Green Functions. January 12, and the Dirac delta function. 1 x x
|
|
- Silvester Martin
- 5 years ago
- Views:
Transcription
1 Gee Fuctios Jauay, 6 The aplacia of a the Diac elta fuctio Cosie the potetial of a isolate poit chage q at x Φ = q 4πɛ x x Fo coveiece, choose cooiates so that x is at the oigi. The i spheical cooiates, the potetial is popotioal to. As a fuctio, f = is efie o the ope iteval,, but ot at the oigi. Its aplacia is also efie o this iteval, a is quickly see to vaish eveywhee, = = = This leas to a ifficulty whe we cosie the ivegece theoe, fo which the volue itegal iclues the oigi 3 x = x ce the ight ha sie is well-efie but the left is ot. Iee, fo a sphee S ε, of aius ε, the itegal o the ight becoes S S ε x = π π = 4π ε ε θθϕ Howeve, the itegal o the left is uefie. The igoous way to hale this is to exte the fuctio f = to a istibutio. A istibutio is efie as the liit of a sequece of fuctios, givig a object which is oly eaigful whe itegate. Thus, if we efie a istibutio f to be the liit f x li a f a x whee f a x is a collectio of fuctios epeig o a paaete a. A istibutio is ofte calle a fuctioal, a we use the two tes itechageably. The itegal of the istibutio is efie as the liit of the well-behave itegals of the seies of fuctios
2 f x x li a f a x x a this ay be pefectly fiite eve if f x is ot a tue fuctio. With this i i, let f a x =. This is efie fo the close iteval [, ], a so is its +a aplacia = + a + a = + a 3/ = 3 + a 3/ = 3 + a 3 4 3/ + a 5/ 3 = + a + 3 3/ + a 5/ 3 = + a 3 + a 5/ + a 5/ 3a = + a 5/ We ay theefoe efie a istibutio to exte δ x = by δ x = li f a x a 3a = li a + a 5/ The itegal is ow well-efie: 3 x li a = π li a a = 4π li 3 x + a a = li a 4πε 3 ε + a 3/ = 4π + a 5/ 3 + a fo ay fiite ε. As a pleasat bous, the ivegece theoe is ow satisfie as log as we uesta to be a istibutio. 3/
3 Notice that the fuctioal δ iveges at = a vaishes fo all >, while its itegal is fiite. Futheoe, if f is ay sooth, boue fuctio of whih vaishes outsie soe copact set, the f 3 x = li f 3 x a + a whee the eaie R satisfies which vaishes as ε, leavig This eas that = 4πf li a = 4πf 4πR = R = li x a < εf li 3 4π li + a 3/ a f a 3/ a f a 3/ 3 x = 4πf f a 3/ is popotioal to a Diac elta fuctioal at the oigi, = 4πδ 3 x x Retuig to a abitay oigi, we ay wite this as x x = 4πδ 3 x x Notice that what is uivesally efee to as the Diac elta fuctio is, popely speakig, a fuctioal. Geeal solutio fo the potetial of a poit chage with bouay coitios I tes of the potetial, Gauss s law i fee space is Φ = ɛ ρ x The chage esity fo a isolate chage q at positio x is ρ x = qδ 3 x x We wish to solve fo the potetial Φ fo this poit souce, a bouay coitios give o soe suface, S. The bouay ay be copise of ultiple pieces. Fo the peceig sectio, we see that the solutio fo the potetial is Φ x = q 4πɛ x x + Φ 3 3
4 whee Φ is ay solutio to the aplace equatio, Φ = Now, we kow that the solutio to the aplace equatio is uique oce we specify bouay coitios, a a foal poof of this will be give below. Suppose we have bouay coitios Φ x S = Φ x S fo ay poit x S o S. The if we equie Φ x S = Φ x S q 4πɛ x S x thee is a uique solutio fo Φ, a theefoe a uique solutio fo Φ satisfyig the give bouay coitios. Alteatively, we ay fi the solutio iectly by solvig Φ = q δ 3 x x 4πɛ with bouay coitios Φ x S. This is oe staightfowa tha it appeas, because the Diac elta fuctio vaishes alost eveywhee. Theefoe, uless x = x, we ae solvig the aplace equatio. As a esult, we ay costuct ou solutio fo Φ fo solutios to the coespoig aplace equatio.. Exaple: Isolate poit chage The siplest exaple is fo a isolate poit chage, with the potetial vaishig at ifiity. aleay show that x x = 4πδ3 x x so we ieiately have Φ x, x = 4πɛ x x + F x We have whee F satisfies the aplace equatio, F =. By uiqueess, the fuctio F ust be eteie by the bouay coitios. I the peset case, we ask fo Φ x, x to vaish at x. Sice the fist te i Φ x, x aleay satisfies this, we equie the sae coitio fo F : F = F = The aguet of the peceeig sectio shows that F x = is the uique solutio to this, so the Gee fuctio fo a isolate poit chage is G x, x = x x. If we have a localize istibutio of chage, ρ x, i epty space, the potetial vaishes at ifiity a we ca use this Gee fuctio to fi the potetial eveywhee by itegatig Φ x = G x, x ρ x 3 x 4πɛ = ρ x ɛ x x 3 x. Exaple: Bouay coitios o a squae.. The aplace equatio Cosie a -iesioal exaple, with bouay coitios give o a squae of sie with oe coe at the oigi. et the bouay at x = have potetial, with the eaiig bouay segets havig 4
