A Lattice Green Function Introduction. Abstract
|
|
- Austin Melton
- 5 years ago
- Views:
Transcription
1 August 5, 25 A Lattice Gree Fuctio Itroductio Stefa Hollos Exstrom Laboratories LLC, 662 Nelso Park Dr, Logmot, Colorado 853, USA Abstract We preset a itroductio to lattice Gree fuctios. Electroic address: stefa@exstrom.com; URL: 1
2 I. INTRODUCTION This is a short cocise itroductio to the cocept of a lattice Gree fuctio (LGF). The LGF is the discrete space couterpart to the more familiar cotiuous space Gree fuctio that has become such a versatile tool i may areas of theoretical physics. Some familiarity with the more commo uses of Gree fuctios, such as i the solutio of partial differetial equatios, is helpful i what follows but ot strictly ecessary. Excellet itroductios to Gree fuctios ca be foud i Barto [1], Duffy [2], ad Ecoomou [3]. The LGF is ofte used i codesed matter, ad statistical physics (radom walk theory). A good discussio of some uses of the LGF ca be foud i Cserti [4]. It also appears, although most ofte ot by ame, whe the fiite differece approximatio is used to solve partial differetial equatios. It is through this applicatio that we will itroduce the cocept of a LGF. I particular we will use the LGF to show how the discretized Poisso equatio ca be solved i a ifiite cubic (3 dimesioal) ad square (2 dimesioal) lattice. It is perhaps appropriate to itroduce the LGF i this way sice solvig the Poisso equatio was George Gree s origial motivatio for developig his epoymous fuctios [5]. A great deal of research has bee doe o lattice Gree fuctios over the last fifty years or so ad other itroductios do exist, see for example Katsura et al [6] ad the two recet papers by Cserti [4, 7]. The hope is that the simple examples give i this itroductio will be accessible to the widest possible audiece. The oly kowledge assumed o the part of the reader is some familiarity with Dirac vector space otatio ad a uderstadig of eigevalues ad eigevectors. II. THREE DIMENSIONAL DISCRETE POISSON EQUATION For the cubic lattice let ˆx 1, ˆx 2, ad ˆx 3 be a set of orthogoal uit vectors, so that ˆx i ˆx j = δ(i, j). If the lattice spacig is a the the primitive lattice vectors are a i = a ˆx i ad all poits i the lattice are give by the lattice vectors r = 1 a a a 3 i = iteger (1) Usig this otatio, the Poisso equatio o a cubic lattice takes the followig form. 3 i=1 [φ( r + a i ) 2φ( r ) + φ( r a i )] = f ( r ) (2) We will refer to this as the discrete Poisso equatio or DPE from here o. To fully defie the equatio, the size of the lattice ad the boudary coditios eed to be specified. 2 These are
3 however ot importat for the curret discussio ad will be specified later. I the limit as the lattice spacig goes to zero this becomes the cotiuous Poisso equatio. 3 2 φ( r) i=1 xi 2 = g( r) (3) Eq. 2 ca be regarded as a fiite differece approximatio of eq. 3 with f ( r ) = a 2 g( r ). Much of the followig developmet will be i terms of Dirac vector space otatio. I this otatio the DPE is Here L deotes the lattice Laplacia operator. L φ = f (4) If we let deote the lattice basis vector associated with the lattice poit r the φ = φ( r ) ad f = f ( r ). I the lattice basis, the vectors φ ad f are ad eq. 4 is φ = I terms of matrix ad vector elemets this becomes φ = φ( r ) (5) f = f = f ( r ) (6) l L φ = l f (7) L l φ( r ) = f ( r l ) (8) The matrix elemets L l ca be idetified by comparig eq. 8 with eq. 2. 6δ(l, ) L l = 1 if r r l = a otherwise (9) This ca also be expressed as follows. L l = 6δ( r l, r ) + Now we wat to solve eq. 4 for φ. At least formally, the solutio is 3 i=1 [δ( r l + a i, r ) + δ( r l a i, r )] (1) φ = L 1 f (11) 3
