x 2,, x N be connected with generl coordintes y, y 2,, y N in E N by reltions x x y,, y N, x 2 x 2 y,, y N, x N x N y,, y N with the Jcobin Jy,, y N x x y N y,, x N y,, x N y N If x P, x P 2,, x P N then nd yp, y P 2,, y P N re coordintes of point P nd if JyP,, y P N, δx x P δx 2 x P 2 δx N x P N Jy,, y N δy y P δy 2 y P 2 δy N y P N 4 If, however, Jy P,, y P N nd the point P is specified by k coordintes yp, y P 2,, y P k tht mens tht N k coordintes y k+, y k+2,, y N re superfluous for the specifiction of the point P, we denote by J k y,, y k Jy,, y N dy k+ dy N the integrl over the N k superfluous coordintes nd it holds
9 A DIRAC DISTRIBUTION 9 δx x P δx 2 x P 2 δx N x P N J k y,, y k δy y P δy 2 y P 2 δy k y P k 5 c Exmple: Polr coordintes in E 2 x r cos ϕ, x 2 r sin ϕ, Jr, ϕ cos ϕ sin ϕ r sin ϕ r cos ϕ r i At points P r P, ϕ P, r P, it is δx x P δx 2 x P 2 δr rp δϕ ϕ P r ii At the point P r P the coordinte ϕ is superfluous nd so tht J r α+2π α r dϕ 2πr, δx δx 2 δr 2πr d Exmple: Sphericl coordintes in E 3 see Fig 8 x r sin ϑ cos ϕ, x 2 r sin ϑ sin ϕ, x 3 r cos ϑ, Figure 8: Sphericl coordintes Jr, ϑ, ϕ x r x 2 r x 3 r x ϑ x 2 ϑ x 3 ϑ x ϕ x 2 ϕ x 3 ϕ sin ϑ cos ϕ r cos ϑ cos ϕ r sin ϑ sin ϕ sin ϑ sin ϕ r cos ϑ sin ϕ r sin ϑ cos ϕ cos ϑ r sin ϑ r 2 sin ϑ i At the point P with coordintes r P, ϑ P, ϑ P π, ie with Jr P, ϑ P, ϕ P, it is
10 A DIRAC DISTRIBUTION δx x P δx 2 x P 2 δx 3 x P 3 δr rp δϑ ϑ P δϕ ϕ P r 2 sin ϑ ii At the point P with coordintes r P, ϑ P, or ϑ P π, the Jcobin Jr P, ϑ P, ϕ P nd Therefore J 2 r, ϑ 2π r 2 sin ϑ dϕ 2πr 2 sin ϑ δx δx 2 δx 3 x P 3 δr rp δϑ 2πr 2 sin ϑ iii At the point P with r P, it is Jr P, ϑ P, ϕ P nd Therefore J r π 2π r 2 sin ϑ dϕ dϑ 4πr 2 δx δx 2 δx 3 δr 4πr 2 e Exmple: Obligue coordintes importnt for the Fourier trnsform of lttices in E N, N 2 Let More explicitly this cn be rewritten s x i ik y k, det ik det A x y + + N y N, x N N y + + NN y N, or in the mtrix form x x N,, N N,, NN y y N, ie x A y As the Jcobin is x i y k ik, Jy,, y N,, N N,, NN det A nd
11 A DIRAC DISTRIBUTION ie δx x P δx N x P N det A δy y P δy N y P N, δ x x P det A δ y yp A6 Notes nd fetures δ p x p 2 + p 2 x 2 3/2 δ p x 22n 3 n!n 2! 2n 3! p π + p 2 x 2, n 2, 3, 2 n 2 3 δ p x p π px δy y y J m xyj m xy x dx 4 δ p x, y p π exp{ p[ exp x2 y 2 ]} 5 δ p x, y p2 π circ p x 2 + y 2 δ p x, y p 2 J p x 2 + y 2 4π p 7 x 2 + y 2 δxy 6 δx + δy x2 + y 2 8 References [] Dirc P A M: The Principles of Quntum Mechnics 4th edition At the Clrendon Press, Oxford 958, 5 [2] Grdshteyn I S, Ryzhik I M: Tble of Integrls, Series, nd Products Acdemic Press, New York nd London 994 [3] Abrmowitz M, Stegun I A: Hndbook of Mthemticl Functions Dover Publictions, Inc, New York 972
63. 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 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 informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 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 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 information12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS
1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility
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 informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
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 informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
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 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 informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
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 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 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 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 informationMapping the delta function and other Radon measures
Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support
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 informationInnerproduct spaces
Innerproduct spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:
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 informationUNIFORM 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 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 informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
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 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 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 informationThe solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr
Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the evectors nd evlues
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationk and v = v 1 j + u 3 i + v 2
ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
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 informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationMath 115 ( ) YumTong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx
Mth 5 20062007) YumTong Siu Lgrnge Multipliers nd Vritionl Problems with Constrints Integrl Constrints. Consider the vritionl problem of finding the extremls for the functionl J[y] = F x,y,y )dx with
More informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
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 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 informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
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 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 informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More information7  Continuous random variables
71 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7  Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin
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 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 informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
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 informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
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 informationSTURMLIOUVILLE BOUNDARY VALUE PROBLEMS
STURMLIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2
More informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
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 informationAPPM 1360 Exam 2 Spring 2016
APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the
More informationF (x) dx = F (x)+c = u + C = du,
35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10
University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin
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 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 informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationAnalytically, vectors will be represented by lowercase boldface Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
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 informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
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 informationSection 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
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 informationPDE Notes. Paul Carnig. January ODE s vs PDE s 1
PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................
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 informationLinear Differential Equations Physics 129a Solutions to Problems Frank Porter Revision F. Porter
Liner Differentil Equtions Physics 19 Solutions to Problems 051018 Frnk Porter Revision 11106 F. Porter 1 Exercises 1. Consider the generl liner second order homogeneous differentil eqution in one dimemsion:
More informationLecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations
18330 Lecture Notes: Orthogonl Polynomils, Gussin Qudrture, nd Integrl Equtions Homer Reid My 1, 2014 In the previous set of notes we rrived t the definition of Chebyshev polynomils T n (x) vi the following
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationChapter 4. Additional Variational Concepts
Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.
More informationMACsolutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
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 informationSturmLiouville Theory
LECTURE 1 SturmLiouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory
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 informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Clculus of One Vrile II Fll 2017 Textook: Single Vrile Clculus: Erly Trnscendentls, 7e, y Jmes Stewrt Unit 2 Skill Set Importnt: Students should expect test questions tht require synthesis of
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 informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More information31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes
Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be
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 informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/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 informationMath 4200: Homework Problems
Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,
More informationMath 3B Final Review
Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:4510:45m SH 1607 Mth Lb Hours: TR 12pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
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 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 informationNumerical Integration
Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture
More informationPractice Problems Solution
Prctice Problems Solution Problem Consier D Simple Hrmonic Oscilltor escribe by the Hmiltonin Ĥ ˆp m + mwˆx Recll the rte of chnge of the expecttion of quntum mechnicl opertor t A ī A, H] + h A t. Let
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
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 information