5 Φ =. The with the gle chage q at x = x, y, the Poisso equatio becoes x + y Φ = q δ x x δ y y 4πɛ We begi by solvig the hoogeeous aplace equatio, x + y Φ = by sepaatig i Catesia cooiates. Witig Φ = X x Y y, the iviig by Φ, the aplace equatio is XY x + y XY = X X x + Y Y y = Sice the fist te epes oly o x a the seco oly o y, each ust be costat, so with the ieiate solutios X X x = Y Y y = α X α x = A α h αx + C α cosh αx Y α y = B α αy + D α cos αy These fuctios will satisfy the bouay coitios fo y a at x = if we set α = π a C α = D α =, leavig X α x = A α h πx Y α y = B α πy Cobiig coefficiets, the geeal solutio is the a su Φ x, y = A h πx The eaiig bouay coitio at x = is fou by settig = Φ, y = πy A h π πy This is just a Fouie seies fo a costat. Multiplyig by πy so that a fially y πy π cos π = = Φ, y = A δ h π π = A h π A = Φ x, y = o A h π π h π a itegatig, y πy 4 πx h π h π πy πy 5
6 .. Fouie epesetatio of the Diac elta fuctio Notig that the y-epeece is escibe by a e seies, we ake use of the fact that the Diac elta fuctioal ay also be witte as a Fouie seies, To fi the A, ultiply by πy δ y y = A πy a itegate to fi the coefficiets, yδ y y πy πy = = y A δ A πy πy y πy = A δ y πy + πy cos = A δ = A δ y cos πy = A so that A = πy Itegatig twice shows that a we have y δ y y = πy πy πy π πy = δ y y..3 A asatz fo a paticula solutio While thee ae systeatic appoaches to solvig the poit paticle Poisso equatio i vaious cooiate systes, we take a siple appoach hee. Suppose we guess that we ca fi a solutio of the fo Φ p x, y = π a substitute eqs. a ito the Poisso equatio x + y f x πy π x + f x πy f x πy πy Φ = q 4πɛ δ x x δ y y 6 = q 4πɛ δ x x πy πy
7 a theefoe, equatig like tes, Now expa f a the seco elta fuctio, a substitute, Theefoe, π Fially, check that x + y B π Φ p x, y = π f x + f = q 4πɛ δ x x f x = δ x x = πx q π 3 ɛ B πx πx + B πx +, Φ p x, y = q π 3 ɛ..4 Bouay coitios, = q π 3 ɛ π πx + πx = q 4πɛ B = q πx 4πɛ πx B = q πx 4πɛ + πx πy π + + π πx πx = q 4πɛ δ x x δ y y πx πy πy πx πy πx πy πy The potetial Φ p x, y is a paticula solutio to ou Poissoi equatio, but it oes ot satisfy the bouay coitio at x =, istea vaishig at all fou bouay lies. To get the coplete solutio, we ee to a the hoogeeous solutio that satisfies the bouay coitios. The full solutio is theefoe Φ x, y = q π 3 ɛ, 3 Supepositio πx + πx πy πy + o The geeal poble of electostatics is to solve the Poissio equatio, Φ = ɛ ρ x 4 πx h π h π πy 7
8 with give chage esity ρ x a bouay coitios Φ x S. Kowig the solutio fo a poit chage at x allows us to o this ieiately by takig the supepositio of ifiitesial chages ρ x 3 x a suig itegatig ove ou etie volue: Φ x = 4πɛ ρ x 3 x x x + Φ x whee Φ satisfies the aplace equatio. We choose Φ so that Φ satisfies the bouay coitio. We will exaie etails of this solutio i the ext Note. 8
W = mgdz = mgh. We can express this potential as a function of z: V ( z) = gz. = mg k. dz dz
Electoagetic Theoy Pof Ruiz, UNC Asheville, doctophys o YouTube Chapte M Notes Laplace's Equatio M Review of Necessay Foe Mateial The Electic Potetial Recall i you study of echaics the usefuless of the
More informationMath 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as
Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More information1. Using Einstein Summation notation, prove the identity: = A
1. Usig Eistei Suatio otatio, pove the idetity: ( B ( B B( + ( B ( B [1 poits] We begi by witig the coss poduct of ad B as: So the ou idetity, C B C ( B C, i ε ik B k We coside ( C ε i ε ik ε iε ik ( ε
More informationAdvanced Physical Geodesy
Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 20
ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More informationa) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.
Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p
More information«A first lesson on Mathematical Induction»
Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio,
More informationGreatest term (numerically) in the expansion of (1 + x) Method 1 Let T
BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationTechnical Report: Bessel Filter Analysis
Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationQuestion 1: The magnetic case
September 6, 018 Corell Uiversity, Departmet of Physics PHYS 337, Advace E&M, HW # 4, due: 9/19/018, 11:15 AM Questio 1: The magetic case I class, we skipped over some details, so here you are asked to
More informationInternational Journal of Mathematics Trends and Technology (IJMTT) Volume 47 Number 1 July 2017
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 Coe Metic Saces, Coe Rectagula Metic Saces ad Coo Fixed Poit Theoes M. Sivastava; S.C. Ghosh Deatet of Matheatics, D.A.V. College
More informationOrthogonal Function Solution of Differential Equations
Royal Holloway Uiversity of Loo Departet of Physics Orthogoal Fuctio Solutio of Differetial Equatios trouctio A give oriary ifferetial equatio will have solutios i ters of its ow fuctios Thus, for eaple,
More informationUsing Difference Equations to Generalize Results for Periodic Nested Radicals
Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =
More informationOn the Basis Property of Eigenfunction. of the Frankl Problem with Nonlocal Parity Conditions. of the Third Kind
It.. Cotemp. Math. Scieces Vol. 9 o. 3 33-38 HIKARI Lt www.m-hikai.com http://x.oi.og/.988/ijcms..33 O the Basis Popety o Eigeuctio o the Fakl Poblem with Nolocal Paity Coitios o the Thi Ki A. Sameipou
More information3. Calculus with distributions
6 RODICA D. COSTIN 3.1. Limits of istributios. 3. Calculus with istributios Defiitio 4. A sequece of istributios {u } coverges to the istributio u (all efie o the same space of test fuctios) if (φ, u )
More informationQuantum Mechanics Lecture Notes 10 April 2007 Meg Noah
The -Patice syste: ˆ H V This is difficut to sove. y V 1 ˆ H V 1 1 1 1 ˆ = ad with 1 1 Hˆ Cete of Mass ˆ fo Patice i fee space He Reative Haitoia eative coodiate of the tota oetu Pˆ the tota oetu tota
More informationTHE LEGENDRE POLYNOMIALS AND THEIR PROPERTIES. r If one now thinks of obtaining the potential of a distributed mass, the solution becomes-
THE LEGENDRE OLYNOMIALS AND THEIR ROERTIES The gravitatioal potetial ψ at a poit A at istace r from a poit mass locate at B ca be represete by the solutio of the Laplace equatio i spherical cooriates.
More information= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!