4 The problem therefore ivolves fidig L 1 = G, where the operator G is what we will call the lattice Gree fuctio. We ca get a expressio for the matrix elemets of G by usig a eigevector expasio. It is easy to show that G ad L have the same eigevectors, ad if λ is a eigevalue of L the 1/λ is a eigevalue of G. The first step the is to fid the eigevalues ad eigevectors of L. Util ow, o assumptios have bee made about the size of the lattice or the boudary coditios. We begi by assumig a fiite lattice with N i poits i the directio a i ad periodic boudary coditios. Periodic boudary coditios mea that for ay lattice fuctio v( r ), the followig will be true v( r + N i a i ) = v( r ) i = 1,2,3 (12) With these assumptios the eigevalue problem for L ca ow be solved. Write the eigevalue equatio for L as follows. I the lattice basis the eigevalue equatio is or i terms of matrix elemets Usig eq. 9 for the matrix elemets of L, eq. 15 becomes. L v m = λ m v m (13) l L v m = λ m l v m (14) L l v m ( r ) = λ m v m ( r l ) (15) v m ( r l + a 1 )+v m ( r l a 1 )+v m ( r l + a 2 )+v m ( r l a 2 )+v m ( r l + a 3 )+v m ( r l a 3 ) 6v m ( r l ) = λ m v m ( r l ) (16) We will ow show that periodic boudary coditios, v m ( r l +N i a i ) = v m ( r l ), require that v m ( r l ) have the followig form v m ( r l ) = Ae i k m r l (17) We set the vector k m equal to k m = m 1 N 1 b 1 + m 2 N 2 b 2 + m 3 N 3 b 3 (18) where m i =,1,2,...,N i 1 ad the vectors b i are reciprocal lattice vectors equal to b i = 2π a ˆx i (19) 4
5 so that we have b i a j = 2πδ(i, j) (2) With this defiitio of k m it is easy to show that eq. 17 obeys the periodic boudary coditios v m ( r l + N i a i ) = Ae i k m r l e i k m N i a i = Ae i k m r l e i2πm i (21) = v m ( r l ) The costat A is chose so that the eigevector is ormalized. v m v m = v m l l v m (22) l = Ae i k m r l Ae i k m r l l = A 2 N 1 N 2 N 3 therefore let A = 1/ N 1 N 2 N 3. We ca ow fid the eigevalues by substitutig eq. 17 ito eq. 16. This gives ( ) λ m = 2 cos k m a 1 + cos k m a 2 + cos k m a 3 3 ( = 2 cos 2πm 1 + cos 2πm 2 + cos 2πm ) 3 3 N 1 N 2 N 3 Now that the eigevalue problem has bee solved we ca express L ad G = L 1 i terms of the eigebasis. For L we have The matrix elemets of L are the (23) L = λ m v m v m (24) m L l = λ m l v m v m (25) m 1 = N 1 N 2 N 3 λ m e i k m ( r l r ) m It is ot too difficult to show that this equatio gives the same results as i eq. 9. For G = L 1 we have ad the matrix elemets are G l = m G = m v m v m λ m (26) l v m v m = 1 e λ m N 1 N 2 N 3 i k m ( r l r ) (27) m λ m 5
6 Note that G l depeds oly o the differece r l r so that G has a circulat matrix represetatio. Let r p = r l r the ad Usig this otatio, eq. 27 becomes 1 G l = G( r p ) = N 1 N 2 N 3 r p = (l 1 1 ) a 1 + (l 2 2 ) a 2 + (l 3 3 ) a 3 (28) = p 1 a 1 + p 2 a 2 + p 3 a 3 k m r p = 2π m 1p 1 N 1 + 2π m 2p 2 N 2 + 2π m 3p 3 N 3 (29) N 1 1 m 1 = N 2 1 m 2 = N 3 1 m 3 = e i 2πm 1 p 1 N 1 e i 2πm 2 p 2 N 2 e i 2πm 3 p 3 N 3 ( ) (3) 2 3 cos 2πm 1 N 1 cos 2πm 2 N 2 cos 2πm 3 N 3 Eq. 3 gives the matrix elemets of the lattice Gree fuctio of the DPE for a fiite lattice with periodic boudary coditios. Note that this is essetially a Fourier series expasio of the matrix elemets which is possible because of the periodic boudary coditios. For other boudary coditios such as G(N i a i ) =, the expasio would have to be i terms of a sie series. We will ow go from a fiite lattice to a ifiite lattice by lettig N i, i = 1,2,3. This meas goig from the Fourier series represetatio of eq. 3 to a Fourier trasform represetatio of the matrix elemets. I eq. 3 let Whe m i is icremeted by 1 the chage i x i is x i = 2πm i N i (31) x i = 2π N i The summatios i eq. 3 ca the be writte as or 1 N i = x i 2π (32) 1 N i N i 1 m i = 2π(1 1 N i ) x i = x i 2π (33) ad i the limit N i the summatio becomes a itegral. For a ifiite lattice eq. 3 the becomes G( r p ) = 1 (2π) 3 Z 2π Z 2π Z 2π 1 2π Z 2π dx i (34) e ix 1 p 1 e ix 2 p 2 e ix 3 p 3 2(3 cosx 1 cosx 2 cosx 3 ) dx 1dx 2 dx 3 (35) 6