0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet
More informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More informationMultivector Functions
I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed
More informationFIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES
IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity
More informationAnalytic Number Theory Solutions
Aalytic Number Theory Solutios Sea Li Corell Uiversity sl6@corell.eu Ja. 03 Itrouctio This ocumet is a work-i-progress solutio maual for Tom Apostol s Itrouctio to Aalytic Number Theory. The solutios were
More informationPARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES
Diola Bagayoko (0 PARTAL DFFERENTAL EQUATONS SEPARATON OF ARABLES. troductio As discussed i previous lectures, partial differetial equatios arise whe the depedet variale, i.e., the fuctio, varies with
More information30 The Electric Field Due to a Continuous Distribution of Charge on a Line
hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line 0 The Electic Field Due to a ontinuous Distibution of hage on a Line Evey integal ust include a diffeential (such as d, dt, dq,
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationSummer MA Lesson 13 Section 1.6, Section 1.7 (part 1)
Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher
More informationMath 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual
Math 66 Week-i-Review - S. Nite // Page of Week i Review #9 (F-F.4, 4.-4.4,.-.) Simple Iteest I = Pt, whee I is the iteest, P is the picipal, is the iteest ate, ad t is the time i yeas. P( + t), whee A
More informationStrong Result for Level Crossings of Random Polynomials
IOSR Joual of haacy ad Biological Scieces (IOSR-JBS) e-issn:78-8, p-issn:19-7676 Volue 11, Issue Ve III (ay - Ju16), 1-18 wwwiosjoualsog Stog Result fo Level Cossigs of Rado olyoials 1 DKisha, AK asigh
More information( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.
Mathematics. Solved Eamples JEE Mai/Boads Eample : Fid the coefficiet of y i c y y Sol: By usig fomula of fidig geeal tem we ca easily get coefficiet of y. I the biomial epasio, ( ) th tem is c T ( y )
More informationBernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes
Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More information1. Hydrogen Atom: 3p State
7633A QUANTUM MECHANICS I - solutio set - autum. Hydroge Atom: 3p State Let us assume that a hydroge atom is i a 3p state. Show that the radial part of its wave fuctio is r u 3(r) = 4 8 6 e r 3 r(6 r).
More information1-D Sampling Using Nonuniform Samples and Bessel Functions
-D Saplig Usig Nouio Saples a Bessel Fuctios Nikolaos E. Myiis *, Mebe, IEEE,Electical Egiee, Ph.D. A & Chistooulos Chazas, Seio Mebe, IEEE, Poesso B A Cultual a Eucatioal Techologies Istitute, Tsiiski
More informationk=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum
0. L Hôpital s rule You alreay kow from Lecture 0 that ay sequece {s k } iuces a sequece of fiite sums {S } through S = s k, a that if s k 0 as k the {S } may coverge to the it k= S = s s s 3 s 4 = s k.
More informationStrong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA
Iteatioal Joual of Reseach i Egieeig ad aageet Techology (IJRET) olue Issue July 5 Available at http://wwwijetco/ Stog Result fo Level Cossigs of Rado olyoials Dipty Rai Dhal D K isha Depatet of atheatics
More informationChapter 2 Transformations and Expectations
Chapter Trasformatios a Epectatios Chapter Distributios of Fuctios of a Raom Variable Problem: Let be a raom variable with cf F ( ) If we efie ay fuctio of, say g( ) g( ) is also a raom variable whose
More informationThe structure of Fourier series
The structure of Fourier series Valery P Dmitriyev Lomoosov Uiversity, Russia Date: February 3, 2011) Fourier series is costructe basig o the iea to moel the elemetary oscillatio 1, +1) by the expoetial
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More information= n which will be written with general term a n Represent this sequence by listing its function values in order:
Sectio 9.: Sequeces a Series (Sums) Remier:, 2, 3,, k 2, k, k, k +, k + 2, The ellipsis ots iicate the sequece is oeig, has ifiitely may terms. I. SEQUENCES Defiitio (page 706): A sequece is a fuctio whose
More informationOVERVIEW OF THE COMBINATORICS FUNCTION TECHNIQUE
OVERVIEW OF THE COMBINATORICS FUNCTION TECHNIQUE Alai J. Phaes Depatet of Physics, Medel Hall, Villaova Uivesity, Villaova, Pesylvaia, 985-699, USA, phaes@eail.villaova.edu Hek F. Aoldus Depatet of Physics
More informationA PROBABILITY PROBLEM
A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,
More informationECE 901 Lecture 4: Estimation of Lipschitz smooth functions
ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig
More informationOn a Problem of Littlewood
Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995
More informationA New Result On A,p n,δ k -Summabilty
OSR Joual of Matheatics (OSR-JM) e-ssn: 2278-5728, p-ssn:239-765x. Volue 0, ssue Ve. V. (Feb. 204), PP 56-62 www.iosjouals.og A New Result O A,p,δ -Suabilty Ripeda Kua &Aditya Kua Raghuashi Depatet of
More informationI PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of
Two Maks Questios I PU MATHEMATIS HAPTER - 08 Biomial Theoem. Epad + usig biomial theoem ad hece fid the coefficiet of y y. Epad usig biomial theoem. Hece fid the costat tem of the epasio.. Simplify +
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationAuchmuty High School Mathematics Department Sequences & Series Notes Teacher Version
equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.