7 Note that the itegrad has a period of 2π i each of the variables so that the limits of itegratio ca be chaged to the more symmetric G( r p ) = 1 (2π) 3 Z π π Z π Z π π π e ix 1 p 1 e ix 2 p 2 e ix 3 p 3 2(3 cosx 1 cosx 2 cosx 3 ) dx 1dx 2 dx 3 (36) The itegral ca be further simplified by lookig at the parity properties of the itegrad. Multiplyig the e ix i p i = cosx i p i + isix i p i factors ad leavig out the resultig odd terms reduces the itegral to G(p 1, p 2, p 3 ) = 1 2π 3 Z π Z π Z π cosx 1 p 1 cosx 2 p 2 cosx 3 p 3 3 cosx 1 cosx 2 cosx 3 dx 1 dx 2 dx 3 (37) Clearly G is a fuctio oly of the parameters p 1, p 2, ad p 3 ad it is a eve fuctio of these parameters. G is also symmetric uder ay permutatio of the parameters. All the uique values of G are therefore cotaied i the wedge p 1 p 2 p 3. We will ow derive a recurrece equatio that the matrix elemets of G obey. By defiitio we have LG = I which i the lattice basis is Substitutig i eq. 1 for L l gives 6G( r l r m ) + l L G m = l m (38) L l G m = δ(l,m) L( r l r )G( r r m ) = δ(l,m) 3 i=1 [G( r l + a i r m ) + G( r l a i r m )] = δ(l,m) (39) Now usig the otatio, r l r m = (l 1 m 1 ) a 1 + (l 2 m 2 ) a 2 + (l 3 m 3 ) a 3 = p 1 a 1 + p 2 a 2 + p 3 a 3, eq. 39 becomes 6G(p 1, p 2, p 3 )+G(p 1 +1, p 2, p 3 )+G(p 1 1, p 2, p 3 )+G(p 1, p 2 +1, p 3 )+G(p 1, p 2 1, p 3 )+G(p 1, p 2, p 3 +1)+ Eq. 4 simplifies cosiderably for some specific values of p 1, p 2, ad p 3. I particular for p 1 = p 2 = p 3 = we get G(1,,) = G(,,) 1 6 Where the symmetry properties of G have bee used, i.e. G(1,,) = G( 1,,) = G(,1,) = G(, 1,) = G(,,1) = G(,, 1). Lettig p 1 = p 2 = p 3 = p i eq. 4, we have (4) (41) 2G(p, p, p) = G(p + 1, p, p) + G(p, p, p 1) (42) 7
8 Lettig p 1 = p, p 2 = p 3 = i eq. 4 gives G(p + 1,,) = 6G(p,,) 4G(p,1,) G(p 1,,) (43) Lettig p 1 = p 2 = p, p 3 = i eq. 4 gives 3G(p, p,) = G(p + 1, p,) + G(p, p 1,) + G(p, p,1) (44) I geeral for p 1 = l, p 2 = m, p 3 = with l, m, ad ot all equal to zero, we have 6G(l,m,) = G(l +1,m,)+G(l 1,m,)+G(l,m+1,)+G(l,m 1,)+G(l,m,+1)+G(l,m, 1) Additioal recursio equatios were developed by Duffi ad Shelly. These recursio equatios, alog with some of those give above ad some relatios due to Horiguchi ad Morita, allowed Glasser ad Boersma to fid a expressio for the geeral matrix elemet G(l,m,) that ivolves kowig oly G(,,), which is give by the itegral G(,,) = 1 2π 3 Z π Z π Z π (45) dx 1 dx 2 dx 3 3 cosx 1 cosx 2 cosx 3 (46) This itegral was first evaluated by Watso i terms of complete elliptic itegrals. It was the show by Glasser ad Zucker to be expressible i terms of gamma fuctios as ( ) ( ) ( ) ( ) G(,,) = 96π 3 Γ Γ Γ Γ A idetity due to Borwei ad Zucker allows this to be simplified to ( ) ( ) G(,,) = 96π 3 Γ2 Γ Joyce [8] has also developed some recursio equatios that allow G(l,m,) to be calculated for arbitrary values of l,m,. He arrives at the same formula as Glasser ad Boersma via a differet method ad also derives a asymptotic formula for G(l,m,). I some very recet work, Joyce [9] gives some formulas that allow the diagoal elemets, G(,, ), to be calculated very accurately for arbitrary values of. He also gives asymptotic formulas for G(,,). (47) (48) III. TWO DIMENSIONAL DISCRETE POISSON EQUATION The same procedure give above ca be used to fid the lattice Gree fuctio for the two dimesioal Poisso equatio. I this case the lattice vectors are r = 1 a a 2 (49) 8
9 The matrix elemets of the lattice Laplacia are 4δ( r l, r ) L l = 1 if r r l = a otherwise (5) Which ca also be expressed as L l = 4δ( r l, r ) + δ( r l + a 1, r ) + δ( r l a 1, r ) + δ( r l + a 2, r ) + δ( r l a 2, r ) (51) The eigevector expasio of the lattice Laplacia matrix elemets are L l = 1 N 1 N 2 λ m e i k m ( r l r ) m (52) k m = m 1 N 1 b 1 + m 2 N 2 b 2 m i = iteger (53) ( ) λ m = 2 cos k m a 1 + cos k m a 2 2 ( = 2 cos 2πm 1 + cos 2πm ) 2 2 N 1 N 2 The matrix elemets of the lattice Gree fuctio are expaded i the eigebasis as which if we let r p = r l r, ca be expressed as G l = G( r p ) = 1 N 1 N 2 For the ifiite lattice this becomes (54) G l = 1 e N 1 N 2 i k m ( r l r ) (55) m λ m N 1 1 m 1 = N 2 1 m 2 = G( r p ) = G(p 1, p 2 ) = 1 2π 2 Z π e i 2πm 1 p 1 N 1 e i 2πm 2 p 2 N 2 ( ) (56) 2 2 cos 2πm 1 N 1 cos 2πm 2 N 2 Z π cosx 1 p 1 cosx 2 p 2 2 cosx 1 cosx 2 dx 1 dx 2 (57) G is a eve fuctio of the parameters p 1 ad p 2, ad it is symmetric uder ay permutatio of the parameters. All the uique values of G are therefore cotaied i the wedge p 1 p 2. There is oe problem with eq. 57. The itegral is diverget for all values of p 1 ad p 2. We ca fix this by usig the shifted Gree fuctio. g(p 1, p 2 ) = G(,) G(p 1, p 2 ) = 1 2π 2 Z π 9 Z π 1 cosx 1 p 1 cosx 2 p 2 2 cosx 1 cosx 2 dx 1 dx 2 (58)
10 The itegral ow exists for all values of p 1 ad p 2. Usig g(p 1, p 2 ) istead of G(p 1, p 2 ) will provide a solutio to the DPE as log as the sum of the source terms, f ( r ), over all the lattice sites is equal to zero. To demostrate this, first ote that the solutio to the DPE i terms of G is give by Now if we have the eq. 59 ca also be writte as φ( r l ) = G l f ( r ) (59) f ( r ) = (6) φ( r l ) = (G ll G l ) f ( r ) (61) where G ll = G( r l r l ) = G(,), G l = G( r l r ) = G(p 1, p 2 ) ad G ll G l = g l. The solutio to the DPE i terms of the shifted Gree fuctio is the where g l = g( r l r ) = g(p 1, p 2 ) = G(,) G(p 1, p 2 ). φ( r l ) = g l f ( r ) (62) From the above discussio, you ca see that i a ubouded two dimesioal space or lattice the DPE is oly solvable if the sources add up to zero. A physical example of this is i two dimesioal electrostatics. The charge uits i two dimesioal electrostatics are actually parallel, ifiite lie charges embedded i a three dimesioal space. For a sigle lie charge, the potetial at ay fiite distace from the lie will be ifiite. For two lies of opposite charge the potetial is fiite i the space surroudig the lies. Note that we are assumig a ubouded space with the zero poit potetial at ifiity. Aother example comes from the theory of radom walks. I oe ad two dimesios a radom walker is guarrateed to evetually retur to its startig positio, while i three dimesios it may ever do so. To see how this is related to the DPE, see the excellet book by Doyle ad Sell [1] o radom walks i electrical etworks. For aother example see Cserti s paper [4] o usig the lattice Gree fuctio to calculate the resistace betwee two poits i a ifiite etwork of resistors. We will ow preset some recurrece equatios that the matrix elemets of the Gree fuctio obey. As i the three dimesioal case these ca easily be foud from the defiig relatio LG = I. This gives the geeral recurrece 4G(p 1, p 2 )+G(p 1 +1, p 2 )+G(p 1 1, p 2 )+G(p 1, p 2 +1)+G(p 1, p 2 1) = δ(p 1,)δ(p 2,) 1 (63)