More informationClassical Electrodynamics
A First Look at Quatum Physics Classical Electroyamics Chapter Itrouctio a Survey Classical Electroyamics Prof. Y. F. Che Cotets A First Look at Quatum Physics. Coulomb s law a electric fiel. Electric
More informationL8b - Laplacians in a circle
L8b - Laplacias i a cicle Rev //04 `Give you evidece,' the Kig epeated agily, `o I'll have you executed, whethe you'e evous o ot.' `I'm a poo ma, you Majesty,' the Hatte bega, i a temblig voice, `--ad
More informationConditional Convergence of Infinite Products
Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this
More informationAnnouncements: The Rydberg formula describes. A Hydrogen-like ion is an ion that
Q: A Hydogelike io is a io that The Boh odel A) is cheically vey siila to Hydoge ios B) has the sae optical spectu as Hydoge C) has the sae ube of potos as Hydoge ) has the sae ube of electos as a Hydoge
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationAsymptotic Expansions of Legendre Wavelet
Asptotic Expasios of Legede Wavelet C.P. Pade M.M. Dixit * Depatet of Matheatics NERIST Nijuli Itaaga Idia. Depatet of Matheatics NERIST Nijuli Itaaga Idia. Astact A e costuctio of avelet o the ouded iteval
More informationAP Calculus BC Review Chapter 12 (Sequences and Series), Part Two. n n th derivative of f at x = 5 is given by = x = approximates ( 6)
AP Calculus BC Review Chapter (Sequeces a Series), Part Two Thigs to Kow a Be Able to Do Uersta the meaig of a power series cetere at either or a arbitrary a Uersta raii a itervals of covergece, a kow
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationOrthogonal Functions
Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they
More informationKEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow
KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you
More information5.6 Binomial Multi-section Matching Transformer
4/14/21 5_6 Bioial Multisectio Matchig Trasforers 1/1 5.6 Bioial Multi-sectio Matchig Trasforer Readig Assiget: pp. 246-25 Oe way to axiize badwidth is to costruct a ultisectio Γ f that is axially flat.
More informationElectron states in a periodic potential. Assume the electrons do not interact with each other. Solve the single electron Schrodinger equation: KJ =
Electo states i a peiodic potetial Assume the electos do ot iteact with each othe Solve the sigle electo Schodige equatio: 2 F h 2 + I U ( ) Ψ( ) EΨ( ). 2m HG KJ = whee U(+R)=U(), R is ay Bavais lattice
More information[ ] = jω µ [ + jω ε E r
Guided Wave Foulation of Maxwell's Equations I. Geneal Theoy: Recapitulation -- fequency doain foulation of the acoscopic Maxwel l equations in a souce-fee egion: cul E H cul H ( ) jω µ ( ) [ I-1a ] (
More informationGeneralization of Horadam s Sequence
Tuish Joual of Aalysis ad Nube Theoy 6 Vol No 3-7 Available olie at http://pubssciepubco/tjat///5 Sciece ad Educatio Publishig DOI:69/tjat---5 Geealizatio of Hoada s Sequece CN Phadte * YS Valaulia Depatet
More informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
More informationFinite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler
Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio
More informationRelation (12.1) states that if two points belong to the convex subset Ω then all the points on the connecting line also belong to Ω.
Lectue 6. Poectio Opeato Deiitio A.: Subset Ω R is cove i [ y Ω R ] λ + λ [ y = z Ω], λ,. Relatio. states that i two poits belog to the cove subset Ω the all the poits o the coectig lie also belog to Ω.