11 For p 1 = p 2 = we have For p 1 = p, p 2 = we have G(,) G(1,) = 1 4 (64) 4G(p,) = G(p + 1,) + G(p 1,) + 2G(p,1) (65) For p 1 = p 2 = p we have 2G(p, p) = G(p + 1, p) + G(p, p 1) (66) Ad i geeral for p 1 = l ad p 2 = m we have 4G(l,m) = G(l + 1,m) + G(l 1,m) + G(l,m + 1) + G(l,m 1) (67) A additioal recurrece equatio for the diagoal elemets is [11] (2 + 1)G( + 1, + 1) 4G(,) + (2 1)G( 1, 1) = (68) Sice the coefficiets i each of these equatios adds to zero you ca see that the shifted Gree fuctio, g(p 1, p 2 ) = G(,) G(p 1, p 2 ) must obey the same recurrece equatios. These equatios for g are listed below g(1,) = 1 4 (69) 4g(p,) = g(p + 1,) + g(p 1,) + 2g(p,1) (7) 2g(p, p) = g(p + 1, p) + g(p, p 1) (71) 4g(l,m) = g(l + 1,m) + g(l 1,m) + g(l,m + 1) + g(l,m 1) (72) (2 + 1)g( + 1, + 1) 4g(,) + (2 1)g( 1, 1) = (73) Ackowledgmets Thaks. [1] G. Barto, Elemets of Gree s Fuctios ad Propagatio : Potetials, Diffusio, ad Waves (Oxford Uiversity Press, 1989). 11
12 [2] D. G. Duffy, Gree s Fuctios with Applicatios (Chapma ad Hall/CRC, 21). [3] E. Ecoomou, Gree s Fuctios i Quatum Physics (Spriger-Verlag, 1983). [4] J. Cserti, Amer. J. Phys. 68, 896 (2), cod-mat/ [5] G. Gree, A Essay o the Applicatio of Mathematical Aalysis to the Theories of Electricity ad Magetism (Nottigham, 1828). [6] S. Katsura, T. Morita, S. Iawashiro, T. Horiguchi, ad Y. Abe, J. Math. Phys. 12, 892 (1971). [7] J. Cserti, G. David, ad A. Piroth, Amer. J. Phys. 7, 153 (22), cod-mat/ [8] G. S. Joyce, J. Phys. A 35, 9811 (22). [9] G. S. Joyce ad R. T. Delves, J. Phys. A 37, 3645 (24). [1] P. G. Doyle ad J. L. Sell, Radom walks ad electric etworks (2), math.pr/157. [11] T. Morita, J. Math. Phys. 12, 1744 (1971). 12
Math 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
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 informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
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 informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
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 informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationEE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More information: Transforms and Partial Differential Equations
Trasforms ad Partial Differetial Equatios 018 SUBJECT NAME : Trasforms ad Partial Differetial Equatios SUBJECT CODE : MA 6351 MATERIAL NAME : Part A questios REGULATION : R013 WEBSITE : wwwharigaeshcom
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationUNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 116C. Problem Set 4. Benjamin Stahl. November 6, 2014
UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 6C Problem Set 4 Bejami Stahl November 6, 4 BOAS, P. 63, PROBLEM.-5 The Laguerre differetial equatio, x y + ( xy + py =, will be solved
More information1 6 = 1 6 = + Factorials and Euler s Gamma function
Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio
More information1 Adiabatic and diabatic representations
1 Adiabatic ad diabatic represetatios 1.1 Bor-Oppeheimer approximatio The time-idepedet Schrödiger equatio for both electroic ad uclear degrees of freedom is Ĥ Ψ(r, R) = E Ψ(r, R), (1) where the full molecular
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More informationBrief Review of Functions of Several Variables
Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(
More informationRotationally invariant integrals of arbitrary dimensions
September 1, 14 Rotatioally ivariat itegrals of arbitrary dimesios James D. Wells Physics Departmet, Uiversity of Michiga, A Arbor Abstract: I this ote itegrals over spherical volumes with rotatioally
More informationx a x a Lecture 2 Series (See Chapter 1 in Boas)
Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio
More informationSECTION 2 Electrostatics
SECTION Electrostatics This sectio, based o Chapter of Griffiths, covers effects of electric fields ad forces i static (timeidepedet) situatios. The topics are: Electric field Gauss s Law Electric potetial
More informationFIR Filter Design: Part II
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we cosider how we might go about desigig FIR filters with arbitrary frequecy resposes, through compositio of multiple sigle-peak
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationQ-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received:
More informationPHY4905: Nearly-Free Electron Model (NFE)
PHY4905: Nearly-Free Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. No-degeerate eigestates
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationCS284A: Representations and Algorithms in Molecular Biology
CS284A: Represetatios ad Algorithms i Molecular Biology Scribe Notes o Lectures 3 & 4: Motif Discovery via Eumeratio & Motif Represetatio Usig Positio Weight Matrix Joshua Gervi Based o presetatios by
More informationEXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES
LE MATEMATICHE Vol. LXXIII 208 Fasc. I, pp. 3 24 doi: 0.448/208.73.. EXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES THOMAS ERNST We preset idetities of various kids for
More informationC. Complex Numbers. x 6x + 2 = 0. This equation was known to have three real roots, given by simple combinations of the expressions
C. Complex Numbers. Complex arithmetic. Most people thik that complex umbers arose from attempts to solve quadratic equatios, but actually it was i coectio with cubic equatios they first appeared. Everyoe
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationAppendix: The Laplace Transform
Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationLecture 6: Integration and the Mean Value Theorem
Math 8 Istructor: Padraic Bartlett Lecture 6: Itegratio ad the Mea Value Theorem Week 6 Caltech - Fall, 2011 1 Radom Questios Questio 1.1. Show that ay positive ratioal umber ca be writte as the sum of
More informationSEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationTaylor polynomial solution of difference equation with constant coefficients via time scales calculus
TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu
More informationINFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is
More informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationMatsubara-Green s Functions
Matsubara-Gree s Fuctios Time Orderig : Cosider the followig operator If H = H the we ca trivially factorise this as, E(s = e s(h+ E(s = e sh e s I geeral this is ot true. However for practical applicatio
More informationExact scattering and bound states solutions for novel hyperbolic potentials with inverse square singularity
Exact scatterig ad boud states solutios for ovel hyperbolic potetials with iverse square sigularity A. D. Alhaidari Saudi Ceter for Theoretical Physics, P. O. Box 37, Jeddah 38, Saudi Arabia Abstract:
More informationLecture 7: Fourier Series and Complex Power Series
Math 1d Istructor: Padraic Bartlett Lecture 7: Fourier Series ad Complex Power Series Week 7 Caltech 013 1 Fourier Series 1.1 Defiitios ad Motivatio Defiitio 1.1. A Fourier series is a series of fuctios
More informationLanczos-Haydock Recursion
Laczos-Haydock Recursio Bor: Feb 893 i Székesehérvár, Hugary- Died: 5 Jue 974 i Budapest, Hugary Corelius Laczos From a abstract mathematical viewpoit, the method for puttig a symmetric matrix i three-diagoal
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More information... and realizing that as n goes to infinity the two integrals should be equal. This yields the Wallis result-
INFINITE PRODUTS Oe defies a ifiite product as- F F F... F x [ F ] Takig the atural logarithm of each side oe has- l[ F x] l F l F l F l F... So that the iitial ifiite product will coverge oly if the sum