More information( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to
Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special
More informationATMO 551a Fall 08. Diffusion
Diffusion Diffusion is a net tanspot of olecules o enegy o oentu o fo a egion of highe concentation to one of lowe concentation by ando olecula) otion. We will look at diffusion in gases. Mean fee path
More informationSOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE
Faulty of Siees ad Matheatis, Uivesity of Niš, Sebia Available at: http://www.pf.i.a.yu/filoat Filoat 22:2 (28), 59 64 SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE Saee Ahad Gupai Abstat. The sequee
More informationphysicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jue 005 5x 3 3. (a) Expess i patial factios. (x 3)( x ) (3) (b) Hece fid the exact value of logaithm. 6 5x 3 dx, givig you aswe as a sigle (x 3)( x ) (5) blak
More informationApplications of the Dirac Sequences in Electrodynamics
Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe 6-7 6 45 Applicatios of the Diac Sequeces i Electodyamics WILHELM W KECS Depatmet of Mathematics
More informationQueueing Theory II. Summary. M/M/1 Output process Networks of Queue Method of Stages. General Distributions
Queueig Theory II Suary M/M/1 Output process Networks of Queue Method of Stages Erlag Distributio Hyperexpoetial Distributio Geeral Distributios Ebedded Markov Chais 1 M/M/1 Output Process Burke s Theore:
More informationSums of Involving the Harmonic Numbers and the Binomial Coefficients
Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS
EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - PROGRESSIONS CONTENTS Be able to apply algebaic techiques Aithmetic pogessio (AP): fist
More information(average number of points per unit length). Note that Equation (9B1) does not depend on the
EE603 Class Notes 9/25/203 Joh Stesby Appeix 9-B: Raom Poisso Poits As iscusse i Chapter, let (t,t 2 ) eote the umber of Poisso raom poits i the iterval (t, t 2 ]. The quatity (t, t 2 ) is a o-egative-iteger-value
More informationand each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.
MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationJacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a
Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi
More informationInhomogeneous Poisson process
Chapter 22 Ihomogeeous Poisso process We coclue our stuy of Poisso processes with the case of o-statioary rates. Let us cosier a arrival rate, λ(t), that with time, but oe that is still Markovia. That
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationx !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationCOMP 2804 Solutions Assignment 1
COMP 2804 Solutios Assiget 1 Questio 1: O the first page of your assiget, write your ae ad studet uber Solutio: Nae: Jaes Bod Studet uber: 007 Questio 2: I Tic-Tac-Toe, we are give a 3 3 grid, cosistig
More informationTHE ANALYTIC LARGE SIEVE
THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig
More informationThe Binomial Multi-Section Transformer
4/15/2010 The Bioial Multisectio Matchig Trasforer preset.doc 1/24 The Bioial Multi-Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where:
More informationRange Symmetric Matrices in Minkowski Space
BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608
More informationMath 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]
Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths
More informationCentral limit theorem for functions of weakly dependent variables
Int. Statistical Inst.: Poc. 58th Wold Statistical Congess, 2011, Dublin (Session CPS058 p.5362 Cental liit theoe fo functions of weakly dependent vaiables Jensen, Jens Ledet Aahus Univesity, Depatent
More informationTHE CLOSED FORMS OF CONVERGENT INFINITE SERIES ESTIMATION OF THE SERIES SUM OF NON-CLOSED FORM ALTERNATING SERIES TO A HIGH DEGREE OF PRECISION.
THE CLSED FRMS F CNERGENT INFINITE SERIES ESTIMATIN F THE SERIES SUM F NN-CLSED FRM ALTERNATING SERIES T A HIGH DEGREE F PRECISIN. Peter G.Bass. PGBass M er..0.0. www.relativityoais.co May 0 Abstract This
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationThe Non-homogeneous Diffusion Equation
The No-hoogeeous Diffusio Equatio The o-hoogeeous diffusio equatio, with sources, has the geeral for, 2 r,t a 2 r,t Fr,t t a 2 is real ad The hoogeeous diffusio equatio, 2 r,t a 2 t r,t ca be solved by
More informationAdvanced Higher Formula List
Advaced Highe Fomula List Note: o fomulae give i eam emembe eveythig! Uit Biomial Theoem Factoial! ( ) ( ) Biomial Coefficiet C!! ( )! Symmety Idetity Khayyam-Pascal Idetity Biomial Theoem ( y) C y 0 0
More informationMinimization of the quadratic test function
Miimizatio of the quadatic test fuctio A quadatic fom is a scala quadatic fuctio of a vecto with the fom f ( ) A b c with b R A R whee A is assumed to be SPD ad c is a scala costat Note: A symmetic mati
More information