More informationR is a scalar defined as follows:
Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS
EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS TUTORIAL 1 - DIFFERENTIATION Use the elemetary rules of calculus arithmetic to solve problems that ivolve differetiatio
More informationSlide 1. Slide 2. Slide 3. Solids of Rotation:
Slide 1 Solids of Rotatio: The Eggplat Experiece Suz Atik Palo Alto High School Palo Alto, Ca EdD; NBCT, AYA Math satik@pausd.org May thaks to my colleague, Kathy Weiss, NBCT, AYA Math, who origially desiged
More informationLecture 6: Integration and the Mean Value Theorem. slope =
Math 8 Istructor: Padraic Bartlett Lecture 6: Itegratio ad the Mea Value Theorem Week 6 Caltech 202 The Mea Value Theorem The Mea Value Theorem abbreviated MVT is the followig result: Theorem. Suppose
More informationOrthogonal transformations
Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear
More information18.01 Calculus Jason Starr Fall 2005
Lecture 18. October 5, 005 Homework. Problem Set 5 Part I: (c). Practice Problems. Course Reader: 3G 1, 3G, 3G 4, 3G 5. 1. Approximatig Riema itegrals. Ofte, there is o simpler expressio for the atiderivative
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationInfinite Series and Improper Integrals
8 Special Fuctios Ifiite Series ad Improper Itegrals Ifiite series are importat i almost all areas of mathematics ad egieerig I additio to umerous other uses, they are used to defie certai fuctios ad to
More informationDifferentiable Convex Functions
Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for
More informationMath 257: Finite difference methods
Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationarxiv: v1 [math-ph] 5 Jul 2017
O eigestates for some sl 2 related Hamiltoia arxiv:1707.01193v1 [math-ph] 5 Jul 2017 Fahad M. Alamrai Faculty of Educatio Sciece Techology & Mathematics, Uiversity of Caberra, Bruce ACT 2601, Australia.,
More informationLecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods
Itroductory lecture otes o Partial Differetial Equatios - c Athoy Peirce. Not to be copied, used, or revised without explicit writte permissio from the copyright ower. 1 Lecture 8: Solvig the Heat, Laplace
More informationLinear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form
Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity
More informationAlgebra of Least Squares
October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationThe Discrete Fourier Transform
The Discrete Fourier Trasform Complex Fourier Series Represetatio Recall that a Fourier series has the form a 0 + a k cos(kt) + k=1 b k si(kt) This represetatio seems a bit awkward, sice it ivolves two
More informationj=1 dz Res(f, z j ) = 1 d k 1 dz k 1 (z c)k f(z) Res(f, c) = lim z c (k 1)! Res g, c = f(c) g (c)
Problem. Compute the itegrals C r d for Z, where C r = ad r >. Recall that C r has the couter-clockwise orietatio. Solutio: We will use the idue Theorem to solve this oe. We could istead use other (perhaps
More information17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)
7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationPROPERTIES OF THE POSITIVE INTEGERS
PROPERTIES OF THE POSITIVE ITEGERS The first itroductio to mathematics occurs at the pre-school level ad cosists of essetially coutig out the first te itegers with oe s figers. This allows the idividuals
More informationReliability and Queueing
Copyright 999 Uiversity of Califoria Reliability ad Queueig by David G. Messerschmitt Supplemetary sectio for Uderstadig Networked Applicatios: A First Course, Morga Kaufma, 999. Copyright otice: Permissio
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More information